introduction to renormalization -...
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![Page 1: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/1.jpg)
Introduction to Renormalization
Vincent Rivasseau
LPT Orsay
Cetraro, Summer 2010
![Page 2: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/2.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theory
What is quantum field theory?
2
![Page 3: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/3.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theory
What is quantum field theory?
Is it putting together quantum mechanics and special relativity?
2
![Page 4: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/4.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theory
What is quantum field theory?
Is it putting together quantum mechanics and special relativity?
Is it functional integrals?
2
![Page 5: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/5.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theory
What is quantum field theory?
Is it putting together quantum mechanics and special relativity?
Is it functional integrals?
Is it Feynman diagrams?
2
![Page 6: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/6.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theory
What is quantum field theory?
Is it putting together quantum mechanics and special relativity?
Is it functional integrals?
Is it Feynman diagrams?
Is it axioms (Wightman... C ⋆ algebras...)?
2
![Page 7: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/7.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theory
What is quantum field theory?
Is it putting together quantum mechanics and special relativity?
Is it functional integrals?
Is it Feynman diagrams?
Is it axioms (Wightman... C ⋆ algebras...)?
Remark/statement/suggestion: quantum field theory has a soul which isrenormalization.
2
![Page 8: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/8.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Trees, Forests, Jungles...
3
![Page 9: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/9.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Trees, Forests, Jungles...
At the core of quantum field theory and renormalization lies thecomputation of connected quantities, hence trees, which are the simplestconnected structures. In fact trees, forests and jungles (ie layered forests)appear in almost endless ways in quantum field theory:
3
![Page 10: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/10.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Trees, Forests, Jungles...
At the core of quantum field theory and renormalization lies thecomputation of connected quantities, hence trees, which are the simplestconnected structures. In fact trees, forests and jungles (ie layered forests)appear in almost endless ways in quantum field theory:
In the parametric representation of Feynman amplitudes(Kirchoff-Symanzik polynomials)
3
![Page 11: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/11.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Trees, Forests, Jungles...
At the core of quantum field theory and renormalization lies thecomputation of connected quantities, hence trees, which are the simplestconnected structures. In fact trees, forests and jungles (ie layered forests)appear in almost endless ways in quantum field theory:
In the parametric representation of Feynman amplitudes(Kirchoff-Symanzik polynomials)
In renormalization theory (Zimmermann’s forests, Gallavotti-Nicolotrees, Connes-Kreimer Hopf algebra...)
3
![Page 12: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/12.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Trees, Forests, Jungles...
At the core of quantum field theory and renormalization lies thecomputation of connected quantities, hence trees, which are the simplestconnected structures. In fact trees, forests and jungles (ie layered forests)appear in almost endless ways in quantum field theory:
In the parametric representation of Feynman amplitudes(Kirchoff-Symanzik polynomials)
In renormalization theory (Zimmermann’s forests, Gallavotti-Nicolotrees, Connes-Kreimer Hopf algebra...)
In constructive field theory (Brydges-Kennedy-Abdesselam-R. forestformula, cluster expansions, Mayer expansions ...)
3
![Page 13: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/13.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Trees, Forests, Jungles...
At the core of quantum field theory and renormalization lies thecomputation of connected quantities, hence trees, which are the simplestconnected structures. In fact trees, forests and jungles (ie layered forests)appear in almost endless ways in quantum field theory:
In the parametric representation of Feynman amplitudes(Kirchoff-Symanzik polynomials)
In renormalization theory (Zimmermann’s forests, Gallavotti-Nicolotrees, Connes-Kreimer Hopf algebra...)
In constructive field theory (Brydges-Kennedy-Abdesselam-R. forestformula, cluster expansions, Mayer expansions ...)
In more exotic renormalization group settings (Fermions in condensedmatter, non-commutative field theory, quantum gravity...)
3
![Page 14: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/14.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Trees, Forests, Jungles...
At the core of quantum field theory and renormalization lies thecomputation of connected quantities, hence trees, which are the simplestconnected structures. In fact trees, forests and jungles (ie layered forests)appear in almost endless ways in quantum field theory:
In the parametric representation of Feynman amplitudes(Kirchoff-Symanzik polynomials)
In renormalization theory (Zimmermann’s forests, Gallavotti-Nicolotrees, Connes-Kreimer Hopf algebra...)
In constructive field theory (Brydges-Kennedy-Abdesselam-R. forestformula, cluster expansions, Mayer expansions ...)
In more exotic renormalization group settings (Fermions in condensedmatter, non-commutative field theory, quantum gravity...)
It has even been proposed by Gurau, Magnen and myself that anyQFT should be considered as a scalar product on the vector spacegenerated by all possible trees (Tree Quantum Field Theory).
3
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theories as weighted species
4
![Page 16: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/16.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theories as weighted species
A quantum field theory F should be the generating function for a certainweighted species, in the sense of combinatorists. It should involve only afew parameters: the space-time dimension, masses, coupling constants.The weights are usually called amplitudes.
4
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theories as weighted species
A quantum field theory F should be the generating function for a certainweighted species, in the sense of combinatorists. It should involve only afew parameters: the space-time dimension, masses, coupling constants.The weights are usually called amplitudes.
Some of these weights may be naively given by divergent integrals, and theseries defining the generating function itself might have zero radius ofconvergence.
4
![Page 18: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/18.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theories as weighted species
A quantum field theory F should be the generating function for a certainweighted species, in the sense of combinatorists. It should involve only afew parameters: the space-time dimension, masses, coupling constants.The weights are usually called amplitudes.
Some of these weights may be naively given by divergent integrals, and theseries defining the generating function itself might have zero radius ofconvergence.
Renormalization theory adresses the first problem. → Couplings movewith scale.
4
![Page 19: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/19.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Quantum Field Theories as weighted species
A quantum field theory F should be the generating function for a certainweighted species, in the sense of combinatorists. It should involve only afew parameters: the space-time dimension, masses, coupling constants.The weights are usually called amplitudes.
Some of these weights may be naively given by divergent integrals, and theseries defining the generating function itself might have zero radius ofconvergence.
Renormalization theory adresses the first problem. → Couplings movewith scale.
Constructive field theory adresses the second problem. Species ofGraphs → Species of Trees.
4
![Page 20: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/20.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalization
5
![Page 21: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/21.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalization
Renormalization in physics is a very general framework to study how asystem changes under change of the observation scale.
5
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalization
Renormalization in physics is a very general framework to study how asystem changes under change of the observation scale.
Mathematically it reshuffles the initial theory with divergent weights into aninfinite iteration of a certain ‘renormalization group map” which involvesonly convergent weights.
5
![Page 23: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/23.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalization
Renormalization in physics is a very general framework to study how asystem changes under change of the observation scale.
Mathematically it reshuffles the initial theory with divergent weights into aninfinite iteration of a certain ‘renormalization group map” which involvesonly convergent weights.
In combinatoric terms roughly speaking it writes F = limGoGo...G , whereG corresponds to a single scale.
5
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalization
Renormalization in physics is a very general framework to study how asystem changes under change of the observation scale.
Mathematically it reshuffles the initial theory with divergent weights into aninfinite iteration of a certain ‘renormalization group map” which involvesonly convergent weights.
In combinatoric terms roughly speaking it writes F = limGoGo...G , whereG corresponds to a single scale.
Single scale constructive theory reshuffles G itself. It relabels the “large”species of graphs in terms of the smaller species of trees. The result istypically only defined at small couplings, where it is the Borel sum of theinitial divergent series.
5
![Page 25: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/25.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalization
Renormalization in physics is a very general framework to study how asystem changes under change of the observation scale.
Mathematically it reshuffles the initial theory with divergent weights into aninfinite iteration of a certain ‘renormalization group map” which involvesonly convergent weights.
In combinatoric terms roughly speaking it writes F = limGoGo...G , whereG corresponds to a single scale.
Single scale constructive theory reshuffles G itself. It relabels the “large”species of graphs in terms of the smaller species of trees. The result istypically only defined at small couplings, where it is the Borel sum of theinitial divergent series.
Multiscale constructive theory tries to combine the two previous steps in aconsistent way, but...5
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The snag
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The snag
Even after the first two problems have been correctly tackled, the flow ofthe composition map delivered by the renormalization group may after awhile wander out of the convergence (Borel) radius delivered byconstructive theory.
6
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The snag
Even after the first two problems have been correctly tackled, the flow ofthe composition map delivered by the renormalization group may after awhile wander out of the convergence (Borel) radius delivered byconstructive theory.
This phenomenon always occur (either at the “infrared” or at the‘ultraviolet” end of the renormalization group) in field theory on ordinaryfour dimensional space time (except possibly for extremely special models).This is somewhat frustrating.
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
R
R
R
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:
R
R
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:
dν =1
Ze−(λ/4!)
R
φ4−(m2/2)R
φ2−(a/2)R
(∂µφ∂µφ)Dφ, (2.1)
where
R
R
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:
dν =1
Ze−(λ/4!)
R
φ4−(m2/2)R
φ2−(a/2)R
(∂µφ∂µφ)Dφ, (2.1)
where
λ is the coupling constant, positive in order for the theory to be stable;
R
R
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:
dν =1
Ze−(λ/4!)
R
φ4−(m2/2)R
φ2−(a/2)R
(∂µφ∂µφ)Dφ, (2.1)
where
λ is the coupling constant, positive in order for the theory to be stable;
m is the mass, which fixes some scale;
R
R
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:
dν =1
Ze−(λ/4!)
R
φ4−(m2/2)R
φ2−(a/2)R
(∂µφ∂µφ)Dφ, (2.1)
where
λ is the coupling constant, positive in order for the theory to be stable;
m is the mass, which fixes some scale;
a is called the ”wave function constant”, in general fixed to 1;
R
R
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:
dν =1
Ze−(λ/4!)
R
φ4−(m2/2)R
φ2−(a/2)R
(∂µφ∂µφ)Dφ, (2.1)
where
λ is the coupling constant, positive in order for the theory to be stable;
m is the mass, which fixes some scale;
a is called the ”wave function constant”, in general fixed to 1;
Z is the normalization, so that this measure is a probability measure;
R
R
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Ordinary φ44
It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:
dν =1
Ze−(λ/4!)
R
φ4−(m2/2)R
φ2−(a/2)R
(∂µφ∂µφ)Dφ, (2.1)
where
λ is the coupling constant, positive in order for the theory to be stable;
m is the mass, which fixes some scale;
a is called the ”wave function constant”, in general fixed to 1;
Z is the normalization, so that this measure is a probability measure;
Dφ is a formal product∏
x∈Rd
dφ(x) of Lebesgue measures at each
point of R4.
7
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The ordinary φ44 propagator
An infinite product of Lebesgue measures is ill-defined. So it is better todefine first the Gaussian part of the measure
dµ(φ) =1
Z0e−(m2/2)
R
φ2−(a/2)R
(∂µφ∂µφ)Dφ. (2.2)
where Z0 is again the normalization factor which makes (2.2) a probabilitymeasure.
8
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The ordinary φ44 propagator
An infinite product of Lebesgue measures is ill-defined. So it is better todefine first the Gaussian part of the measure
dµ(φ) =1
Z0e−(m2/2)
R
φ2−(a/2)R
(∂µφ∂µφ)Dφ. (2.2)
where Z0 is again the normalization factor which makes (2.2) a probabilitymeasure.The covariance of dµ is called the (free) propagator
C (p) =1
(2π)21
p2 + m2, C (x , y) =
∫ ∞
0dαe−αm2 e−|x−y |2/4α
α2, (2.3)
8
![Page 39: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/39.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The ordinary φ44 propagator
An infinite product of Lebesgue measures is ill-defined. So it is better todefine first the Gaussian part of the measure
dµ(φ) =1
Z0e−(m2/2)
R
φ2−(a/2)R
(∂µφ∂µφ)Dφ. (2.2)
where Z0 is again the normalization factor which makes (2.2) a probabilitymeasure.The covariance of dµ is called the (free) propagator
C (p) =1
(2π)21
p2 + m2, C (x , y) =
∫ ∞
0dαe−αm2 e−|x−y |2/4α
α2, (2.3)
where we recognize the heat kernel.
8
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
The full interacting measure may now be written as the multiplication ofthe Gaussian measure dµ(φ) by the interaction factor:
dν =1
Ze−(λ/4!)
R
φ4(x)dxdµ(φ) (2.4)
9
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
The full interacting measure may now be written as the multiplication ofthe Gaussian measure dµ(φ) by the interaction factor:
dν =1
Ze−(λ/4!)
R
φ4(x)dxdµ(φ) (2.4)
and the Schwinger functions are the normalized moments of this measure:
SN(z1, ..., zN) =
∫
φ(z1)...φ(zN)dν(φ). (2.5)
9
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
The full interacting measure may now be written as the multiplication ofthe Gaussian measure dµ(φ) by the interaction factor:
dν =1
Ze−(λ/4!)
R
φ4(x)dxdµ(φ) (2.4)
and the Schwinger functions are the normalized moments of this measure:
SN(z1, ..., zN) =
∫
φ(z1)...φ(zN)dν(φ). (2.5)
Expanding the exponential as a formal power series in the coupling constantλ we get perturbative field theory:
SN(z1, ..., zN) =1
Z
∞∑
n=0
(−λ)n
n!
∫
[
∫
φ4(x)dx
4!
]nφ(z1)...φ(zN)dµ(φ) (2.6)
9
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman RulesBy Wick theorem, SN is a sum over “Wick contractions schemes”, i.e. waysof pairing together 4n + N fields into 2n + N/2 pairs. There are exactly(4n + N − 1)(4n + N − 3)...5.3.1 = (4n + N)!! such contraction schemes.
10
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman RulesBy Wick theorem, SN is a sum over “Wick contractions schemes”, i.e. waysof pairing together 4n + N fields into 2n + N/2 pairs. There are exactly(4n + N − 1)(4n + N − 3)...5.3.1 = (4n + N)!! such contraction schemes.
1
x
x
x
1
1
Sources
z2
z
x
x
x
z
zz
z1
2
3
1
1
1
3
z1
z4
4
Vertices A Wick contraction
Figure: A contraction scheme
10
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
R
11
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
Each internal position x1, ..., xn is associated to a vertex of degree 4 andeach external position zi to a vertex of degree 1.
R
11
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
Each internal position x1, ..., xn is associated to a vertex of degree 4 andeach external position zi to a vertex of degree 1.The Feynman amplitude is in position space
AG (z1, ..., zN) =
∫ n∏
v=1
ddxv
∏
ℓ
C (xℓ, x′ℓ) (2.7)
R
11
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
Each internal position x1, ..., xn is associated to a vertex of degree 4 andeach external position zi to a vertex of degree 1.The Feynman amplitude is in position space
AG (z1, ..., zN) =
∫ n∏
v=1
ddxv
∏
ℓ
C (xℓ, x′ℓ) (2.7)
There is an interesting combinatoric factor in front, which counts howmany ”Wick schemes” lead to that graph and multiplies by 1
n!(−λ4! )n.
R
11
![Page 49: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/49.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Feynman Rules
Each internal position x1, ..., xn is associated to a vertex of degree 4 andeach external position zi to a vertex of degree 1.The Feynman amplitude is in position space
AG (z1, ..., zN) =
∫ n∏
v=1
ddxv
∏
ℓ
C (xℓ, x′ℓ) (2.7)
There is an interesting combinatoric factor in front, which counts howmany ”Wick schemes” lead to that graph and multiplies by 1
n!(−λ4! )n.
Feynman amplitudes are functions (in fact distributions) of the externalpositions z1, ..., zN . They may diverge either because of integration over allof R4 or because of the singularity in the propagator C (x , y) at x = y .
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A Recipe for Renormalization
12
![Page 51: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/51.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A Recipe for Renormalization
In ordinary field theory renormalization relies on the combination of threeingredients:
12
![Page 52: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/52.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A Recipe for Renormalization
In ordinary field theory renormalization relies on the combination of threeingredients:
A scale decomposition
12
![Page 53: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/53.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A Recipe for Renormalization
In ordinary field theory renormalization relies on the combination of threeingredients:
A scale decomposition
A locality principle
12
![Page 54: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/54.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A Recipe for Renormalization
In ordinary field theory renormalization relies on the combination of threeingredients:
A scale decomposition
A locality principle
A power counting
12
![Page 55: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/55.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A Recipe for Renormalization
In ordinary field theory renormalization relies on the combination of threeingredients:
A scale decomposition
A locality principle
A power counting
The first two elements are quite universal. The third depends on the detailsof the model.
12
![Page 56: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/56.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Scale decomposition
13
![Page 57: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/57.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Scale decomposition
It is convenient to perform it using the parametric representation of thepropagator:
13
![Page 58: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/58.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Scale decomposition
It is convenient to perform it using the parametric representation of thepropagator:
C =∑
i∈N
C i , (2.8)
C i (x , y) =
∫ M−2(i+1)
M−2i
dαe−αm2 e−‖x−y‖2/4α
α2(2.9)
6 KM2ie−cM i‖x−y‖ (2.10)
where M is a fixed integer.
13
![Page 59: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/59.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Scale decomposition
It is convenient to perform it using the parametric representation of thepropagator:
C =∑
i∈N
C i , (2.8)
C i (x , y) =
∫ M−2(i+1)
M−2i
dαe−αm2 e−‖x−y‖2/4α
α2(2.9)
6 KM2ie−cM i‖x−y‖ (2.10)
where M is a fixed integer.
Higher and higher values of the scale index i probe shorter and shorterdistances.
13
![Page 60: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/60.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Scale attributions, high subgraphs
Decomposing the propagator means that for any graph we have to sumover an independent scale index for each line of the graph.
14
![Page 61: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/61.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Scale attributions, high subgraphs
Decomposing the propagator means that for any graph we have to sumover an independent scale index for each line of the graph.
At fixed scale attribution, some subgraphs play an essential role. They arethe connected subgraphs whose internal lines all have higher scale indexthan all the external lines of the subgraph. Let’s call them the ”high”subgraphs. They form a forest for the inclusion relation.
14
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Locality principle
The locality principle is independent of the dimension: it simply remarksthat every ”high” subgraph looks more and more local as the gap betweenthe smallest internal and the largest external scale grows.
15
![Page 63: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/63.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Locality principle
The locality principle is independent of the dimension: it simply remarksthat every ”high” subgraph looks more and more local as the gap betweenthe smallest internal and the largest external scale grows.Let’s visualize an example
15
![Page 64: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/64.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Locality principle
The locality principle is independent of the dimension: it simply remarksthat every ”high” subgraph looks more and more local as the gap betweenthe smallest internal and the largest external scale grows.Let’s visualize an example
15
![Page 65: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/65.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Locality principle
The locality principle is independent of the dimension: it simply remarksthat every ”high” subgraph looks more and more local as the gap betweenthe smallest internal and the largest external scale grows.Let’s visualize an example
15
![Page 66: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/66.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Power counting
16
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Power counting
Power counting depends on the dimension d .
16
![Page 68: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/68.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Power counting
Power counting depends on the dimension d .The key question is whether after spatial integration of the internal verticesof a high subgraph, save one, the sum over the gap between the lowestinternal and highest external scale converges or diverges.
16
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Power counting
Power counting depends on the dimension d .The key question is whether after spatial integration of the internal verticesof a high subgraph, save one, the sum over the gap between the lowestinternal and highest external scale converges or diverges.
In four dimension by the previous estimates of a single scale propagator C ,power counting delivers a factor M2i per line and M−4i per vertexintegration
∫
d4x . There are n − 1 ”internal” integrations to perform tocompare a high connected subgraph to a local vertex.
16
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
17
![Page 71: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/71.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
For a connected φ44 graph, the net factor is 2l(G )−4(n(G )−1) = 4−N(G )
(because 4n = 2l + N). When this factor is strictly negative, the sum isgeometrically convergent, otherwise it diverges.
17
![Page 72: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/72.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
For a connected φ44 graph, the net factor is 2l(G )−4(n(G )−1) = 4−N(G )
(because 4n = 2l + N). When this factor is strictly negative, the sum isgeometrically convergent, otherwise it diverges.
17
![Page 73: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/73.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
For a connected φ44 graph, the net factor is 2l(G )−4(n(G )−1) = 4−N(G )
(because 4n = 2l + N). When this factor is strictly negative, the sum isgeometrically convergent, otherwise it diverges.
For instance for this graph the sum over the red scale i at fixed blue scalediverges (logarithmically) because there are two line factors M2i and asingle internal integration M−4i .
17
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
18
![Page 75: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/75.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
High subgraphs with N = 2 and 4 diverge when inserted into ordinary ones.
18
![Page 76: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/76.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
High subgraphs with N = 2 and 4 diverge when inserted into ordinary ones.
However this divergence can be absorbed into a change of the threeparameters (coupling constant, mass and wave function) which appeared inthe initial model.
18
![Page 77: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/77.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
High subgraphs with N = 2 and 4 diverge when inserted into ordinary ones.
However this divergence can be absorbed into a change of the threeparameters (coupling constant, mass and wave function) which appeared inthe initial model.
This means physically that the parameters of the model do change with theobservation scale but not the structure of the model itself. This is a kind ofsophisticated self-similarity.
18
![Page 78: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/78.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Perturbative renormalisability of φ44
High subgraphs with N = 2 and 4 diverge when inserted into ordinary ones.
However this divergence can be absorbed into a change of the threeparameters (coupling constant, mass and wave function) which appeared inthe initial model.
This means physically that the parameters of the model do change with theobservation scale but not the structure of the model itself. This is a kind ofsophisticated self-similarity.
Such models are called (perturbatively) renormalizable. But...
18
![Page 79: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/79.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The flow
Every ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.
19
![Page 80: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/80.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The flow
Every ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.In the case of the φ4
4 theory the evolution of the coupling constant λ underchange of scale is mainly due to the first non trivial one-particle irreduciblegraph
19
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The flow
Every ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.In the case of the φ4
4 theory the evolution of the coupling constant λ underchange of scale is mainly due to the first non trivial one-particle irreduciblegraph
−λi−1 = −λi + β(−λi )2,
dλi
di= +β(λi )
2,
19
![Page 82: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/82.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The flowEvery ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.In the case of the φ4
4 theory the evolution of the coupling constant λ underchange of scale is mainly due to the first non trivial one-particle irreduciblegraph
−λi−1 = −λi + β(−λi )2,
dλi
di= +β(λi )
2,
19
![Page 83: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/83.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
20
![Page 84: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/84.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
Expressing the perturbation theory in terms of the last (renormalized)coupling leads to the renormalized expansion.
20
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
Expressing the perturbation theory in terms of the last (renormalized)coupling leads to the renormalized expansion.
The renormalized expansion requires some complicated book-keeping(Zimmermann’s forests, Connes-Kreimer Hopf algebra).
20
![Page 86: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/86.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
Expressing the perturbation theory in terms of the last (renormalized)coupling leads to the renormalized expansion.
The renormalized expansion requires some complicated book-keeping(Zimmermann’s forests, Connes-Kreimer Hopf algebra).
It subtracts local pieces of divergent subgraphs irrespective of whetherthey are high or not.
20
![Page 87: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/87.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
Expressing the perturbation theory in terms of the last (renormalized)coupling leads to the renormalized expansion.
The renormalized expansion requires some complicated book-keeping(Zimmermann’s forests, Connes-Kreimer Hopf algebra).
It subtracts local pieces of divergent subgraphs irrespective of whetherthey are high or not.
There is a price to pay, called renormalons.
20
![Page 88: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/88.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalons
21
![Page 89: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/89.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalons
Let us write the bubble graph in momentum space
AG (k) =
∫
d4p1
(p2 + m2)(p + k)2 + m2)
21
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalons
Let us write the bubble graph in momentum space
AG (k) =
∫
d4p1
(p2 + m2)(p + k)2 + m2)
AeffG (k) =
∫
d4p1
(p2 + m2)(p + k)2 + m2)−
∫
|p|≥|k|
1
(p2 + m2)2
is bounded: |AeffG (k)| ≤ const.
21
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalons
Let us write the bubble graph in momentum space
AG (k) =
∫
d4p1
(p2 + m2)(p + k)2 + m2)
AeffG (k) =
∫
d4p1
(p2 + m2)(p + k)2 + m2)−
∫
|p|≥|k|
1
(p2 + m2)2
is bounded: |AeffG (k)| ≤ const.
ArenG (k) =
∫
d4p1
(p2 + m2)(p + k)2 + m2)−
∫
1
(p2 + m2)2
is finite but unbounded: |AeffG (k)| ∼|k|→∞ c log |k/m|.
21
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalons, II
22
![Page 93: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/93.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Renormalons, II
A chain of n such graphs as above behaves as [log |q|]n. Inserting them in aconvergent loop leads to a total amplitude of Pn
∫
[log |q|]nd4q
[q2 + m2]3≃n→∞ cnn!
which cannot be summed over n.
22
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
23
![Page 95: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/95.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
Expressing the theory in terms of all the running couplings leads to theeffective expansion (which is not a power series in a single coupling).
23
![Page 96: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/96.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
Expressing the theory in terms of all the running couplings leads to theeffective expansion (which is not a power series in a single coupling).
Because there is a single forest subtracted there is no book-keeping,(no need for Zimmermann’s forests nor Connes-Kreimer Hopf algebras)
23
![Page 97: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/97.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Effective versus renormalized series
Expressing the theory in terms of all the running couplings leads to theeffective expansion (which is not a power series in a single coupling).
Because there is a single forest subtracted there is no book-keeping,(no need for Zimmermann’s forests nor Connes-Kreimer Hopf algebras)
There are also no renormalons, so the effective expansion is good forconstructive purpose.
23
![Page 98: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/98.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The unavoidable Landau ghost?
24
![Page 99: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/99.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The unavoidable Landau ghost?
In the case of the φ44 theory the evolution of the coupling constant λ under
change of scale is mainly due to the first non trivial one-particle irreduciblegraph we already saw.
24
![Page 100: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/100.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The unavoidable Landau ghost?
In the case of the φ44 theory the evolution of the coupling constant λ under
change of scale is mainly due to the first non trivial one-particle irreduciblegraph we already saw.
24
![Page 101: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/101.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The unavoidable Landau ghost?
In the case of the φ44 theory the evolution of the coupling constant λ under
change of scale is mainly due to the first non trivial one-particle irreduciblegraph we already saw.
It gives the flow equation
−λi−1 = −λi + β(−λi )2,
dλi
di= +β(λi )
2, (2.11)
whose sign cannot be changed without losing stability.This flow diverges ina finite time!
24
![Page 102: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/102.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The unavoidable Landau ghost?
In the case of the φ44 theory the evolution of the coupling constant λ under
change of scale is mainly due to the first non trivial one-particle irreduciblegraph we already saw.
It gives the flow equation
−λi−1 = −λi + β(−λi )2,
dλi
di= +β(λi )
2, (2.11)
whose sign cannot be changed without losing stability.This flow diverges ina finite time! In the 60’s all known field theories sufferered from thisLandau ghost.
24
![Page 103: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/103.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Asymptotic Freedom
25
![Page 104: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/104.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Asymptotic Freedom
In fact field theory and renormalization made in the early 70’s a spectacularcomeback:
25
![Page 105: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/105.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Asymptotic Freedom
In fact field theory and renormalization made in the early 70’s a spectacularcomeback:
Weinberg and Salam unified the weak and electromagnetic interactionsinto the formalism of Yang and Mills of non-Abelian gauge theories,which are based on an internal non commutative symmetry.
25
![Page 106: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/106.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Asymptotic Freedom
In fact field theory and renormalization made in the early 70’s a spectacularcomeback:
Weinberg and Salam unified the weak and electromagnetic interactionsinto the formalism of Yang and Mills of non-Abelian gauge theories,which are based on an internal non commutative symmetry.
’tHooft and Veltmann succeeded to show that these theories are stillrenormalisable. They used a new technical tool calleddimensionalrenormalization.
25
![Page 107: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/107.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Asymptotic Freedom
In fact field theory and renormalization made in the early 70’s a spectacularcomeback:
Weinberg and Salam unified the weak and electromagnetic interactionsinto the formalism of Yang and Mills of non-Abelian gauge theories,which are based on an internal non commutative symmetry.
’tHooft and Veltmann succeeded to show that these theories are stillrenormalisable. They used a new technical tool calleddimensionalrenormalization.
’t Hooft (1972, unpublished), Politzer, Gross and Wilczek (1973)discovered that these theories did not suffer from the Landau ghost.Gross and Wilczek then developed a theory of this type, QCD todescribe strong interactions (nuclear forces).
25
![Page 108: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/108.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Asymptotic Freedom
In fact field theory and renormalization made in the early 70’s a spectacularcomeback:
Weinberg and Salam unified the weak and electromagnetic interactionsinto the formalism of Yang and Mills of non-Abelian gauge theories,which are based on an internal non commutative symmetry.
’tHooft and Veltmann succeeded to show that these theories are stillrenormalisable. They used a new technical tool calleddimensionalrenormalization.
’t Hooft (1972, unpublished), Politzer, Gross and Wilczek (1973)discovered that these theories did not suffer from the Landau ghost.Gross and Wilczek then developed a theory of this type, QCD todescribe strong interactions (nuclear forces).
Around the same time K. Wilson enlarged considerably the realm ofrenormalization, under the name of the renormalization group.
25
![Page 109: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/109.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Asymptotic Freedom
In fact field theory and renormalization made in the early 70’s a spectacularcomeback:
Weinberg and Salam unified the weak and electromagnetic interactionsinto the formalism of Yang and Mills of non-Abelian gauge theories,which are based on an internal non commutative symmetry.
’tHooft and Veltmann succeeded to show that these theories are stillrenormalisable. They used a new technical tool calleddimensionalrenormalization.
’t Hooft (1972, unpublished), Politzer, Gross and Wilczek (1973)discovered that these theories did not suffer from the Landau ghost.Gross and Wilczek then developed a theory of this type, QCD todescribe strong interactions (nuclear forces).
Around the same time K. Wilson enlarged considerably the realm ofrenormalization, under the name of the renormalization group.
Happy end, all the people in red in this page got the Nobel prize...25
![Page 110: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/110.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive Theory
26
![Page 111: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/111.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive Theory
Functional integration looks good for global stability bounds(|
∫
e−λφ4dµ(φ)| ≤ 1)
26
![Page 112: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/112.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive Theory
Functional integration looks good for global stability bounds(|
∫
e−λφ4dµ(φ)| ≤ 1) but bad to compute connected functions
26
![Page 113: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/113.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive Theory
Functional integration looks good for global stability bounds(|
∫
e−λφ4dµ(φ)| ≤ 1) but bad to compute connected functions
Feynman graphs have the opposite properties. Connectivity is readeasily on Feynman graphs, but these graphs proliferate too fast atlarge order for convergence
26
![Page 114: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/114.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive Theory
Functional integration looks good for global stability bounds(|
∫
e−λφ4dµ(φ)| ≤ 1) but bad to compute connected functions
Feynman graphs have the opposite properties. Connectivity is readeasily on Feynman graphs, but these graphs proliferate too fast atlarge order for convergence
Constructive theory is a search for a good compromise between functionalintegral and Feynman graphs.
26
![Page 115: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/115.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive Theory
Functional integration looks good for global stability bounds(|
∫
e−λφ4dµ(φ)| ≤ 1) but bad to compute connected functions
Feynman graphs have the opposite properties. Connectivity is readeasily on Feynman graphs, but these graphs proliferate too fast atlarge order for convergence
Constructive theory is a search for a good compromise between functionalintegral and Feynman graphs.
Trees are at the core of this compromise because they are not too manyand they ensure connectedness. Cycles/loops should be kept in functionalintegral form. So constructive theory is about finding trees and resummingloops.
26
![Page 116: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/116.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A partial history of constructive field theory
27
![Page 117: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/117.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A partial history of constructive field theory
70’s: superrenormalizable theories, eg φ42, φ4
3... with non-canonicalcluster and Mayer expansions (lattices of cubes...)
27
![Page 118: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/118.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A partial history of constructive field theory
70’s: superrenormalizable theories, eg φ42, φ4
3... with non-canonicalcluster and Mayer expansions (lattices of cubes...)
80’s: tools generalized into multiscale expansions to treatrenormalizable theories such as GN2.
27
![Page 119: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/119.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A partial history of constructive field theory
70’s: superrenormalizable theories, eg φ42, φ4
3... with non-canonicalcluster and Mayer expansions (lattices of cubes...)
80’s: tools generalized into multiscale expansions to treatrenormalizable theories such as GN2.
Many applications developed in statistical mechanics
27
![Page 120: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/120.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A partial history of constructive field theory
70’s: superrenormalizable theories, eg φ42, φ4
3... with non-canonicalcluster and Mayer expansions (lattices of cubes...)
80’s: tools generalized into multiscale expansions to treatrenormalizable theories such as GN2.
Many applications developed in statistical mechanics
90’s: cluster expansions were realized unnecessary for Fermions. Thissimplification was important to constructive analysis of interactingFermions in condensed matter, with many results in the 00’s.
27
![Page 121: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/121.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A partial history of constructive field theory
70’s: superrenormalizable theories, eg φ42, φ4
3... with non-canonicalcluster and Mayer expansions (lattices of cubes...)
80’s: tools generalized into multiscale expansions to treatrenormalizable theories such as GN2.
Many applications developed in statistical mechanics
90’s: cluster expansions were realized unnecessary for Fermions. Thissimplification was important to constructive analysis of interactingFermions in condensed matter, with many results in the 00’s.
in the late 00’s new canonical methods have been found to dispense ofcluster/Mayer expansions for Bosons as well (loop vertex expansions,tree QFT...) and new domains opened (NCQFT, Quantum Gravity...)
27
![Page 122: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/122.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A partial history of constructive field theory
70’s: superrenormalizable theories, eg φ42, φ4
3... with non-canonicalcluster and Mayer expansions (lattices of cubes...)
80’s: tools generalized into multiscale expansions to treatrenormalizable theories such as GN2.
Many applications developed in statistical mechanics
90’s: cluster expansions were realized unnecessary for Fermions. Thissimplification was important to constructive analysis of interactingFermions in condensed matter, with many results in the 00’s.
in the late 00’s new canonical methods have been found to dispense ofcluster/Mayer expansions for Bosons as well (loop vertex expansions,tree QFT...) and new domains opened (NCQFT, Quantum Gravity...)
Still no full-fledged constructive four dimensional field theorycompleted. φ4
4 has a Landau ghost, non-Abelian gauge theory hasGribov ambiguities and infrared confinement (1M$ problem...). →Grosse-Wulkenhaar model
27
![Page 123: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/123.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive QFT in Single Slid(c)e
28
![Page 124: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/124.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive QFT in Single Slid(c)e
Perturbative QFTS =
∑
G
AG
28
![Page 125: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/125.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive QFT in Single Slid(c)e
Perturbative QFTS =
∑
G
AG
Problem: it does not converge:
∑
G
|AG | = +∞
28
![Page 126: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/126.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive QFT in Single Slid(c)e
Perturbative QFTS =
∑
G
AG
Problem: it does not converge:
∑
G
|AG | = +∞
Constructive QFT = clever rewriting
28
![Page 127: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/127.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive QFT in Single Slid(c)e
Perturbative QFTS =
∑
G
AG
Problem: it does not converge:
∑
G
|AG | = +∞
Constructive QFT = clever rewriting
S =∑
T
AT
28
![Page 128: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/128.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive QFT in Single Slid(c)e
Perturbative QFTS =
∑
G
AG
Problem: it does not converge:
∑
G
|AG | = +∞
Constructive QFT = clever rewriting
S =∑
T
AT
Solution: it converges!∑
T
|AT | < +∞
28
![Page 129: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/129.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive rewriting
29
![Page 130: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/130.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive rewriting
The forest formula provides canonical barycentric weights w(G ,T ) for thevarious trees T in a connected graph G .
29
![Page 131: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/131.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive rewriting
The forest formula provides canonical barycentric weights w(G ,T ) for thevarious trees T in a connected graph G .
w(G ,T ) =
∫ 1
0
∏
ℓ∈T
dwℓ
∏
ℓ 6∈T
xTℓ (w)
29
![Page 132: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/132.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive rewriting
The forest formula provides canonical barycentric weights w(G ,T ) for thevarious trees T in a connected graph G .
w(G ,T ) =
∫ 1
0
∏
ℓ∈T
dwℓ
∏
ℓ 6∈T
xTℓ (w)
xTℓ (w) = inf
ℓ′wℓ′ , ℓ′ ∈ unique path in T joining ends of ℓ
29
![Page 133: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/133.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Constructive rewriting
The forest formula provides canonical barycentric weights w(G ,T ) for thevarious trees T in a connected graph G .
w(G ,T ) =
∫ 1
0
∏
ℓ∈T
dwℓ
∏
ℓ 6∈T
xTℓ (w)
xTℓ (w) = inf
ℓ′wℓ′ , ℓ′ ∈ unique path in T joining ends of ℓ
Barycentric weights means
∑
T∈G
w(G ,T ) = 1 ∀G .
29
![Page 134: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/134.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
An Example
30
![Page 135: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/135.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
An Example
11
l2
l3
l4v
1
v2
v3
30
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
An Example
11
l2
l3
l4v
1
v2
v3
w(G , T12) =
∫ 1
0dw1
∫ 1
0dw2[inf(w1,w2)]
2 = 1/6
30
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
An Example
11
l2
l3
l4v
1
v2
v3
w(G , T12) =
∫ 1
0dw1
∫ 1
0dw2[inf(w1,w2)]
2 = 1/6
w(G ,T13) = w(G ,T14) = w(G , T23) = w(G , T24)
=
∫ 1
0dw1
∫ 1
0dw3
[
inf(w1, w3)]
w3 = 5/24
30
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
An Example
11
l2
l3
l4v
1
v2
v3
w(G , T12) =
∫ 1
0dw1
∫ 1
0dw2[inf(w1,w2)]
2 = 1/6
w(G ,T13) = w(G ,T14) = w(G , T23) = w(G , T24)
=
∫ 1
0dw1
∫ 1
0dw3
[
inf(w1, w3)]
w3 = 5/24
1
6+ 4 · 5
24= 1
30
![Page 139: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/139.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Fermions/Bosons
31
![Page 140: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/140.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Fermions/Bosons
The constructive rewriting is nothing but
S =∑
G
[
∑
T⊂G
w(G ,T )
]
AG =∑
T
AT , AT =∑
G⊃T
w(G ,T )AG
31
![Page 141: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/141.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Fermions/Bosons
The constructive rewriting is nothing but
S =∑
G
[
∑
T⊂G
w(G ,T )
]
AG =∑
T
AT , AT =∑
G⊃T
w(G ,T )AG
For Fermions one can apply this method directly to the graphs of theordinary perturbative expansion.
31
![Page 142: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/142.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Fermions/Bosons
The constructive rewriting is nothing but
S =∑
G
[
∑
T⊂G
w(G ,T )
]
AG =∑
T
AT , AT =∑
G⊃T
w(G ,T )AG
For Fermions one can apply this method directly to the graphs of theordinary perturbative expansion.
For Bosons one should apply this method to the graphs of the intermediatefield expansion (Loop Vertex Expansion).
31
![Page 143: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/143.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The Forest Formula or “constructive swiss knife”
32
![Page 144: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/144.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The Forest Formula or “constructive swiss knife”
Let F be a smooth function of n(n − 1)/2 line variables xℓ, ℓ = (i , j),1 ≤ i < j ≤ n. The forest formula states
F (1, ..., 1) =∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∏
ℓ∈F
∂
∂xℓF
[
xF (w)]
, where
32
![Page 145: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/145.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The Forest Formula or “constructive swiss knife”
Let F be a smooth function of n(n − 1)/2 line variables xℓ, ℓ = (i , j),1 ≤ i < j ≤ n. The forest formula states
F (1, ..., 1) =∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∏
ℓ∈F
∂
∂xℓF
[
xF (w)]
, where
the sum over F is over all forests over n vertices,
32
![Page 146: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/146.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The Forest Formula or “constructive swiss knife”
Let F be a smooth function of n(n − 1)/2 line variables xℓ, ℓ = (i , j),1 ≤ i < j ≤ n. The forest formula states
F (1, ..., 1) =∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∏
ℓ∈F
∂
∂xℓF
[
xF (w)]
, where
the sum over F is over all forests over n vertices,
the ‘weakening parameter” xFℓ (w) is 0 if ℓ = (i , j) with i and j in
different connected components with respect to F ; otherwise it is theinfimum of the wℓ′ for ℓ′ running over the unique path from i to j in F .
32
![Page 147: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/147.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The Forest Formula or “constructive swiss knife”
Let F be a smooth function of n(n − 1)/2 line variables xℓ, ℓ = (i , j),1 ≤ i < j ≤ n. The forest formula states
F (1, ..., 1) =∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∏
ℓ∈F
∂
∂xℓF
[
xF (w)]
, where
the sum over F is over all forests over n vertices,
the ‘weakening parameter” xFℓ (w) is 0 if ℓ = (i , j) with i and j in
different connected components with respect to F ; otherwise it is theinfimum of the wℓ′ for ℓ′ running over the unique path from i to j in F .
Furthermore the real symmetric matrix xFi ,j(w) (completed by 1 on
the diagonal i = j) is positive.
32
![Page 148: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/148.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
The Forest Formula or “constructive swiss knife”
Let F be a smooth function of n(n − 1)/2 line variables xℓ, ℓ = (i , j),1 ≤ i < j ≤ n. The forest formula states
F (1, ..., 1) =∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∏
ℓ∈F
∂
∂xℓF
[
xF (w)]
, where
the sum over F is over all forests over n vertices,
the ‘weakening parameter” xFℓ (w) is 0 if ℓ = (i , j) with i and j in
different connected components with respect to F ; otherwise it is theinfimum of the wℓ′ for ℓ′ running over the unique path from i to j in F .
Furthermore the real symmetric matrix xFi ,j(w) (completed by 1 on
the diagonal i = j) is positive.
32
![Page 149: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/149.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Five different ways to compute a log
33
![Page 150: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/150.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Five different ways to compute a log
F (λ) =
∫ +∞
−∞e−λx4−x2/2 dx√
2π
is Borel summable. How to compute G (λ) = log F (λ) (and prove it is alsoBorel summable)?
33
![Page 151: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/151.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Five different ways to compute a log
F (λ) =
∫ +∞
−∞e−λx4−x2/2 dx√
2π
is Borel summable. How to compute G (λ) = log F (λ) (and prove it is alsoBorel summable)?
Composition of series (XIXth century)
33
![Page 152: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/152.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Five different ways to compute a log
F (λ) =
∫ +∞
−∞e−λx4−x2/2 dx√
2π
is Borel summable. How to compute G (λ) = log F (λ) (and prove it is alsoBorel summable)?
Composition of series (XIXth century)
A la Feynman (1950)
33
![Page 153: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/153.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Five different ways to compute a log
F (λ) =
∫ +∞
−∞e−λx4−x2/2 dx√
2π
is Borel summable. How to compute G (λ) = log F (λ) (and prove it is alsoBorel summable)?
Composition of series (XIXth century)
A la Feynman (1950)
‘Classical Constructive”, a la Glimm-Jaffe-Spencer (1970...)
33
![Page 154: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/154.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Five different ways to compute a log
F (λ) =
∫ +∞
−∞e−λx4−x2/2 dx√
2π
is Borel summable. How to compute G (λ) = log F (λ) (and prove it is alsoBorel summable)?
Composition of series (XIXth century)
A la Feynman (1950)
‘Classical Constructive”, a la Glimm-Jaffe-Spencer (1970...)
With loop vertices (2007)...
33
![Page 155: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/155.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Five different ways to compute a log
F (λ) =
∫ +∞
−∞e−λx4−x2/2 dx√
2π
is Borel summable. How to compute G (λ) = log F (λ) (and prove it is alsoBorel summable)?
Composition of series (XIXth century)
A la Feynman (1950)
‘Classical Constructive”, a la Glimm-Jaffe-Spencer (1970...)
With loop vertices (2007)...
Tree QFT (2008)...
33
![Page 156: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/156.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Borel Summability
34
![Page 157: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/157.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Borel Summability
Borel summability of a series an means existence of a function f with twoproperties
34
![Page 158: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/158.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Borel Summability
Borel summability of a series an means existence of a function f with twoproperties
Analyticity in a disk tangent at the origin to the imaginary axis
34
![Page 159: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/159.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Borel Summability
Borel summability of a series an means existence of a function f with twoproperties
Analyticity in a disk tangent at the origin to the imaginary axis
plus uniform remainder estimates:
|f (λ) −N
∑
n=0
anλn| ≤ KN |λ|N+1N! (4.12)
34
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Borel Summability
Borel summability of a series an means existence of a function f with twoproperties
Analyticity in a disk tangent at the origin to the imaginary axis
plus uniform remainder estimates:
|f (λ) −N
∑
n=0
anλn| ≤ KN |λ|N+1N! (4.12)
Given any series an, there is at most one such function f . When there isone, it is called the Borel sum, and it can be computed from the series toarbitrary accuracy.
34
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Composition of series
35
![Page 162: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/162.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Composition of series
F = 1 + H, H =∑
p≥1
ap(−λ)p, ap =(4p)!!
p!
log(1 + x) =∞
∑
n=1
(−1)n+1 xn
n
G =
∞∑
n=1
(−1)n+1 H(λ)n
n=
∑
k≥1
bk(−λ)k ,
bk =
k∑
n=1
(−1)n+1
n
∑
p1,..,pn≥1p1+...+pn=k
∏
j
(4pj)!!
pj !
35
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Composition of series
F = 1 + H, H =∑
p≥1
ap(−λ)p, ap =(4p)!!
p!
log(1 + x) =∞
∑
n=1
(−1)n+1 xn
n
G =
∞∑
n=1
(−1)n+1 H(λ)n
n=
∑
k≥1
bk(−λ)k ,
bk =
k∑
n=1
(−1)n+1
n
∑
p1,..,pn≥1p1+...+pn=k
∏
j
(4pj)!!
pj !
Borel summability is unclear. Even the sign of bk is unclear.
35
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A la Feynman
36
![Page 165: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/165.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A la Feynman
F = 1 + H, H =∑
p≥1
ap(−λ)p, ap =1
p!#vacuum graphs on p vertices
36
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A la Feynman
F = 1 + H, H =∑
p≥1
ap(−λ)p, ap =1
p!#vacuum graphs on p vertices
G =
∞∑
k=1
(−λ)kbk , bk =1
k!#vacuum connected graphs on k vertices
36
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A la Feynman
F = 1 + H, H =∑
p≥1
ap(−λ)p, ap =1
p!#vacuum graphs on p vertices
G =
∞∑
k=1
(−λ)kbk , bk =1
k!#vacuum connected graphs on k vertices
b1 = 3, b2 = 48, b3 = 1584...
36
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
A la Feynman
F = 1 + H, H =∑
p≥1
ap(−λ)p, ap =1
p!#vacuum graphs on p vertices
G =
∞∑
k=1
(−λ)kbk , bk =1
k!#vacuum connected graphs on k vertices
b1 = 3, b2 = 48, b3 = 1584...
Borel summability unclear. bk ≥ 0 clear.
36
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive
37
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive
Cluster expansion = Taylor-Lagrange expansion of the functional integral:
F = 1 + H, H = −λ
∫ 1
0dt
∫ +∞
−∞x4e−λtx4−x2/2 dx√
2π
37
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive
Cluster expansion = Taylor-Lagrange expansion of the functional integral:
F = 1 + H, H = −λ
∫ 1
0dt
∫ +∞
−∞x4e−λtx4−x2/2 dx√
2π
Mayer expansion: define Hi = −λ∫ 10 dt
∫ +∞−∞ x4
i e−λtx4i −x2
i /2 dxi√2π
= H ∀i ,
εij = 0 ∀i , j and write
F = 1 + H =∞
∑
n=0
n∏
i=1
Hi (λ)∏
1≤i<j≤n
εij
Defining ηij = −1, εij = 1 + ηij = 1 + xijηij |xij=1 and apply swiss knife.
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive Field Theory, II
38
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive Field Theory, II
F =∞
∑
n=0
1
n!
∑
F
n∏
i=1
Hi (λ)
∏
ℓ∈F
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈F
[
1 + ηℓxFℓ (w)
]
38
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive Field Theory, II
F =∞
∑
n=0
1
n!
∑
F
n∏
i=1
Hi (λ)
∏
ℓ∈F
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈F
[
1 + ηℓxFℓ (w)
]
G =
∞∑
n=1
1
n!
∑
T
n∏
i=1
Hi (λ)
∏
ℓ∈T
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈T
[
1 + ηℓxTℓ (w)
]
38
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive Field Theory, II
F =∞
∑
n=0
1
n!
∑
F
n∏
i=1
Hi (λ)
∏
ℓ∈F
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈F
[
1 + ηℓxFℓ (w)
]
G =
∞∑
n=1
1
n!
∑
T
n∏
i=1
Hi (λ)
∏
ℓ∈T
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈T
[
1 + ηℓxTℓ (w)
]
where the second sum runs over trees!
38
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive Field Theory, II
F =∞
∑
n=0
1
n!
∑
F
n∏
i=1
Hi (λ)
∏
ℓ∈F
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈F
[
1 + ηℓxFℓ (w)
]
G =
∞∑
n=1
1
n!
∑
T
n∏
i=1
Hi (λ)
∏
ℓ∈T
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈T
[
1 + ηℓxTℓ (w)
]
where the second sum runs over trees!
Convergence easy because each Hi contains a different ”copy”∫
dxi offunctional integration.
38
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Classical Constructive Field Theory, II
F =∞
∑
n=0
1
n!
∑
F
n∏
i=1
Hi (λ)
∏
ℓ∈F
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈F
[
1 + ηℓxFℓ (w)
]
G =
∞∑
n=1
1
n!
∑
T
n∏
i=1
Hi (λ)
∏
ℓ∈T
[
∫ 1
0dwℓ
]
ηℓ
∏
ℓ6∈T
[
1 + ηℓxTℓ (w)
]
where the second sum runs over trees!
Convergence easy because each Hi contains a different ”copy”∫
dxi offunctional integration.
Borel summability now easy from the Borel summability of H. But thismethod does not extend to noncommutative theory.
38
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices
39
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices
Intermediate field representation
39
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices
Intermediate field representation
F =
∫ +∞
−∞e−λx4−x2/2 dx√
2π=
∫ +∞
−∞
∫ +∞
−∞e−i
√2λσx2−x2/2−σ2/2 dx√
2π
dσ√2π
=
∫ +∞
−∞e−
12
log[1+i√
8λσ]−σ2/2 dσ√2π
=
∫ +∞
−∞
∞∑
n=0
V n
n!dµ(σ) (4.13)
39
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices
Intermediate field representation
F =
∫ +∞
−∞e−λx4−x2/2 dx√
2π=
∫ +∞
−∞
∫ +∞
−∞e−i
√2λσx2−x2/2−σ2/2 dx√
2π
dσ√2π
=
∫ +∞
−∞e−
12
log[1+i√
8λσ]−σ2/2 dσ√2π
=
∫ +∞
−∞
∞∑
n=0
V n
n!dµ(σ) (4.13)
Apply swiss knife by making copies: V n(σ) → ∏ni=1 Vi (σi ),
dµ(σ) → dµC (σi), Cij = 1 = xij |xij=1.
39
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices, II
40
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices, II
F =
∞∑
n=0
1
n!
∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∫
∏
ℓ∈F
∂
∂σi(ℓ)
∂
∂σj(ℓ)
n∏
i=1
V (σi )
dµCF
where CFij = xF
ℓ (w) if i < j , CFii = 1.
40
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices, II
F =
∞∑
n=0
1
n!
∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∫
∏
ℓ∈F
∂
∂σi(ℓ)
∂
∂σj(ℓ)
n∏
i=1
V (σi )
dµCF
where CFij = xF
ℓ (w) if i < j , CFii = 1.
G =
∞∑
n=1
1
n!
∑
T
∏
ℓ∈T
[
∫ 1
0dwℓ
]
∫
∏
ℓ∈T
∂
∂σi(ℓ)
∂
∂σj(ℓ)
n∏
i=1
V (σi )
dµCT
40
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Loop Vertices, II
F =
∞∑
n=0
1
n!
∑
F
∏
ℓ∈F
[
∫ 1
0dwℓ
]
∫
∏
ℓ∈F
∂
∂σi(ℓ)
∂
∂σj(ℓ)
n∏
i=1
V (σi )
dµCF
where CFij = xF
ℓ (w) if i < j , CFii = 1.
G =
∞∑
n=1
1
n!
∑
T
∏
ℓ∈T
[
∫ 1
0dwℓ
]
∫
∏
ℓ∈T
∂
∂σi(ℓ)
∂
∂σj(ℓ)
n∏
i=1
V (σi )
dµCT
where the second sum runs over trees!
40
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Advantages
41
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Advantages
One can picture the result as a sum over trees on loops, or ”cacti”. Since
∂k
∂σklog[1 + i
√8λσ] = −(k − 1)!(−i
√8λ)k [1 + i
√8λσ]−k ,
41
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Advantages
One can picture the result as a sum over trees on loops, or ”cacti”. Since
∂k
∂σklog[1 + i
√8λσ] = −(k − 1)!(−i
√8λ)k [1 + i
√8λσ]−k ,
Convergence is easy because |[1 + i√
8λσ]−k | ≤ 1.
41
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Advantages
One can picture the result as a sum over trees on loops, or ”cacti”. Since
∂k
∂σklog[1 + i
√8λσ] = −(k − 1)!(−i
√8λ)k [1 + i
√8λσ]−k ,
Convergence is easy because |[1 + i√
8λσ]−k | ≤ 1.
Borel summability is easy.
41
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Advantages
One can picture the result as a sum over trees on loops, or ”cacti”. Since
∂k
∂σklog[1 + i
√8λσ] = −(k − 1)!(−i
√8λ)k [1 + i
√8λσ]−k ,
Convergence is easy because |[1 + i√
8λσ]−k | ≤ 1.
Borel summability is easy.
This method extends to non commutative field theory.
41
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
42
![Page 192: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/192.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.
42
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.
the universal space E algebraic span of marked trees,
42
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.
the universal space E algebraic span of marked trees,
the canonical (BKAR) forest formula,
42
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.
the universal space E algebraic span of marked trees,
the canonical (BKAR) forest formula,
a positive operator H, which glues one more branch on any tree:
42
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.
the universal space E algebraic span of marked trees,
the canonical (BKAR) forest formula,
a positive operator H, which glues one more branch on any tree:
x x* =
42
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.
the universal space E algebraic span of marked trees,
the canonical (BKAR) forest formula,
a positive operator H, which glues one more branch on any tree:
x x* =
Each QFT corresponds to a scalar product on E through the tree formula.Correlation functions are expressed in terms of the resolvent 1
1+H.
42
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Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay
What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension
Tree Quantum Field Theory
This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.
the universal space E algebraic span of marked trees,
the canonical (BKAR) forest formula,
a positive operator H, which glues one more branch on any tree:
x x* =
Each QFT corresponds to a scalar product on E through the tree formula.Correlation functions are expressed in terms of the resolvent 1
1+H.
This formalism still in development could eg allow continuous interpolationbetween QFT’s (even in different space-time dimensions).
42
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Renormalization and Condensed Matter
Vincent Rivasseau
LPT Orsay
Cetraro, Summer 2010, Lecture II
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann variables
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann variables
Independent Grassmann variables ψ1, ..., ψn satisfy completeanticommutation relations
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann variables
Independent Grassmann variables ψ1, ..., ψn satisfy completeanticommutation relations
ψiψj = −ψjψi ∀i , j
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann variables
Independent Grassmann variables ψ1, ..., ψn satisfy completeanticommutation relations
ψiψj = −ψjψi ∀i , j
plus a rule called Grassmann integration which is
∫
dψi = 0,
∫
ψidψi = 1.
and the rule that dψ symbols also anticommute between themselves andwith all ψ variables.
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann variables
Independent Grassmann variables ψ1, ..., ψn satisfy completeanticommutation relations
ψiψj = −ψjψi ∀i , j
plus a rule called Grassmann integration which is
∫
dψi = 0,
∫
ψidψi = 1.
and the rule that dψ symbols also anticommute between themselves andwith all ψ variables.
Any function of Grassmann variables is a polynomial with highestdegree one in each variable.
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann variables
Independent Grassmann variables ψ1, ..., ψn satisfy completeanticommutation relations
ψiψj = −ψjψi ∀i , j
plus a rule called Grassmann integration which is
∫
dψi = 0,
∫
ψidψi = 1.
and the rule that dψ symbols also anticommute between themselves andwith all ψ variables.
Any function of Grassmann variables is a polynomial with highestdegree one in each variable.
Pfaffians and determinants can be nicely written as Grassmannintegrals.
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
The main important fact is that the determinant of any n by n matrix canbe expressed as a Grassmann Gaussian integral over 2n independentGrassmann variables.
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
The main important fact is that the determinant of any n by n matrix canbe expressed as a Grassmann Gaussian integral over 2n independentGrassmann variables.
It is convenient to name these variables as ψ1, . . . , ψn, ψ1, . . . , ψn, althoughthe bars have nothing to do with complex conjugation.
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
The main important fact is that the determinant of any n by n matrix canbe expressed as a Grassmann Gaussian integral over 2n independentGrassmann variables.
It is convenient to name these variables as ψ1, . . . , ψn, ψ1, . . . , ψn, althoughthe bars have nothing to do with complex conjugation.
The formula is
detM =
∫
∏
i
dψidψie−
P
ij ψiMijψj .
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
The main important fact is that the determinant of any n by n matrix canbe expressed as a Grassmann Gaussian integral over 2n independentGrassmann variables.
It is convenient to name these variables as ψ1, . . . , ψn, ψ1, . . . , ψn, althoughthe bars have nothing to do with complex conjugation.
The formula is
detM =
∫
∏
i
dψidψie−
P
ij ψiMijψj .
Remember that for ordinary commuting variables and a positive n by n
Hermitian matrix M
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
The main important fact is that the determinant of any n by n matrix canbe expressed as a Grassmann Gaussian integral over 2n independentGrassmann variables.
It is convenient to name these variables as ψ1, . . . , ψn, ψ1, . . . , ψn, althoughthe bars have nothing to do with complex conjugation.
The formula is
detM =
∫
∏
i
dψidψie−
P
ij ψiMijψj .
Remember that for ordinary commuting variables and a positive n by n
Hermitian matrix M
1
πn
∫ +∞
−∞
∏
i
d φidφie−
P
ij φiMijφj = det−1M.
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann Gaussian Integrals
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann Gaussian Integrals
Normalized Grassmann Gaussian measures may be written formally as
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann Gaussian Integrals
Normalized Grassmann Gaussian measures may be written formally as
dµM =
∏
dψidψie−
P
ij ψiM−1ijψj
∫∏
dψidψie−
P
ij ψiM−1ijψj
.
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann Gaussian Integrals
Normalized Grassmann Gaussian measures may be written formally as
dµM =
∏
dψidψie−
P
ij ψiM−1ijψj
∫∏
dψidψie−
P
ij ψiM−1ijψj
.
and again are characterized by their two point function or covariance
∫
ψiψjdµM = Mij .
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Grassmann Gaussian Integrals
Normalized Grassmann Gaussian measures may be written formally as
dµM =
∏
dψidψie−
P
ij ψiM−1ijψj
∫∏
dψidψie−
P
ij ψiM−1ijψj
.
and again are characterized by their two point function or covariance
∫
ψiψjdµM = Mij .
plus the Grassmann-Wick rule that n-point functions are expressed as sumover Wick contractions with signs.
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
In short Grassmann Gaussian measures are simpler than ordinary Gaussianmeasures for two main reasons:
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
In short Grassmann Gaussian measures are simpler than ordinary Gaussianmeasures for two main reasons:
Grassmann Gaussian measures are associated to any matrix M, there isno positivity requirement for M like for ordinary Gaussian measures.
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Determinants
In short Grassmann Gaussian measures are simpler than ordinary Gaussianmeasures for two main reasons:
Grassmann Gaussian measures are associated to any matrix M, there isno positivity requirement for M like for ordinary Gaussian measures.
their normalization directly computes the determinant of M, not theinverse (square-root of) the determinant of M. This is essential inmany areas where factoring out this determinant is desirable; it explainsin particular the success of Grassmann and supersymmetric functionalintegrals in the study of disordered systems → Tom Spencer’s lectures.
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Pfaffians
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by
detA = [Pf(A)]2.
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by
detA = [Pf(A)]2.
We can express the Pfaffian as:
Pf(A) =
∫
dχ1...dχne−
P
i<j χiAijχj =
∫
dχ1...dχne− 1
2
P
i,j χiAijχj .
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by
detA = [Pf(A)]2.
We can express the Pfaffian as:
Pf(A) =
∫
dχ1...dχne−
P
i<j χiAijχj =
∫
dχ1...dχne− 1
2
P
i,j χiAijχj .
Indeed we write
detA =
∫
∏
i
dψidψie−
P
ij ψiAijψj .
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by
detA = [Pf(A)]2.
We can express the Pfaffian as:
Pf(A) =
∫
dχ1...dχne−
P
i<j χiAijχj =
∫
dχ1...dχne− 1
2
P
i,j χiAijχj .
Indeed we write
detA =
∫
∏
i
dψidψie−
P
ij ψiAijψj .
Performing the change of variables (which a posteriori justifies the complexnotation)
ψi =1√2(χi − iωi ), ψi =
1√2(χi + iωi ),
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by
detA = [Pf(A)]2.
We can express the Pfaffian as:
Pf(A) =
∫
dχ1...dχne−
P
i<j χiAijχj =
∫
dχ1...dχne− 1
2
P
i,j χiAijχj .
Indeed we write
detA =
∫
∏
i
dψidψie−
P
ij ψiAijψj .
Performing the change of variables (which a posteriori justifies the complexnotation)
ψi =1√2(χi − iωi ), ψi =
1√2(χi + iωi ),
shows why detA is a perfect square and proves the red formula.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by
detA = [Pf(A)]2.
We can express the Pfaffian as:
Pf(A) =
∫
dχ1...dχne−
P
i<j χiAijχj =
∫
dχ1...dχne− 1
2
P
i,j χiAijχj .
Indeed we write
detA =
∫
∏
i
dψidψie−
P
ij ψiAijψj .
Performing the change of variables (which a posteriori justifies the complexnotation)
ψi =1√2(χi − iωi ), ψi =
1√2(χi + iωi ),
shows why detA is a perfect square and proves the red formula.Reference on Grassmann calculus in QFT: J. Feldman, RenormalizationGroup and Fermionic Functional Integrals.6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Fields
The Hamiltonian Fock space formalism for electrons in condensed matter isbetter reexpressed in terms of Grasmannian fields.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Fields
The Hamiltonian Fock space formalism for electrons in condensed matter isbetter reexpressed in terms of Grasmannian fields.
We consider independent Grasmmann valued fields ψ(ξ), ψ(ξ) with
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Fields
The Hamiltonian Fock space formalism for electrons in condensed matter isbetter reexpressed in terms of Grasmannian fields.
We consider independent Grasmmann valued fields ψ(ξ), ψ(ξ) with
ξ = (~x , t), ~x ∈ Rd , t ∈ [− ~
kT,
~
kT].
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Fields
The Hamiltonian Fock space formalism for electrons in condensed matter isbetter reexpressed in terms of Grasmannian fields.
We consider independent Grasmmann valued fields ψ(ξ), ψ(ξ) with
ξ = (~x , t), ~x ∈ Rd , t ∈ [− ~
kT,
~
kT].
The propagator is
Cab(k) = δab1
ik0 − [ε(~k) − µ]
where a, b are the spin indices.
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Fields
The Hamiltonian Fock space formalism for electrons in condensed matter isbetter reexpressed in terms of Grasmannian fields.
We consider independent Grasmmann valued fields ψ(ξ), ψ(ξ) with
ξ = (~x , t), ~x ∈ Rd , t ∈ [− ~
kT,
~
kT].
The propagator is
Cab(k) = δab1
ik0 − [ε(~k) − µ]
where a, b are the spin indices.
The momentum vector ~k has d spatial dimensions and ε(~k) is the energyfor a single electron of momentum ~k. The parameter µ corresponds to thechemical potential. The (spatial) Fermi surface is the manifold ε(~k) = µ.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Jellium Model
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Jellium Model
For a jellium isotropic model the energy function is invariant under spatialrotations
ε(~k) =~k2
2m
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Jellium Model
For a jellium isotropic model the energy function is invariant under spatialrotations
ε(~k) =~k2
2m
where m is some effective or “dressed” electron mass. In this case theFermi surface is simply a sphere. This jellium isotropic model is realistic inthe limit of weak electron densities, where the Fermi surface becomesapproximately spherical.
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Jellium Model
For a jellium isotropic model the energy function is invariant under spatialrotations
ε(~k) =~k2
2m
where m is some effective or “dressed” electron mass. In this case theFermi surface is simply a sphere. This jellium isotropic model is realistic inthe limit of weak electron densities, where the Fermi surface becomesapproximately spherical.
In general a propagator with a more complicated energy function ε(~k) hasto be considered to take into account eg the effect of the lattice of ions in asolid.
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Hubbard model in d = 2, H2
( ( ( ((
(
(
(
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Hubbard model in d = 2, H2
The position variable x lives on the lattice Z2, hence
ε(k) = cos k1 + cos k2
( ( ( ((
(
(
(
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Hubbard model in d = 2, H2
The position variable x lives on the lattice Z2, hence
ε(k) = cos k1 + cos k2
(! (" (# ($ % $ # " !(!
("
(#
($
%
$
#
"
!
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Hubbard model in d = 2, H2
The position variable x lives on the lattice Z2, hence
ε(k) = cos k1 + cos k2
(! (" (# ($ % $ # " !(!
("
(#
($
%
$
#
"
!
At µ = 0 the Fermi surface is a square of side size√
2π, joining the points(π, 0), (0, π) in the first Brillouin zone. The particle-hole symmetry makesthe Fermi surface invariant under RG flow.
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Matsubara Frequencies
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Matsubara Frequencies
The Matsubara frequencies are:
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Matsubara Frequencies
The Matsubara frequencies are:
k0 = ±2n + 1
β~π
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Matsubara Frequencies
The Matsubara frequencies are:
k0 = ±2n + 1
β~π
so the integral over k0 is really a discrete sum over n.
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Matsubara Frequencies
The Matsubara frequencies are:
k0 = ±2n + 1
β~π
so the integral over k0 is really a discrete sum over n.
For any n we have k0 6= 0, so that the denominator in C (k) can never be 0.This is why the temperature provides a natural infrared cut-off. But whenT → 0, k0 becomes a continuous variable and the propagator diverges onthe “space-time” Fermi surface, defined by k0 = 0 and ε(~k) = µ.
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Interaction
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Interaction
The electron-electron interaction is complicated; in solids it is mostly due tothe underlying lattice (phonons). To study the long range behavior of thesystem it can be efficiently modeled again by a local term:
SΛ = λ
∫
Λdd+1ξ (
∑
a∈↑,↓
ψaψa)2(ξ) .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Interaction
The electron-electron interaction is complicated; in solids it is mostly due tothe underlying lattice (phonons). To study the long range behavior of thesystem it can be efficiently modeled again by a local term:
SΛ = λ
∫
Λdd+1ξ (
∑
a∈↑,↓
ψaψa)2(ξ) .
Remark that this term is quite unique, as a, the spin index, takes only twovalues.
11
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Renormalization in Condensed Matter
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Renormalization in Condensed Matter
The general renormalization ideas (slice decomposition, locality, powercounting) and even constructive techniques have to be subtly adapted tothe case the electrons of condensed matter.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Scale decomposition
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Scale decomposition
The scales are again defined by slicing gently the propagator according toits spectrum. For instance:
C =∞
∑
j=1
Cj ; Cj(k) =fj(k)
ik0 − (ε(~k) − µ)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Scale decomposition
The scales are again defined by slicing gently the propagator according toits spectrum. For instance:
C =∞
∑
j=1
Cj ; Cj(k) =fj(k)
ik0 − (ε(~k) − µ)
where the slice function fj(k) effectively forces |ik0 − (ε(~k) − µ)| ∼ M−j ,
for some fixed parameter M > 1.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Slices around J2 Fermi Surface
14
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Slices around J2 Fermi Surface
These slices pinch more and more the Fermi surface as j → ∞.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Slices around J2 Fermi Surface
These slices pinch more and more the Fermi surface as j → ∞.
k
k
k
1-12
0
1=0
k2
1
k0=0
k1
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Slices around J2 Fermi Surface
These slices pinch more and more the Fermi surface as j → ∞.
k
k
k
1-12
0
1=0
k2
1
k0=0
k1
The multiscale analysis now relies on finding the subgraphs which haveinternal scales j lower than external scales (analog of an infrared QFTproblem), to compute an effective theory for degrees of freedom closer andcloser to the Fermi surface.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
Moving propagators around indeed leads to a non trivial phase factor:
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
Moving propagators around indeed leads to a non trivial phase factor:
Clow (y , ..) = Clow (x , ...)+ [ ~(x − y) ·~pF + ~(x − y) · (~∇−~pF )]Clow (x , ...)+ · · ·
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
Moving propagators around indeed leads to a non trivial phase factor:
Clow (y , ..) = Clow (x , ...)+ [ ~(x − y) ·~pF + ~(x − y) · (~∇−~pF )]Clow (x , ...)+ · · ·
However the most divergent part of eg the bubble graph
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
Moving propagators around indeed leads to a non trivial phase factor:
Clow (y , ..) = Clow (x , ...)+ [ ~(x − y) ·~pF + ~(x − y) · (~∇−~pF )]Clow (x , ...)+ · · ·
However the most divergent part of eg the bubble graph
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
Moving propagators around indeed leads to a non trivial phase factor:
Clow (y , ..) = Clow (x , ...)+ [ ~(x − y) ·~pF + ~(x − y) · (~∇−~pF )]Clow (x , ...)+ · · ·
However the most divergent part of eg the bubble graph
occurs when the two pairs of entering momenta at both ends approximatelyadds to zero,
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
Moving propagators around indeed leads to a non trivial phase factor:
Clow (y , ..) = Clow (x , ...)+ [ ~(x − y) ·~pF + ~(x − y) · (~∇−~pF )]Clow (x , ...)+ · · ·
However the most divergent part of eg the bubble graph
occurs when the two pairs of entering momenta at both ends approximatelyadds to zero,in which case the two factors, ~pF · ~(x − y) and −~pF · ~(x − y),approximately cancel each other!
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle
It is surprising and subtle that the locality principle still holds!
Moving propagators around indeed leads to a non trivial phase factor:
Clow (y , ..) = Clow (x , ...)+ [ ~(x − y) ·~pF + ~(x − y) · (~∇−~pF )]Clow (x , ...)+ · · ·
However the most divergent part of eg the bubble graph
occurs when the two pairs of entering momenta at both ends approximatelyadds to zero,in which case the two factors, ~pF · ~(x − y) and −~pF · ~(x − y),approximately cancel each other!
This is why the theory is still renormalizable.
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle, II
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle, II
More precisely, in momentum space
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle, II
More precisely, in momentum space
AG (p1, p2, p3, p4) =
∫
dk0d~k1
ik0 +~k2
2m− µ
1
iη(k0 + q0) + (~k+~q)2
2m− µ
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle, II
More precisely, in momentum space
AG (p1, p2, p3, p4) =
∫
dk0d~k1
ik0 +~k2
2m− µ
1
iη(k0 + q0) + (~k+~q)2
2m− µ
where q = p1 + p2 = −(p3 + p4) and η = ±1, depending on the arrows forthe lines (from ψ to ψ).
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle, II
More precisely, in momentum space
AG (p1, p2, p3, p4) =
∫
dk0d~k1
ik0 +~k2
2m− µ
1
iη(k0 + q0) + (~k+~q)2
2m− µ
where q = p1 + p2 = −(p3 + p4) and η = ±1, depending on the arrows forthe lines (from ψ to ψ).
The maximal value occurs for q = 0 and arrows in the same direction(η = −1). This algebraic combination corresponds to the so-called Cooperpairs.
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Locality Principle, II
More precisely, in momentum space
AG (p1, p2, p3, p4) =
∫
dk0d~k1
ik0 +~k2
2m− µ
1
iη(k0 + q0) + (~k+~q)2
2m− µ
where q = p1 + p2 = −(p3 + p4) and η = ±1, depending on the arrows forthe lines (from ψ to ψ).
The maximal value occurs for q = 0 and arrows in the same direction(η = −1). This algebraic combination corresponds to the so-called Cooperpairs.
In the case of an attractive BCS interaction the main effect is growth of thecoupling constant with j and a small change in the Fermi radius.
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Power Counting is Independent of Dimension
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Power Counting is Independent of Dimension
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Power Counting is Independent of Dimension
In fact power counting gives a just renormalizable theory in any dimension!
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Power Counting is Independent of Dimension
In fact power counting gives a just renormalizable theory in any dimension!
Since the slices are insensitive to the directions tangent to the Fermisphere, the theory has the power counting of a two dimensional model witha 1/|p| kind of propagator and a φ4 kind of interaction, hence is justrenormalizable.
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Power Counting is Independent of Dimension
In fact power counting gives a just renormalizable theory in any dimension!
Since the slices are insensitive to the directions tangent to the Fermisphere, the theory has the power counting of a two dimensional model witha 1/|p| kind of propagator and a φ4 kind of interaction, hence is justrenormalizable.
More precisely in momentum representation a vacuum graph with n verticeshas n + 1 loop lines, hence n + 1 loop integrals, each over a M−2j volume;and it has l = 2n propagators, each of which of size M j . The result isneutral in n, which indicates just renormalizability.
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The BCS phase transition
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The BCS phase transition
It occurs when the coupling constant become of order 1. The RG flow isgoverned by the bubble B(p) and leads to an effective coupling
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The BCS phase transition
It occurs when the coupling constant become of order 1. The RG flow isgoverned by the bubble B(p) and leads to an effective coupling
λeff = λbare/(1 − λbareB(p)).
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The BCS phase transition
It occurs when the coupling constant become of order 1. The RG flow isgoverned by the bubble B(p) and leads to an effective coupling
λeff = λbare/(1 − λbareB(p)).
When λB(0) = 1 we get divergence. Expanding
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The BCS phase transition
It occurs when the coupling constant become of order 1. The RG flow isgoverned by the bubble B(p) and leads to an effective coupling
λeff = λbare/(1 − λbareB(p)).
When λB(0) = 1 we get divergence. Expanding
λB(p) = 1 + p2λB”(0)...
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The BCS phase transition
It occurs when the coupling constant become of order 1. The RG flow isgoverned by the bubble B(p) and leads to an effective coupling
λeff = λbare/(1 − λbareB(p)).
When λB(0) = 1 we get divergence. Expanding
λB(p) = 1 + p2λB”(0)...
we get the Cooper pair boson propagator in 1/λB”(0)p2, which has apower counting depending of the dimension.
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
RG and Phase Transition
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
RG and Phase Transition
Expanding a theory around a given vacuum is a saddle pointapproximation. The quadratic part, or Hessian approximation, definesthe propagators of the particles; the rest is their interactions.
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
RG and Phase Transition
Expanding a theory around a given vacuum is a saddle pointapproximation. The quadratic part, or Hessian approximation, definesthe propagators of the particles; the rest is their interactions.
A phase transition can occur when the RG flow has moved the theoryout of the convergent region near the initial vacuum.
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
RG and Phase Transition
Expanding a theory around a given vacuum is a saddle pointapproximation. The quadratic part, or Hessian approximation, definesthe propagators of the particles; the rest is their interactions.
A phase transition can occur when the RG flow has moved the theoryout of the convergent region near the initial vacuum.
This means that another saddle point (or orbit) has to be computedwith a different Hessian hence usually completely different particlesand physics.
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Some lessons to remember
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Some lessons to remember
Nature seems to love renormalizable theories!
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Some lessons to remember
Nature seems to love renormalizable theories!
Particles and even the nature of their associated renormalization groupcan change completely across a phase transition!
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Fermi Liquid Problem
x
x
( ⁄y ( ⁄y
⁄
⁄
‘
¼ ¼
N’
’’ F
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Fermi Liquid Problem
Physics Textbooks Definition Says:
x
x
( ⁄y ( ⁄y
⁄
⁄
‘
¼ ¼
N’
’’ F
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Fermi Liquid Problem
Physics Textbooks Definition Says:
A Fermi liquid is an interacting system of Fermions whose density of statesat zero temperature, like in the free case, is discontinuous at a certainvalue. This value is called the interacting Fermi radius.
x
x
( ⁄y ( ⁄y
⁄
⁄
‘
¼ ¼
N’
’’ F
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Fermi Liquid Problem
Physics Textbooks Definition Says:
A Fermi liquid is an interacting system of Fermions whose density of statesat zero temperature, like in the free case, is discontinuous at a certainvalue. This value is called the interacting Fermi radius.
x
--.
x
( ⁄y ( ⁄y
⁄
⁄
‘
¼ ¼
N’
’’ F
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Fermi Liquid Problem
Physics Textbooks Definition Says:
A Fermi liquid is an interacting system of Fermions whose density of statesat zero temperature, like in the free case, is discontinuous at a certainvalue. This value is called the interacting Fermi radius.
x
--.
x
( ⁄y ( ⁄y
⁄
⁄
‘
¼ ¼
N’
’’ F
Problem: this discontinuity never really occurs for parity-invariant modelsbecause BCS or similar phase transition are generic at low temperatures(Kohn-Luttinger).
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
The Fermi Liquid Problem
Physics Textbooks Definition Says:
A Fermi liquid is an interacting system of Fermions whose density of statesat zero temperature, like in the free case, is discontinuous at a certainvalue. This value is called the interacting Fermi radius.
x
--.
x
( ⁄y ( ⁄y
⁄
⁄
‘
¼ ¼
N’
’’ F
Problem: this discontinuity never really occurs for parity-invariant modelsbecause BCS or similar phase transition are generic at low temperatures(Kohn-Luttinger).
“An interacting Fermi liquid maybe a figment of the imagination” (P.W.Anderson).21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Salmhofer’s CriterionSalmhofer proposed a useful mathematical criterion of Fermi liquidbehavior. Perhaps this is what the standard textbooks “had in mind”.
x
x
( ⁄y ( ⁄y
⁄
⁄
‘
¼ ¼
N’
’’ F
22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Salmhofer’s CriterionSalmhofer proposed a useful mathematical criterion of Fermi liquidbehavior. Perhaps this is what the standard textbooks “had in mind”.
x
x
(>?⁄y%@A& (>?⁄
y%@A&
/
⁄* <
⁄
‘
¼ ¼
N’
’’ F
22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Salmhofer’s CriterionSalmhofer proposed a useful mathematical criterion of Fermi liquidbehavior. Perhaps this is what the standard textbooks “had in mind”.
x
x
(>?⁄y%@A& (>?⁄
y%@A&
/
⁄* <
⁄
‘
¼ ¼
N’
’’ F
Salmhofer’s Criterion:
22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Salmhofer’s CriterionSalmhofer proposed a useful mathematical criterion of Fermi liquidbehavior. Perhaps this is what the standard textbooks “had in mind”.
x
x
(>?⁄y%@A& (>?⁄
y%@A&
/
⁄* <
⁄
‘
¼ ¼
N’
’’ F
Salmhofer’s Criterion:
The Schwinger functions are analytic in λ in a domain|λ| ≤ K/| log T |,
22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Salmhofer’s CriterionSalmhofer proposed a useful mathematical criterion of Fermi liquidbehavior. Perhaps this is what the standard textbooks “had in mind”.
x
x
(>?⁄y%@A& (>?⁄
y%@A&
/
⁄* <
⁄
‘
¼ ¼
N’
’’ F
Salmhofer’s Criterion:
The Schwinger functions are analytic in λ in a domain|λ| ≤ K/| log T |,The self-energy as function of the momentum is bounded uniformlytogether with its first and second derivatives in that domain.
22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Salmhofer’s CriterionSalmhofer proposed a useful mathematical criterion of Fermi liquidbehavior. Perhaps this is what the standard textbooks “had in mind”.
x
x
(>?⁄y%@A& (>?⁄
y%@A&
/
⁄* <
⁄
‘
¼ ¼
N’
’’ F
Salmhofer’s Criterion:
The Schwinger functions are analytic in λ in a domain|λ| ≤ K/| log T |,The self-energy as function of the momentum is bounded uniformlytogether with its first and second derivatives in that domain.
A key question after discovery of high Tc supraconductors was theproperties of 2 − d interacting Fermi liquids.22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy (Single Scale) Fermionic Model
23
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
23
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
Consider eg a Fermionic d-dimensional QFT in an infrared slice with N
colors. Suppose the propagator is diagonal in color space and satisfies thebound
|Cj ,ab(x , y)| ≤ δabM−dj/2
√N
e−M−j |x−y |
23
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
Consider eg a Fermionic d-dimensional QFT in an infrared slice with N
colors. Suppose the propagator is diagonal in color space and satisfies thebound
|Cj ,ab(x , y)| ≤ δabM−dj/2
√N
e−M−j |x−y |
We say that the interaction is of the vector type (or Gross-Neveu type) if itis of the form
V = λ
∫
ddx(
N∑
a=1
ψa(x)ψa(x))(
N∑
b=1
ψb(x)ψb(x))
23
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
Consider eg a Fermionic d-dimensional QFT in an infrared slice with N
colors. Suppose the propagator is diagonal in color space and satisfies thebound
|Cj ,ab(x , y)| ≤ δabM−dj/2
√N
e−M−j |x−y |
We say that the interaction is of the vector type (or Gross-Neveu type) if itis of the form
V = λ
∫
ddx(
N∑
a=1
ψa(x)ψa(x))(
N∑
b=1
ψb(x)ψb(x))
where λ is the coupling constant.23
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy Fermionic Model, II
We claim that
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy Fermionic Model, II
We claim that
The perturbation theory for the connected functions of this single slicemodel has a radius of convergence in λ which is uniform in j and N.
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Toy Fermionic Model, II
We claim that
The perturbation theory for the connected functions of this single slicemodel has a radius of convergence in λ which is uniform in j and N.
The J2 model is roughly similar to that model with the role of colorspayed by N = M j angular sectors around the Fermi surface.
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Tree Expansion
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
p = limΛ→∞
1
|Λ|(
∫
dµC (ψ, ψ)eSΛ(ψa,ψa))
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
p = limΛ→∞
1
|Λ|(
∫
dµC (ψ, ψ)eSΛ(ψa,ψa))
=∞
∑
n=0
(λn/n!)N
∑
a1,...,an,b1,...,bn=1
∑
T
∑
Ω
ε(T ,Ω)(
∏
ℓ∈T
∫ 1
0dwℓ
)
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
p = limΛ→∞
1
|Λ|(
∫
dµC (ψ, ψ)eSΛ(ψa,ψa))
=∞
∑
n=0
(λn/n!)N
∑
a1,...,an,b1,...,bn=1
∑
T
∑
Ω
ε(T ,Ω)(
∏
ℓ∈T
∫ 1
0dwℓ
)
∫
Rnd
dx1...dxnδ(x1 = 0)∏
ℓ∈T
(
Cj ,ab(xℓ, yℓ))
× det[Cj ,ab()]remaining
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
Cj(xk , ym) < fj ,k , gj ,m >L2
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
Cj(xk , ym) < fj ,k , gj ,m >L2
(this is realized through fj ,k = fj(xk , ·) and gj ,m = gj(·, ym) if
fj(p).gj(p) = Cj(p)).
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
Cj(xk , ym) < fj ,k , gj ,m >L2
(this is realized through fj ,k = fj(xk , ·) and gj ,m = gj(·, ym) if
fj(p).gj(p) = Cj(p)). Then we have the Gram inequality:
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
Cj(xk , ym) < fj ,k , gj ,m >L2
(this is realized through fj ,k = fj(xk , ·) and gj ,m = gj(·, ym) if
fj(p).gj(p) = Cj(p)). Then we have the Gram inequality:
|det[Cj ,ab()]remaining| ≤∏
anti−fields k
||fj ,k ||∏
fields m
||gj ,m||
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
Cj(xk , ym) < fj ,k , gj ,m >L2
(this is realized through fj ,k = fj(xk , ·) and gj ,m = gj(·, ym) if
fj(p).gj(p) = Cj(p)). Then we have the Gram inequality:
|det[Cj ,ab()]remaining| ≤∏
anti−fields k
||fj ,k ||∏
fields m
||gj ,m||
Proof: The w dependence disappears by extracting the symmetric squareroot v of the positive matrix xT so that xT
km =∑n
k=1 vTknv
Tnm.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in j for the Toy Model
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in j for the Toy Model
There is a factor M−dj/2 per line, or M−dj/4 per field ie entry of theloop determinant. This gives a factor M−dj per vertex
27
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in j for the Toy Model
There is a factor M−dj/2 per line, or M−dj/4 per field ie entry of theloop determinant. This gives a factor M−dj per vertex
There is a factor M+dj per vertex spatial integration (save one)
27
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in j for the Toy Model
There is a factor M−dj/2 per line, or M−dj/4 per field ie entry of theloop determinant. This gives a factor M−dj per vertex
There is a factor M+dj per vertex spatial integration (save one)
Hence the λ radius of convergence is uniform in j .
27
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in N for the Toy Model
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
There is a factor N per vertex (plus one)
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
There is a factor N per vertex (plus one)
The last item is not obvious to prove, because we don’t know all the graph,but only a tree To prove it we organize the sum over the colors from leavesto root of the tree. In this way the pay a factor N at each leaf to know thecolor index which does not go towards the root, then prune the leaf anditerate. The last vertex (the root) is the only special one as it costs two N
factors.
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
There is a factor N per vertex (plus one)
The last item is not obvious to prove, because we don’t know all the graph,but only a tree To prove it we organize the sum over the colors from leavesto root of the tree. In this way the pay a factor N at each leaf to know thecolor index which does not go towards the root, then prune the leaf anditerate. The last vertex (the root) is the only special one as it costs two N
factors.
Hence the λ radius of convergence is uniform in N.
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Model in a RG Slice
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
(using Gevrey cutoffs fj to get fractional exponential decay).
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
(using Gevrey cutoffs fj to get fractional exponential decay).
This is much worse than the factor M−3j/2 that would be needed.
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
(using Gevrey cutoffs fj to get fractional exponential decay).
This is much worse than the factor M−3j/2 that would be needed.
But the situation improves if we cut the Fermi slice into smaller pieces(called sectors).
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Sectors
30
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.
30
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
30
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
(using again Gevrey cutoffs fja for fractional power decay).
30
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
(using again Gevrey cutoffs fja for fractional power decay).But since N = M j
M−2j =M−3j/2
√N
30
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
(using again Gevrey cutoffs fja for fractional power decay).But since N = M j
M−2j =M−3j/2
√N
so that the bound is identical to that of the toy model.
30
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Momentum Conservation
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
31
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
Hence an approximate rhombus should be an approximate parallelogram.
31
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
Hence an approximate rhombus should be an approximate parallelogram.
Momentum conservation δ(p1 + p2 + p3 + p4) at each vertex follows fromtranslation invariance of J2. Hence p1, p2, p3, p4 form a quadrilateral. For j
large we have |pk | ≃√
2Mµ hence the quadrilateral is an approximaterhombus. Hence the four sectors to which p1, p2, p3 and p4 should beroughly equal two by two (parallelogram condition).
31
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
Hence an approximate rhombus should be an approximate parallelogram.
Momentum conservation δ(p1 + p2 + p3 + p4) at each vertex follows fromtranslation invariance of J2. Hence p1, p2, p3, p4 form a quadrilateral. For j
large we have |pk | ≃√
2Mµ hence the quadrilateral is an approximaterhombus. Hence the four sectors to which p1, p2, p3 and p4 should beroughly equal two by two (parallelogram condition).
It means that the interaction is roughly of the color (or Gross-Neveu) typewith respect to these angular sectors:
(
∑
a
ψaψa
)(
∑
b
ψbψb
)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Anisotropic Sectors
32
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
This is the source of painful technical complications (anisotropic angularsectors) which were developed by Feldman, Magnen, Trubowitz and myself.
32
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
This is the source of painful technical complications (anisotropic angularsectors) which were developed by Feldman, Magnen, Trubowitz and myself.
One should use in fact M j/2 longer sectors in the tangential direction (oflength M−j/2). The corresponding propagators have dual decay because thesectors are still aprroximately flat.
32
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
This is the source of painful technical complications (anisotropic angularsectors) which were developed by Feldman, Magnen, Trubowitz and myself.
One should use in fact M j/2 longer sectors in the tangential direction (oflength M−j/2). The corresponding propagators have dual decay because thesectors are still aprroximately flat.
Ultimately the conclusion is unchanged: the radius of convergence of J2 ina slice is independent of the slice index j .
32
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Results on 2d Interacting Fermi Liquid
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
In 2002 Benfatto, Giuliani and Mastropietro extended our analysis tothe case of non-rotation invariant curves close to the jellium case.
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
In 2002 Benfatto, Giuliani and Mastropietro extended our analysis tothe case of non-rotation invariant curves close to the jellium case.
Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
In 2002 Benfatto, Giuliani and Mastropietro extended our analysis tothe case of non-rotation invariant curves close to the jellium case.
Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.Following this road, Feldman, Knorrer and Trubowitz built in greatdetail a 2D Fermi liquids with non-parity invariant surfaces in animpressive series of papers completed around 2003-2004.
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Next Models
34
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Next Models
There are not many sectors for H2; it is not a Fermi liquid in the sense ofSalmhofer but a “Luttinger liquid” with logarithmic corrections.(Afchain-Magnen-R., 2004).
34
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Next Models
There are not many sectors for H2; it is not a Fermi liquid in the sense ofSalmhofer but a “Luttinger liquid” with logarithmic corrections.(Afchain-Magnen-R., 2004).
As the Hubbard filling factor moves from zero to half-filling, there is acrossover between Fermi and Luttinger behavior (Benfatto, Giuliani,Mastropietro)
34
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay
Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model
Next Models
There are not many sectors for H2; it is not a Fermi liquid in the sense ofSalmhofer but a “Luttinger liquid” with logarithmic corrections.(Afchain-Magnen-R., 2004).
As the Hubbard filling factor moves from zero to half-filling, there is acrossover between Fermi and Luttinger behavior (Benfatto, Giuliani,Mastropietro)
There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).
34
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Introduction to Group Field Theory
Vincent Rivasseau
LPT Orsay
Cetraro, Summer 2010, Lecture III
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
A scale decomposition
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
A scale decomposition
The locality principle
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
A scale decomposition
The locality principle
Power counting
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
A scale decomposition
The locality principle
Power counting
However a quantum theory of space-time and gravity may have to startwithout any fixed space time background metric. So what should be thecorresponding background-invariant notion of scale?
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
A scale decomposition
The locality principle
Power counting
However a quantum theory of space-time and gravity may have to startwithout any fixed space time background metric. So what should be thecorresponding background-invariant notion of scale?
We propose
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
A scale decomposition
The locality principle
Power counting
However a quantum theory of space-time and gravity may have to startwithout any fixed space time background metric. So what should be thecorresponding background-invariant notion of scale?
We propose
Definition: A scale is a slice of eigenvalues of a propagator (slicing gentlyaccording to a geometric progression).
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Scale
In quantum field theory (QFT) we said renormalization group relies on
A scale decomposition
The locality principle
Power counting
However a quantum theory of space-time and gravity may have to startwithout any fixed space time background metric. So what should be thecorresponding background-invariant notion of scale?
We propose
Definition: A scale is a slice of eigenvalues of a propagator (slicing gentlyaccording to a geometric progression).
So we turned the problem into that of finding the right propagator withnon-trivial spectrum for quantum gravity.
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Non Commutative Quantum Field Theory
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Non Commutative Quantum Field Theory
The Grosse-Wulkenhaar φ⋆4 model on the Moyal space R4 with1/(p2 + Ωx2) propagator is just renormalizable. But itsrenormalization relies on
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Non Commutative Quantum Field Theory
The Grosse-Wulkenhaar φ⋆4 model on the Moyal space R4 with1/(p2 + Ωx2) propagator is just renormalizable. But itsrenormalization relies on
A new scale decomposition (which mixes ultraviolet and infrared),
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Non Commutative Quantum Field Theory
The Grosse-Wulkenhaar φ⋆4 model on the Moyal space R4 with1/(p2 + Ωx2) propagator is just renormalizable. But itsrenormalization relies on
A new scale decomposition (which mixes ultraviolet and infrared),A new locality principle (Moyality),
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Non Commutative Quantum Field Theory
The Grosse-Wulkenhaar φ⋆4 model on the Moyal space R4 with1/(p2 + Ωx2) propagator is just renormalizable. But itsrenormalization relies on
A new scale decomposition (which mixes ultraviolet and infrared),A new locality principle (Moyality),A new power counting (under which only regular graphs diverge).
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Non Commutative Quantum Field Theory
The Grosse-Wulkenhaar φ⋆4 model on the Moyal space R4 with1/(p2 + Ωx2) propagator is just renormalizable. But itsrenormalization relies on
A new scale decomposition (which mixes ultraviolet and infrared),A new locality principle (Moyality),A new power counting (under which only regular graphs diverge).
A big unexpected bonus is the existence of a non trivial fixed pointwhich makes the GW model essentially the prime candidate for a fullysolvable 4D field theory.
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The new multiscale analysis
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The new multiscale analysis
The propagator 1/[p2 + Ω2(x)2] has parametric representation in x space
C (x , y) =θ
4Ω
( Ω
πθ
)
∫ ∞
0dα e−
µ20θ
4Ωα
1
(sinhα)2exp
(
−Ω
θ sinhα‖x − y‖2 −
Ω
θtanh
α
2(‖x‖2+‖y‖2)
)
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The new multiscale analysis
The propagator 1/[p2 + Ω2(x)2] has parametric representation in x space
C (x , y) =θ
4Ω
( Ω
πθ
)
∫ ∞
0dα e−
µ20θ
4Ωα
1
(sinhα)2exp
(
−Ω
θ sinhα‖x − y‖2 −
Ω
θtanh
α
2(‖x‖2+‖y‖2)
)
involving the Mehler kernel rather than the heat kernel.
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The new multiscale analysis
The propagator 1/[p2 + Ω2(x)2] has parametric representation in x space
C (x , y) =θ
4Ω
( Ω
πθ
)
∫ ∞
0dα e−
µ20θ
4Ωα
1
(sinhα)2exp
(
−Ω
θ sinhα‖x − y‖2 −
Ω
θtanh
α
2(‖x‖2+‖y‖2)
)
involving the Mehler kernel rather than the heat kernel.
We slice it according to the same method
C i (x , y) =
∫ M−2(i−1)
M−2i
dα · · · 6 KM2ie−c1M2i‖x−y‖2−c2M
−2i (‖x‖2+‖y‖2).
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The new multiscale analysis
The propagator 1/[p2 + Ω2(x)2] has parametric representation in x space
C (x , y) =θ
4Ω
( Ω
πθ
)
∫ ∞
0dα e−
µ20θ
4Ωα
1
(sinhα)2exp
(
−Ω
θ sinhα‖x − y‖2 −
Ω
θtanh
α
2(‖x‖2+‖y‖2)
)
involving the Mehler kernel rather than the heat kernel.
We slice it according to the same method
C i (x , y) =
∫ M−2(i−1)
M−2i
dα · · · 6 KM2ie−c1M2i‖x−y‖2−c2M
−2i (‖x‖2+‖y‖2).
The corresponding scales correspond to a mixture of the previous ultraviolet
and infrared notions.
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyal vertex
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyal vertex
The Moyal vertex in direct space is proportional to
∫ 4∏
i=1
d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(
2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))
.
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyal vertex
The Moyal vertex in direct space is proportional to
∫ 4∏
i=1
d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(
2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))
.
This vertex is non-local and oscillates. It has a parallelogram shape, and itsoscillation is proportional to its area:
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyal vertex
The Moyal vertex in direct space is proportional to
∫ 4∏
i=1
d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(
2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))
.
This vertex is non-local and oscillates. It has a parallelogram shape, and itsoscillation is proportional to its area:
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyal vertex
The Moyal vertex in direct space is proportional to
∫ 4∏
i=1
d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(
2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))
.
This vertex is non-local and oscillates. It has a parallelogram shape, and itsoscillation is proportional to its area:
5
![Page 394: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/394.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyal vertex
The Moyal vertex in direct space is proportional to
∫ 4∏
i=1
d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(
2ıθ−1(x1 ∧ x2 + x3 ∧ x4))
.
This vertex is non-local and oscillates. It has a parallelogram shape, and itsoscillation is proportional to its area:
5
![Page 395: Introduction to Renormalization - php.math.unifi.itphp.math.unifi.it/users/cimeCourses/2010/03/Talks/Rivasseau-slide.p… · connected structures. In fact trees, forests and jungles](https://reader033.vdocuments.site/reader033/viewer/2022050407/5f847c4e1336427e1c0af5ac/html5/thumbnails/395.jpg)
Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyality principle
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyality principle
It replaces locality:
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyality principle
It replaces locality:
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyality principle
It replaces locality:
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Moyality principle
It replaces locality:
This principle applies only to regular, high subgraphs.
6
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Noncommutative Renormalization
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Noncommutative Renormalization
Renormalizability of the GW model combines the three elements:
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Noncommutative Renormalization
Renormalizability of the GW model combines the three elements:
Power counting tells us that only regular graphs with two and four externallegs must be renormalized.
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Noncommutative Renormalization
Renormalizability of the GW model combines the three elements:
Power counting tells us that only regular graphs with two and four externallegs must be renormalized.
The Moyality principle says that such “high” regular graphs look like Moyalproducts. The corresponding counterterms are therefore of the form of theinitial theory.
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Noncommutative Renormalization
Renormalizability of the GW model combines the three elements:
Power counting tells us that only regular graphs with two and four externallegs must be renormalized.
The Moyality principle says that such “high” regular graphs look like Moyalproducts. The corresponding counterterms are therefore of the form of theinitial theory.
Again this means that the theory is renormalizable.
7
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Group Field Theory
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Group Field Theory
Group field theory (GFT) lies at the crossroads between
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Group Field Theory
Group field theory (GFT) lies at the crossroads between
The discretization of Cartan’s 1rst order formalism by loop quantumgravity
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Group Field Theory
Group field theory (GFT) lies at the crossroads between
The discretization of Cartan’s 1rst order formalism by loop quantumgravity
The higher dimensional tensorial extension of the matrix modelsrelevant for 2D quantum gravity
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Group Field Theory
Group field theory (GFT) lies at the crossroads between
The discretization of Cartan’s 1rst order formalism by loop quantumgravity
The higher dimensional tensorial extension of the matrix modelsrelevant for 2D quantum gravity
The “Regge” or simplicial approach to quantum gravity
8
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Advantages of Group Field Theory
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Advantages of Group Field Theory
There are several enticing factors about GFT:
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Advantages of Group Field Theory
There are several enticing factors about GFT:
GFT is an attempt to quantize completely space-time, summing at thesame time over metrics and topologies.
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Advantages of Group Field Theory
There are several enticing factors about GFT:
GFT is an attempt to quantize completely space-time, summing at thesame time over metrics and topologies.
GFT could provide a scenario for emergent large, manifold-like classicalspace-time as a condensed phase of more elementary space-timequanta.
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Advantages of Group Field Theory
There are several enticing factors about GFT:
GFT is an attempt to quantize completely space-time, summing at thesame time over metrics and topologies.
GFT could provide a scenario for emergent large, manifold-like classicalspace-time as a condensed phase of more elementary space-timequanta.
If that is the case, there is no reason that some GFT’s cannothopefully be renormalized through a suitably adapted multiscale RG.
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Advantages of Group Field Theory
There are several enticing factors about GFT:
GFT is an attempt to quantize completely space-time, summing at thesame time over metrics and topologies.
GFT could provide a scenario for emergent large, manifold-like classicalspace-time as a condensed phase of more elementary space-timequanta.
If that is the case, there is no reason that some GFT’s cannothopefully be renormalized through a suitably adapted multiscale RG.
The spin foams of loop gravity are exactly Feynman amplitudes ofgroup field theory.
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Objections to GFTThe main objections to GFT’s that I heard during the last two years are
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Objections to GFTThe main objections to GFT’s that I heard during the last two years are
1) The GFT’s are tensor generalizations of matrix models; but whereasmatrix models really triangulate 2D Riemann surfaces, GFT’striangulate much more singular objects (not even pseudo-manifoldswith local singularities).
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Objections to GFTThe main objections to GFT’s that I heard during the last two years are
1) The GFT’s are tensor generalizations of matrix models; but whereasmatrix models really triangulate 2D Riemann surfaces, GFT’striangulate much more singular objects (not even pseudo-manifoldswith local singularities).
2) The natural GFT’s action does not look positive. Hence thenon-perturbative meaning of the theory is unclear.
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Objections to GFTThe main objections to GFT’s that I heard during the last two years are
1) The GFT’s are tensor generalizations of matrix models; but whereasmatrix models really triangulate 2D Riemann surfaces, GFT’striangulate much more singular objects (not even pseudo-manifoldswith local singularities).
2) The natural GFT’s action does not look positive. Hence thenon-perturbative meaning of the theory is unclear.
3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time.
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Objections to GFTThe main objections to GFT’s that I heard during the last two years are
1) The GFT’s are tensor generalizations of matrix models; but whereasmatrix models really triangulate 2D Riemann surfaces, GFT’striangulate much more singular objects (not even pseudo-manifoldswith local singularities).
2) The natural GFT’s action does not look positive. Hence thenon-perturbative meaning of the theory is unclear.
3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time.
4) GFT’s have infinities and it is not clear whether these infinitiesshould or can be absorbed in a renormalization process;
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Objections to GFTThe main objections to GFT’s that I heard during the last two years are
1) The GFT’s are tensor generalizations of matrix models; but whereasmatrix models really triangulate 2D Riemann surfaces, GFT’striangulate much more singular objects (not even pseudo-manifoldswith local singularities).
2) The natural GFT’s action does not look positive. Hence thenon-perturbative meaning of the theory is unclear.
3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time.
4) GFT’s have infinities and it is not clear whether these infinitiesshould or can be absorbed in a renormalization process;
5) GFT’s dont predict space-time dimension, nor the standard model,hence are not a TOE.
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Answering the Objections
11
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Answering the Objections
1 and 2) Colored Field theory could be an answer.
11
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Answering the Objections
1 and 2) Colored Field theory could be an answer.
3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time. I believe quantum gravitymight be a theory with ultraviolet/infrared mixing. Experience withNCQFT may help.
11
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Answering the Objections
1 and 2) Colored Field theory could be an answer.
3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time. I believe quantum gravitymight be a theory with ultraviolet/infrared mixing. Experience withNCQFT may help.
4) GFT’s have infinities and it is not clear whether these infinitiesshould or can be absorbed in a renormalization process. We havestarted in Orsay a systematic study of this point.
11
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Answering the Objections
1 and 2) Colored Field theory could be an answer.
3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time. I believe quantum gravitymight be a theory with ultraviolet/infrared mixing. Experience withNCQFT may help.
4) GFT’s have infinities and it is not clear whether these infinitiesshould or can be absorbed in a renormalization process. We havestarted in Orsay a systematic study of this point.
5) GFT’s dont predict space-time dimension, nor the standard model,hence are not a TOE. Here I have nothing to say, except it is not clearto me that competition (string theory) is faring really so much betteron this point.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT
12
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT
A colored GFT (Gurau, arXiv:0907.2582). is a model with two types ofvertices, one with only incoming and the other with only outgoing fields.
12
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT
A colored GFT (Gurau, arXiv:0907.2582). is a model with two types ofvertices, one with only incoming and the other with only outgoing fields.
The D + 1 fields of the vertex each have a distinct color and the propagatorjoins only identical colors.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT
A colored GFT (Gurau, arXiv:0907.2582). is a model with two types ofvertices, one with only incoming and the other with only outgoing fields.
The D + 1 fields of the vertex each have a distinct color and the propagatorjoins only identical colors.
Faces are 2-D objects which are the connected bicolor components of thegraphs; their length is therefore even. Higher p-dimensional ”bubbles” arethe connected components with p colors. Therefore colored graphs are trueD-dimensional complexes, with a canonical D dimensional homology.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT may solve the first two objections
13
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT may solve the first two objections
Colored GFT’s amplitudes are dual to manifolds with only localisolated singularities.arXiv:1006.0714 Lost in Translation: Topological Singularities in Group Field Theory, Razvan Gurau
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT may solve the first two objections
Colored GFT’s amplitudes are dual to manifolds with only localisolated singularities.arXiv:1006.0714 Lost in Translation: Topological Singularities in Group Field Theory, Razvan Gurau
The Fermionic version of colored GFT seems more natural. This couldessentially also solve the stability problem.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT may solve the first two objections
Colored GFT’s amplitudes are dual to manifolds with only localisolated singularities.arXiv:1006.0714 Lost in Translation: Topological Singularities in Group Field Theory, Razvan Gurau
The Fermionic version of colored GFT seems more natural. This couldessentially also solve the stability problem.
A complete study of the power counting and renormalization is easierin colored GFT’s.
13
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT may solve the first two objections
Colored GFT’s amplitudes are dual to manifolds with only localisolated singularities.arXiv:1006.0714 Lost in Translation: Topological Singularities in Group Field Theory, Razvan Gurau
The Fermionic version of colored GFT seems more natural. This couldessentially also solve the stability problem.
A complete study of the power counting and renormalization is easierin colored GFT’s.
Fermionic D-dimensional colored GFT has a natural but puzzlingSU(D) symmetry. Hence it may be a more promising starting point toinclude standard model matter fields.
13
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Colored GFT may solve the first two objections
Colored GFT’s amplitudes are dual to manifolds with only localisolated singularities.arXiv:1006.0714 Lost in Translation: Topological Singularities in Group Field Theory, Razvan Gurau
The Fermionic version of colored GFT seems more natural. This couldessentially also solve the stability problem.
A complete study of the power counting and renormalization is easierin colored GFT’s.
Fermionic D-dimensional colored GFT has a natural but puzzlingSU(D) symmetry. Hence it may be a more promising starting point toinclude standard model matter fields.
13
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
GFT as theories of Holonomies
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
GFT as theories of Holonomies
Consider a manifold M with a connection Γ (eg Levi-Civita of a metric). Allinformation is encoded in the holonomies along closed curves γ in M.If X is the vector field solution of the parallel transport equation
dγν
dt∂νX
µ + Γµ
νσXνdγσ
dt= 0 , X(0) = X0
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
GFT as theories of Holonomies
Consider a manifold M with a connection Γ (eg Levi-Civita of a metric). Allinformation is encoded in the holonomies along closed curves γ in M.If X is the vector field solution of the parallel transport equation
dγν
dt∂νX
µ + Γµ
νσXνdγσ
dt= 0 , X(0) = X0
then X(T ) = gX0 for some g ∈ GL(TMγ(0)). g (independent of X0) is theholonomy along the curve γ.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
GFT as theories of Holonomies
Consider a manifold M with a connection Γ (eg Levi-Civita of a metric). Allinformation is encoded in the holonomies along closed curves γ in M.If X is the vector field solution of the parallel transport equation
dγν
dt∂νX
µ + Γµ
νσXνdγσ
dt= 0 , X(0) = X0
then X(T ) = gX0 for some g ∈ GL(TMγ(0)). g (independent of X0) is theholonomy along the curve γ.
Suppose we discretize M with flat n dimensional simplices. Their boundary(n − 1 dimensional) is also flat. The curvature is located at the “joints” ofthese blocks, that is at n − 2 dimensional cells.
Hence holonomies h are associated to blocks of codimension 2.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Discretized Surfaces and Matrix Models
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Discretized Surfaces and Matrix Models
Consider a two dimensional surface M , and fix a triangulation of M.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Discretized Surfaces and Matrix Models
Consider a two dimensional surface M , and fix a triangulation of M.
The holonomy group of a surface is G = U(1).
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Discretized Surfaces and Matrix Models
Consider a two dimensional surface M , and fix a triangulation of M.
The holonomy group of a surface is G = U(1). To all vertices (points) inour triangulation we associate a holonomy g , namely parallel transportalong any small curve encircling the vertex.
g
g’
g’’
..........
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Discretized Surfaces and Matrix Models
Consider a two dimensional surface M , and fix a triangulation of M.
The holonomy group of a surface is G = U(1). To all vertices (points) inour triangulation we associate a holonomy g , namely parallel transportalong any small curve encircling the vertex.
g
g’
g’’
..........
The surface and its metric are specified by the gluing of the triangles andby the holonomies g . The associated weight is a function F (g , g ′, . . . )
15
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Discretized Surfaces and Matrix Models
Consider a two dimensional surface M , and fix a triangulation of M.
The holonomy group of a surface is G = U(1). To all vertices (points) inour triangulation we associate a holonomy g , namely parallel transportalong any small curve encircling the vertex.
g
g’
g’’
..........
The surface and its metric are specified by the gluing of the triangles andby the holonomies g . The associated weight is a function F (g , g ′, . . . )
Quantization of geometry should sum both over metrics compatible with atriangulation and over all triangulations.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Dual Graph
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Dual GraphThe discrete information about the triangulation of the surface can beencoded in a ribbon (2-stranded) graph.
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Dual GraphThe discrete information about the triangulation of the surface can beencoded in a ribbon (2-stranded) graph.
The ribbon vertices of the graph are dual to triangles and the ribbon linesof the graph are dual to edges.
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Dual GraphThe discrete information about the triangulation of the surface can beencoded in a ribbon (2-stranded) graph.
The ribbon vertices of the graph are dual to triangles and the ribbon linesof the graph are dual to edges.
g1
g2
g3
g4
1
3
g
g
16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Dual GraphThe discrete information about the triangulation of the surface can beencoded in a ribbon (2-stranded) graph.
The ribbon vertices of the graph are dual to triangles and the ribbon linesof the graph are dual to edges.
g1
g2
g3
g4
1
3
g
g
In the dual graph the group elements g are associatedto the sides of the ribbons, also called strands. Wedistribute them on all ribbon vertices sharing the samestrand.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Dual GraphThe discrete information about the triangulation of the surface can beencoded in a ribbon (2-stranded) graph.
The ribbon vertices of the graph are dual to triangles and the ribbon linesof the graph are dual to edges.
g1
g2
g3
g4
1
3
g
g
In the dual graph the group elements g are associatedto the sides of the ribbons, also called strands. Wedistribute them on all ribbon vertices sharing the samestrand.
Suppose that the weight function F factors into contributions of dualvertices and dual lines:
F =∏
V
V (g1, g2, g3)∏
L
K (g1, g2)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Dual GraphThe discrete information about the triangulation of the surface can beencoded in a ribbon (2-stranded) graph.
The ribbon vertices of the graph are dual to triangles and the ribbon linesof the graph are dual to edges.
g1
g2
g3
g4
1
3
g
g
In the dual graph the group elements g are associatedto the sides of the ribbons, also called strands. Wedistribute them on all ribbon vertices sharing the samestrand.
Suppose that the weight function F factors into contributions of dualvertices and dual lines:
F =∏
V
V (g1, g2, g3)∏
L
K (g1, g2)
Then a 2-stranded graph is a ribbon Feynman graph and its weight is theintegrand of the associated Feynman amplitude.16
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
2D Group Field Theory
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
2D Group Field Theory
The associated matrix model action is (φ(g1, g2) = φ∗(g2, g1))
S =1
2
∫
G×G
φ(g1, g2)K−1(g1, g2)φ
∗(g1, g2)
+λ
∫
G×G×G
V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
2D Group Field Theory
The associated matrix model action is (φ(g1, g2) = φ∗(g2, g1))
S =1
2
∫
G×G
φ(g1, g2)K−1(g1, g2)φ
∗(g1, g2)
+λ
∫
G×G×G
V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,
and the complete correlation function is
< φ(g1, g2) . . . φ(g2n−1, g2n) >=
∫
[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
2D Group Field Theory
The associated matrix model action is (φ(g1, g2) = φ∗(g2, g1))
S =1
2
∫
G×G
φ(g1, g2)K−1(g1, g2)φ
∗(g1, g2)
+λ
∫
G×G×G
V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,
and the complete correlation function is
< φ(g1, g2) . . . φ(g2n−1, g2n) >=
∫
[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .
The insertions φ(g1, g2) fix a boundary triangulation and metric.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
2D Group Field Theory
The associated matrix model action is (φ(g1, g2) = φ∗(g2, g1))
S =1
2
∫
G×G
φ(g1, g2)K−1(g1, g2)φ
∗(g1, g2)
+λ
∫
G×G×G
V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,
and the complete correlation function is
< φ(g1, g2) . . . φ(g2n−1, g2n) >=
∫
[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .
The insertions φ(g1, g2) fix a boundary triangulation and metric.
Each Feynman graph fixes a bulk triangulation.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
2D Group Field Theory
The associated matrix model action is (φ(g1, g2) = φ∗(g2, g1))
S =1
2
∫
G×G
φ(g1, g2)K−1(g1, g2)φ
∗(g1, g2)
+λ
∫
G×G×G
V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,
and the complete correlation function is
< φ(g1, g2) . . . φ(g2n−1, g2n) >=
∫
[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .
The insertions φ(g1, g2) fix a boundary triangulation and metric.
Each Feynman graph fixes a bulk triangulation. The amplitude of a graph isthe sum over bulk metrics compatible with the triangulation.
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
2D Group Field Theory
The associated matrix model action is (φ(g1, g2) = φ∗(g2, g1))
S =1
2
∫
G×G
φ(g1, g2)K−1(g1, g2)φ
∗(g1, g2)
+λ
∫
G×G×G
V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,
and the complete correlation function is
< φ(g1, g2) . . . φ(g2n−1, g2n) >=
∫
[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .
The insertions φ(g1, g2) fix a boundary triangulation and metric.
Each Feynman graph fixes a bulk triangulation. The amplitude of a graph isthe sum over bulk metrics compatible with the triangulation. Thecorrelation function automatically performs both sums!
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Matrix Models
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Matrix Models
Take K ,V = 1 and develop φ in Fourier series
φ(g1, g2) =∑
Z×Z
eımg1e−ıng2φmn .
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Matrix Models
Take K ,V = 1 and develop φ in Fourier series
φ(g1, g2) =∑
Z×Z
eımg1e−ıng2φmn .
The action takes the more familiar form in Fourier space
S =1
2
∑
mn
φmnφ∗mn + λ
∑
mnk
φmnφnkφkm .
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Matrix Models
Take K ,V = 1 and develop φ in Fourier series
φ(g1, g2) =∑
Z×Z
eımg1e−ıng2φmn .
The action takes the more familiar form in Fourier space
S =1
2
∑
mn
φmnφ∗mn + λ
∑
mnk
φmnφnkφkm .
Summing all Feynman graphs generated by the matrix model action sumsover different topologies. This is 2D quantum gravity (David, Ginzparg).
18
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT
In three dimensions we must use the holonomy group G = SU(2).
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT
In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2).
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT
In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2). Eachfield φ is associated to a triangular face of a tetrahedron, therefore it hasthree arguments.
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT
In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2). Eachfield φ is associated to a triangular face of a tetrahedron, therefore it hasthree arguments. The propagator K is the inverse of the quadratic part
1
2
∫
G3
φ(ga, gb, gc)K−1(ga, gb, gc)φ
∗(ga, gb, gc)
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT
In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2). Eachfield φ is associated to a triangular face of a tetrahedron, therefore it hasthree arguments. The propagator K is the inverse of the quadratic part
1
2
∫
G3
φ(ga, gb, gc)K−1(ga, gb, gc)φ
∗(ga, gb, gc)
The vertex is dual to a tetrahedron. A tetrahedron is bounded by fourtriangles therefore the vertex is a φ4 term
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT
In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2). Eachfield φ is associated to a triangular face of a tetrahedron, therefore it hasthree arguments. The propagator K is the inverse of the quadratic part
1
2
∫
G3
φ(ga, gb, gc)K−1(ga, gb, gc)φ
∗(ga, gb, gc)
The vertex is dual to a tetrahedron. A tetrahedron is bounded by fourtriangles therefore the vertex is a φ4 term
λ
∫
G6
V (g , . . . , g)φ(g03, g02, g01)φ(g01, g13, g12)φ(g12, g02, g23)φ(g23, g13, g03)
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Vertex
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT VertexWhat we call the vertex in QFT usually obeys some kind of locality property.
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT VertexWhat we call the vertex in QFT usually obeys some kind of locality property.
Proposed Definition A vertex joining 2p strands is called simple if it hasfor kernel in direct group space a product of p delta functions matchingstrands two by two in different half-lines.
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT VertexWhat we call the vertex in QFT usually obeys some kind of locality property.
Proposed Definition A vertex joining 2p strands is called simple if it hasfor kernel in direct group space a product of p delta functions matchingstrands two by two in different half-lines.
Then the natural 3D GFT tetrahedron vertex in 3 dimensions is simple(with p = 6) as it writes
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT VertexWhat we call the vertex in QFT usually obeys some kind of locality property.
Proposed Definition A vertex joining 2p strands is called simple if it hasfor kernel in direct group space a product of p delta functions matchingstrands two by two in different half-lines.
Then the natural 3D GFT tetrahedron vertex in 3 dimensions is simple(with p = 6) as it writes
V [φ] = λ
∫
(
12∏
i=1
dgi
)
φ(g1, g2, g3)φ(g4, g5, g6)
φ(g7, g8, g9)φ(g10, g11, g12)Q(g1, ..g12),
with a kernel
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT VertexWhat we call the vertex in QFT usually obeys some kind of locality property.
Proposed Definition A vertex joining 2p strands is called simple if it hasfor kernel in direct group space a product of p delta functions matchingstrands two by two in different half-lines.
Then the natural 3D GFT tetrahedron vertex in 3 dimensions is simple(with p = 6) as it writes
V [φ] = λ
∫
(
12∏
i=1
dgi
)
φ(g1, g2, g3)φ(g4, g5, g6)
φ(g7, g8, g9)φ(g10, g11, g12)Q(g1, ..g12),
with a kernel
Q(g1, ..g12) = δ(g3g−14 )δ(g2g
−18 )δ(g6g
−17 )δ(g9g
−110 )δ(g5g
−111 )δ(g1g
−112 )
20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT VertexWhat we call the vertex in QFT usually obeys some kind of locality property.
Proposed Definition A vertex joining 2p strands is called simple if it hasfor kernel in direct group space a product of p delta functions matchingstrands two by two in different half-lines.
Then the natural 3D GFT tetrahedron vertex in 3 dimensions is simple(with p = 6) as it writes
V [φ] = λ
∫
(
12∏
i=1
dgi
)
φ(g1, g2, g3)φ(g4, g5, g6)
φ(g7, g8, g9)φ(g10, g11, g12)Q(g1, ..g12),
with a kernel
Q(g1, ..g12) = δ(g3g−14 )δ(g2g
−18 )δ(g6g
−17 )δ(g9g
−110 )δ(g5g
−111 )δ(g1g
−112 )
satisfying to our definition.20
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Vertex is dual of a tetrahedron
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Vertex is dual of a tetrahedron
The tetrahedron is dual to a vertex, its triangles are dual to halflines and itsedges are dual to strands. Hence we could label the 3D GFT vertex as
(123)
(023)
(013)
(012)
(02)
(23) (02) (03)
(03)
(13)
(01)
(01)(12)
(12)
(13)
(23)
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Vertex is dual of a tetrahedron
The tetrahedron is dual to a vertex, its triangles are dual to halflines and itsedges are dual to strands. Hence we could label the 3D GFT vertex as
(123)
(023)
(013)
(012)
(02)
(23) (02) (03)
(03)
(13)
(01)
(01)(12)
(12)
(13)
(23)
The GFT lines connect two vertices,thus are formed of three strands.
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Vertex is dual of a tetrahedron
The tetrahedron is dual to a vertex, its triangles are dual to halflines and itsedges are dual to strands. Hence we could label the 3D GFT vertex as
(123)
(023)
(013)
(012)
(02)
(23) (02) (03)
(03)
(13)
(01)
(01)(12)
(12)
(13)
(23)
The GFT lines connect two vertices,thus are formed of three strands.The graph built with such verticesand lines is a 3-stranded graph.
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Vertex is dual of a tetrahedron
The tetrahedron is dual to a vertex, its triangles are dual to halflines and itsedges are dual to strands. Hence we could label the 3D GFT vertex as
(123)
(023)
(013)
(012)
(02)
(23) (02) (03)
(03)
(13)
(01)
(01)(12)
(12)
(13)
(23)
The GFT lines connect two vertices,thus are formed of three strands.The graph built with such verticesand lines is a 3-stranded graph.
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Vertex is dual of a tetrahedron
The tetrahedron is dual to a vertex, its triangles are dual to halflines and itsedges are dual to strands. Hence we could label the 3D GFT vertex as
(123)
(023)
(013)
(012)
(02)
(23) (02) (03)
(03)
(13)
(01)
(01)(12)
(12)
(13)
(23)
The GFT lines connect two vertices,thus are formed of three strands.The graph built with such verticesand lines is a 3-stranded graph.
This corresponds to the kernel written before
Q(g1, ..g12) = δ(g3g−14 )δ(g2g
−18 )δ(g6g
−17 )δ(g9g
−110 )δ(g5g
−111 )δ(g1g
−112 )
21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
How to choose the 3D GFT propagator?
22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
How to choose the 3D GFT propagator?In Cartan’s formalism, 3D quantum gravity is expressed in terms of adreibein e
e i (x) = e ia(x)dxa
and a spin connection ω with values in the so(3) Lie algebra
ωi (x) = ωia(x)dxa
22
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
How to choose the 3D GFT propagator?In Cartan’s formalism, 3D quantum gravity is expressed in terms of adreibein e
e i (x) = e ia(x)dxa
and a spin connection ω with values in the so(3) Lie algebra
ωi (x) = ωia(x)dxa
The action is
S(e, ω) =
∫
e i ∧ F (ω)i
where F is the curvature of ω :
F (ω) = dω + ω ∧ ω
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3d Gravity
23
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3d Gravity
Varying ω gives Cartan’s equation De = 0. Varying e gives F = 0, hence aflat space-time.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3d Gravity
Varying ω gives Cartan’s equation De = 0. Varying e gives F = 0, hence aflat space-time.
Hence 3D gravity is a topological theory with only global observables, nopropagating degrees of freedom. This is confirmed by perturbation aroundflat space, which leads to the Chern-Simons theory, also topological.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3d Gravity
Varying ω gives Cartan’s equation De = 0. Varying e gives F = 0, hence aflat space-time.
Hence 3D gravity is a topological theory with only global observables, nopropagating degrees of freedom. This is confirmed by perturbation aroundflat space, which leads to the Chern-Simons theory, also topological.
Nevertheless the theory is physically interesting. Matter can be added;point particles do not curve space but induce an angular deficit proportionalto their mass. Therefore in 3D gravity there is a limit (2π) to anypoint-particle’s mass.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3d Gravity
Varying ω gives Cartan’s equation De = 0. Varying e gives F = 0, hence aflat space-time.
Hence 3D gravity is a topological theory with only global observables, nopropagating degrees of freedom. This is confirmed by perturbation aroundflat space, which leads to the Chern-Simons theory, also topological.
Nevertheless the theory is physically interesting. Matter can be added;point particles do not curve space but induce an angular deficit proportionalto their mass. Therefore in 3D gravity there is a limit (2π) to anypoint-particle’s mass.
The 3D GFT propagator should just implement this flatness condition.
23
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT Propagator
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT PropagatorThe following propagator
[Cφ](g1, g2, g3) =
∫
dhφ(hg1, hg2, hg3)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT PropagatorThe following propagator
[Cφ](g1, g2, g3) =
∫
dhφ(hg1, hg2, hg3)
is a projector onto gauge invariant fields φ(hg , hg ′, hg”) = φ(g , g ′, g”). Ithas kernel
C =
∫
dh δ(g1hg ′−11 ) δ(g2hg ′−1
2 ) δ(g3hg ′−13 )
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT PropagatorThe following propagator
[Cφ](g1, g2, g3) =
∫
dhφ(hg1, hg2, hg3)
is a projector onto gauge invariant fields φ(hg , hg ′, hg”) = φ(g , g ′, g”). Ithas kernel
C =
∫
dh δ(g1hg ′−11 ) δ(g2hg ′−1
2 ) δ(g3hg ′−13 )
Performing the integrations over all group elements of the vertices andkeeping the h elements unintegrated gives for each triangulation, or3-stranded graph G a Feynman amplitude
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT PropagatorThe following propagator
[Cφ](g1, g2, g3) =
∫
dhφ(hg1, hg2, hg3)
is a projector onto gauge invariant fields φ(hg , hg ′, hg”) = φ(g , g ′, g”). Ithas kernel
C =
∫
dh δ(g1hg ′−11 ) δ(g2hg ′−1
2 ) δ(g3hg ′−13 )
Performing the integrations over all group elements of the vertices andkeeping the h elements unintegrated gives for each triangulation, or3-stranded graph G a Feynman amplitude
ZG =
∫
∏
e
dhe
∏
f
δ(gf )
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT PropagatorThe following propagator
[Cφ](g1, g2, g3) =
∫
dhφ(hg1, hg2, hg3)
is a projector onto gauge invariant fields φ(hg , hg ′, hg”) = φ(g , g ′, g”). Ithas kernel
C =
∫
dh δ(g1hg ′−11 ) δ(g2hg ′−1
2 ) δ(g3hg ′−13 )
Performing the integrations over all group elements of the vertices andkeeping the h elements unintegrated gives for each triangulation, or3-stranded graph G a Feynman amplitude
ZG =
∫
∏
e
dhe
∏
f
δ(gf )
where gf = ~∏e∈f he is the holonomy along the face f .
24
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT = Boulatov Model
Therefore 3D GFT with the projector C as propagator implements exactlythe correct flatness conditions of 3D gravity!
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT = Boulatov Model
Therefore 3D GFT with the projector C as propagator implements exactlythe correct flatness conditions of 3D gravity!
This was the discovery of Boulatov (1992).
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
3D GFT = Boulatov Model
Therefore 3D GFT with the projector C as propagator implements exactlythe correct flatness conditions of 3D gravity!
This was the discovery of Boulatov (1992).
Fourier transforming the model, one gets Ponzano-Regge amplitudes.
25
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Properties of 3D GFT
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Properties of 3D GFT
This 3D GFT is topological and the amplitude of a graph changesthrough a global multiplicative factor under (1-4) or (2-3) Pachnermoves.
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Properties of 3D GFT
This 3D GFT is topological and the amplitude of a graph changesthrough a global multiplicative factor under (1-4) or (2-3) Pachnermoves.
Amplitudes may be infinite, for instance the tetraedron graph orcomplete graph K4 diverges as Λ3 at zero external data, if Λ is theultraviolet cutoff on the size of j ’s.
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Properties of 3D GFT
This 3D GFT is topological and the amplitude of a graph changesthrough a global multiplicative factor under (1-4) or (2-3) Pachnermoves.
Amplitudes may be infinite, for instance the tetraedron graph orcomplete graph K4 diverges as Λ3 at zero external data, if Λ is theultraviolet cutoff on the size of j ’s.
Regularization by going to a quantum group at a root of unity leads towell-defined topological invariants of the triangulation, namely theTuraev-Viro invariants.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Properties of 3D GFT
This 3D GFT is topological and the amplitude of a graph changesthrough a global multiplicative factor under (1-4) or (2-3) Pachnermoves.
Amplitudes may be infinite, for instance the tetraedron graph orcomplete graph K4 diverges as Λ3 at zero external data, if Λ is theultraviolet cutoff on the size of j ’s.
Regularization by going to a quantum group at a root of unity leads towell-defined topological invariants of the triangulation, namely theTuraev-Viro invariants.
However the theory is unsuited for a RG analysis, as the propagatorhas spectrum limited to 0 and 1. How to make slices with such aspectrum?
26
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The 4 Dimensional Case
27
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The 4 Dimensional Case
In the first order Cartan formalism, the action is∫
⋆[e ∧ e] ∧ F where againthe vierbein e and the spin connection ω are considered independentvariables.
27
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The 4 Dimensional Case
In the first order Cartan formalism, the action is∫
⋆[e ∧ e] ∧ F where againthe vierbein e and the spin connection ω are considered independentvariables.
Not all two-forms B in four dimenions are of the form e ∧ e. So BF theoryis not gravity in 4 dimension. The difference is expressed through a set ofconstraints classically called the Plebanski constraints.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The 4 Dimensional Case
In the first order Cartan formalism, the action is∫
⋆[e ∧ e] ∧ F where againthe vierbein e and the spin connection ω are considered independentvariables.
Not all two-forms B in four dimenions are of the form e ∧ e. So BF theoryis not gravity in 4 dimension. The difference is expressed through a set ofconstraints classically called the Plebanski constraints.
These constraints render 4D gravity much more complicated and interestingsince they are responsible for the local propagating degrees of freedom, thegravitational waves. (Constraints on B allow richer set of F ’s than justF = 0).
27
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT Vertex
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT Vertex
The vertex is the easy part as it should be again given by gluing rules forthe five tetraedra which join into pentachores:
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT Vertex
The vertex is the easy part as it should be again given by gluing rules forthe five tetraedra which join into pentachores:
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT Vertex
The vertex is the easy part as it should be again given by gluing rules forthe five tetraedra which join into pentachores:
The pentachore
28
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT
The corresponding graphs are made of vertices dual to the pentachores, offour-stranded edges dual to tetraedra, and of faces or closed thread circuitsdual to triangles.
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT
The corresponding graphs are made of vertices dual to the pentachores, offour-stranded edges dual to tetraedra, and of faces or closed thread circuitsdual to triangles.
If we keep the same propagator than in the Boulatov theory
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT
The corresponding graphs are made of vertices dual to the pentachores, offour-stranded edges dual to tetraedra, and of faces or closed thread circuitsdual to triangles.
If we keep the same propagator than in the Boulatov theory
[Cφ](g1, g2, g3, g4) =
∫
dhφ(hg1, hg2, hg3, hg4) (2.1)
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT
The corresponding graphs are made of vertices dual to the pentachores, offour-stranded edges dual to tetraedra, and of faces or closed thread circuitsdual to triangles.
If we keep the same propagator than in the Boulatov theory
[Cφ](g1, g2, g3, g4) =
∫
dhφ(hg1, hg2, hg3, hg4) (2.1)
we get a φ5 GFT with 4-stranded graphs called the Ooguri group fieldtheory. It is not 4D gravity, but a discretization of the 4D BF theory, ie thePlebanski constraints are missing.
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
4D GFT
The corresponding graphs are made of vertices dual to the pentachores, offour-stranded edges dual to tetraedra, and of faces or closed thread circuitsdual to triangles.
If we keep the same propagator than in the Boulatov theory
[Cφ](g1, g2, g3, g4) =
∫
dhφ(hg1, hg2, hg3, hg4) (2.1)
we get a φ5 GFT with 4-stranded graphs called the Ooguri group fieldtheory. It is not 4D gravity, but a discretization of the 4D BF theory, ie thePlebanski constraints are missing.
Again such a topological version of gravity is unsuited for RG analysis.
29
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Barrett-Crane proposal
30
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Barrett-Crane proposal
A first attempt to implement the Plebanski constraints in this language isdue to Barrett and Crane. They suggested one should restrict to simplerepresentations of SU(2) × SU(2), namely those satisfying j+ = j−.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Barrett-Crane proposal
A first attempt to implement the Plebanski constraints in this language isdue to Barrett and Crane. They suggested one should restrict to simplerepresentations of SU(2) × SU(2), namely those satisfying j+ = j−.
However this does not seem to work because the Barrett-Crane proposalimplements the Plebanski constraints too strongly. It does not seem inparticular to include correctly the angular degrees of freedom of thegraviton.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Holst Action
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Holst Action
Still another action classically equivalent to the Einstein-Hilbert action isthe Holst action:
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Holst Action
Still another action classically equivalent to the Einstein-Hilbert action isthe Holst action:
S = −1
8πG
∫
[⋆(e ∧ e) +1
γ(e ∧ e)] ∧ F
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Holst Action
Still another action classically equivalent to the Einstein-Hilbert action isthe Holst action:
S = −1
8πG
∫
[⋆(e ∧ e) +1
γ(e ∧ e)] ∧ F
where the first term is the Palatini action, and the second one, the Holstterm, is often called topological since it does not affect the equations ofmotion. The parameter γ is called the (Barbero)-Immirzi parameter, andplays an essential role in Loop quantum gravity.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The Holst Action
Still another action classically equivalent to the Einstein-Hilbert action isthe Holst action:
S = −1
8πG
∫
[⋆(e ∧ e) +1
γ(e ∧ e)] ∧ F
where the first term is the Palatini action, and the second one, the Holstterm, is often called topological since it does not affect the equations ofmotion. The parameter γ is called the (Barbero)-Immirzi parameter, andplays an essential role in Loop quantum gravity.
We can then rewrite this action a la Plebanski and get thePalatini-Holst-Plebanski functional integral, loosely written as:
dν =1
Ze− 1
8πG
R
[⋆B+ 1γB]∧F
δ(B = e ∧ e) DB Dω
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The EPR(LS)/FK theory
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.It seems to incoprorate better the angular degrees of freedom of thegraviton. It also reproduces the correct Einstein-Hilbert action in the limitof large spins (Barrett et al, Conrady-Freidel, 2009).
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.It seems to incoprorate better the angular degrees of freedom of thegraviton. It also reproduces the correct Einstein-Hilbert action in the limitof large spins (Barrett et al, Conrady-Freidel, 2009).These spin foams translate into an new proposal for a 4D GFT with a newimproved propagator which incorporates
32
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.It seems to incoprorate better the angular degrees of freedom of thegraviton. It also reproduces the correct Einstein-Hilbert action in the limitof large spins (Barrett et al, Conrady-Freidel, 2009).These spin foams translate into an new proposal for a 4D GFT with a newimproved propagator which incorporates
a SU(2) × SU(2) averaging C at both ends like in Ooguri theory
32
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.It seems to incoprorate better the angular degrees of freedom of thegraviton. It also reproduces the correct Einstein-Hilbert action in the limitof large spins (Barrett et al, Conrady-Freidel, 2009).These spin foams translate into an new proposal for a 4D GFT with a newimproved propagator which incorporates
a SU(2) × SU(2) averaging C at both ends like in Ooguri theory
a simplicity constraint which depends on the value of the Immirziparameter. For instance for 0 < γ < 1 it restricts representations to:
j+ =1 + γ
2j ; j− =
1 − γ
2j (2.2)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.It seems to incoprorate better the angular degrees of freedom of thegraviton. It also reproduces the correct Einstein-Hilbert action in the limitof large spins (Barrett et al, Conrady-Freidel, 2009).These spin foams translate into an new proposal for a 4D GFT with a newimproved propagator which incorporates
a SU(2) × SU(2) averaging C at both ends like in Ooguri theory
a simplicity constraint which depends on the value of the Immirziparameter. For instance for 0 < γ < 1 it restricts representations to:
j+ =1 + γ
2j ; j− =
1 − γ
2j (2.2)
a new projector S averaging on a single SU(2) residual gaugeinvariance in the middle of the propagator.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Renormalization group, at last?
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Renormalization group, at last?
The main new feature is that the full propagator can therefore be written asK = CSC with C 2 = C and S2 = S , hence C and S are two projectors, butthey do not commute, hence K has non-trivial spectrum!
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Renormalization group, at last?
The main new feature is that the full propagator can therefore be written asK = CSC with C 2 = C and S2 = S , hence C and S are two projectors, butthey do not commute, hence K has non-trivial spectrum!
This means that a RG analysis become possible! That’s our program.
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Renormalization group, at last?
The main new feature is that the full propagator can therefore be written asK = CSC with C 2 = C and S2 = S , hence C and S are two projectors, butthey do not commute, hence K has non-trivial spectrum!
This means that a RG analysis become possible! That’s our program.
In 2008 Perini, Rovelli and Speziale found in some natural normalization ofthe theory a Λ6 divergence for the graph G2
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Renormalization group, at last?
The main new feature is that the full propagator can therefore be written asK = CSC with C 2 = C and S2 = S , hence C and S are two projectors, butthey do not commute, hence K has non-trivial spectrum!
This means that a RG analysis become possible! That’s our program.
In 2008 Perini, Rovelli and Speziale found in some natural normalization ofthe theory a Λ6 divergence for the graph G2
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Renormalization group, at last?
The main new feature is that the full propagator can therefore be written asK = CSC with C 2 = C and S2 = S , hence C and S are two projectors, butthey do not commute, hence K has non-trivial spectrum!
This means that a RG analysis become possible! That’s our program.
In 2008 Perini, Rovelli and Speziale found in some natural normalization ofthe theory a Λ6 divergence for the graph G2
and a tantalizing logarithmic divergence for a radiative correction to thecoupling constant.
33
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far
34
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far
This section is based on joint work with collaborators J. Ben Geloun, T.Krajewski, Karim Noui, Jacques Magnen, Matteo Smerlak, A. Tanasa andP. Vitale.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far
This section is based on joint work with collaborators J. Ben Geloun, T.Krajewski, Karim Noui, Jacques Magnen, Matteo Smerlak, A. Tanasa andP. Vitale.
Our program aims at renormalizing group field theories, eg of theEPR(LS)/FK type; we have performed only some preliminary steps yet
34
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far
This section is based on joint work with collaborators J. Ben Geloun, T.Krajewski, Karim Noui, Jacques Magnen, Matteo Smerlak, A. Tanasa andP. Vitale.
Our program aims at renormalizing group field theories, eg of theEPR(LS)/FK type; we have performed only some preliminary steps yet
We have investigated the power counting of the Boulatov model, andfound the first uniform bounds for its Freidel-Louapre constructiveregularization, using the loop vertex expansion technique.arXiv:0906.5477, Scaling behaviour of three-dimensional group field theory (MNRS, Class. Quant. Gravity .26
(2009)185012)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, II
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, II
We have computed in the Abelian case the power counting of graphsfor BF theory, and found how to relate them to the number of bubblesof the graphs in the colored case.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, II
We have computed in the Abelian case the power counting of graphsfor BF theory, and found how to relate them to the number of bubblesof the graphs in the colored case.
The Abelian amplitudes lead to new class of topological polynomialsfor stranded graphs.
35
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, II
We have computed in the Abelian case the power counting of graphsfor BF theory, and found how to relate them to the number of bubblesof the graphs in the colored case.
The Abelian amplitudes lead to new class of topological polynomialsfor stranded graphs.arXiv:0911.1719 Bosonic Colored Group Field Theory (BMR)
arXiv:1002.3592, Linearized Group Field Theory and Power Counting Theorems (BKMR, Class. Quant. Grav. 27 (2010)
155012)
35
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, II
We have computed in the Abelian case the power counting of graphsfor BF theory, and found how to relate them to the number of bubblesof the graphs in the colored case.
The Abelian amplitudes lead to new class of topological polynomialsfor stranded graphs.arXiv:0911.1719 Bosonic Colored Group Field Theory (BMR)
arXiv:1002.3592, Linearized Group Field Theory and Power Counting Theorems (BKMR, Class. Quant. Grav. 27 (2010)
155012)
See also the related work of Bonzom and Smerlak
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, III
36
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, III
We have started the study of the general superficial degree ofdivergence of EPR(LS)/FK graphs through saddle point analysis usingcoherent states techniques. We recover in many cases the Abeliancounting, and find also corrections to it in general.
36
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, III
We have started the study of the general superficial degree ofdivergence of EPR(LS)/FK graphs through saddle point analysis usingcoherent states techniques. We recover in many cases the Abeliancounting, and find also corrections to it in general.
But saddle points may come in many varieties, degenerate or not. Thisrichness is interesting but a major challenge.
36
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, III
We have started the study of the general superficial degree ofdivergence of EPR(LS)/FK graphs through saddle point analysis usingcoherent states techniques. We recover in many cases the Abeliancounting, and find also corrections to it in general.
But saddle points may come in many varieties, degenerate or not. Thisrichness is interesting but a major challenge.
We recovered the Λ6 divergence of the EPR(LS)/FK graph G2 in thecase of non-degenerate saddle point configurations, but also a Λ9
divergence for the maximally degenerate saddle points. Our methodalso works at non zero external spins.
36
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, III
We have started the study of the general superficial degree ofdivergence of EPR(LS)/FK graphs through saddle point analysis usingcoherent states techniques. We recover in many cases the Abeliancounting, and find also corrections to it in general.
But saddle points may come in many varieties, degenerate or not. Thisrichness is interesting but a major challenge.
We recovered the Λ6 divergence of the EPR(LS)/FK graph G2 in thecase of non-degenerate saddle point configurations, but also a Λ9
divergence for the maximally degenerate saddle points. Our methodalso works at non zero external spins.
arXiv:1007. 3150 Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models (KMRTV)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, IV
37
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, IV
We have written still a more compact representation recently for theEPR/FK propagator in terms of traces
37
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, IV
We have written still a more compact representation recently for theEPR/FK propagator in terms of traces
We hope that raising eventually the EPR/FK propagator to the rightpower should lead to a just renormalizable model of 4D colored groupfield theory.
37
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What we have done so far, IV
We have written still a more compact representation recently for theEPR/FK propagator in terms of traces
We hope that raising eventually the EPR/FK propagator to the rightpower should lead to a just renormalizable model of 4D colored groupfield theory.
EPRL/FK Group Field Theory, arXiv:1008.0354 (BGR)
37
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What remains to be done in our program
Essentially everything!
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What remains to be done in our program
Essentially everything!
We need to investigate better the dominance of type 1 or ball-likegraphs, perhaps in the simpler colored BF models and establish at leastrough bounds on all other graphs showing they are smaller.
38
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What remains to be done in our program
Essentially everything!
We need to investigate better the dominance of type 1 or ball-likegraphs, perhaps in the simpler colored BF models and establish at leastrough bounds on all other graphs showing they are smaller.
We need to cut propagators with non trivial spectra like theEPR(LS)/FK into slices according to their sepctrum. This requires twocutoffs. This is required to identify the ”high” subgraphs, and performa true multiscale RG analysis. We have to find the new localityprinciple and the correct power counting. Only then can we find ifsome of these models are renormalizable or not.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What remains to be done in our program
39
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What remains to be done in our program
We need to find out whether symmetries such as the tantalizingtopological BF symmetry recovered at γ = 1 create fixed points of theRG analysis.
39
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
What remains to be done in our program
We need to find out whether symmetries such as the tantalizingtopological BF symmetry recovered at γ = 1 create fixed points of theRG analysis.
We need to get a better glimpse of whether a phase transition to acondensed phase can lead to emergence of a large smooth space-time.
39
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Conclusion
40
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Conclusion
EPR(LS)/FK group field theory is an interesting 4D group field theory.It has the right gluing rules plus a non-trivial propagator spectrum.
40
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Conclusion
EPR(LS)/FK group field theory is an interesting 4D group field theory.It has the right gluing rules plus a non-trivial propagator spectrum.
I feel the discovery of the noncommutative RG associated to the GWmodel, which is nonlocal and mixes ordinary scales, is an encouragingstep towards finding a similar renormalization group for group fieldtheories.
40
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Conclusion
EPR(LS)/FK group field theory is an interesting 4D group field theory.It has the right gluing rules plus a non-trivial propagator spectrum.
I feel the discovery of the noncommutative RG associated to the GWmodel, which is nonlocal and mixes ordinary scales, is an encouragingstep towards finding a similar renormalization group for group fieldtheories.
I hope black holes on one side, and possibly some mean fieldtransplanckian RG fixed point might bring constraints on GFT- basedscenarios for geometrogenesis and cosmology.
40
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay
GFT Group Field Theory
Conclusion
EPR(LS)/FK group field theory is an interesting 4D group field theory.It has the right gluing rules plus a non-trivial propagator spectrum.
I feel the discovery of the noncommutative RG associated to the GWmodel, which is nonlocal and mixes ordinary scales, is an encouragingstep towards finding a similar renormalization group for group fieldtheories.
I hope black holes on one side, and possibly some mean fieldtransplanckian RG fixed point might bring constraints on GFT- basedscenarios for geometrogenesis and cosmology.
However ”geometrogenesis” phase transition may be of a higher tensornature, hence harder to analyze than confinement, which is still notwell analytically understood. So computer studies may becomenecessary at some point.
40
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Hubbard Model, Three Dimensions
Vincent Rivasseau
LPT Orsay
Cetraro, Summer 2010, Lecture IV
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy (Single Scale) Fermionic Model
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
Consider eg a Fermionic d-dimensional QFT in an infrared slice with N
colors. Suppose the propagator is diagonal in color space and satisfies thebound
|Cj ,ab(x , y)| ≤ δabM−dj/2
√N
e−M−j |x−y |
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
Consider eg a Fermionic d-dimensional QFT in an infrared slice with N
colors. Suppose the propagator is diagonal in color space and satisfies thebound
|Cj ,ab(x , y)| ≤ δabM−dj/2
√N
e−M−j |x−y |
We say that the interaction is of the vector type (or Gross-Neveu type) if itis of the form
V = λ
∫
ddx(
N∑
a=1
ψa(x)ψa(x))(
N∑
b=1
ψb(x)ψb(x))
2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy (Single Scale) Fermionic Model
There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.
Consider eg a Fermionic d-dimensional QFT in an infrared slice with N
colors. Suppose the propagator is diagonal in color space and satisfies thebound
|Cj ,ab(x , y)| ≤ δabM−dj/2
√N
e−M−j |x−y |
We say that the interaction is of the vector type (or Gross-Neveu type) if itis of the form
V = λ
∫
ddx(
N∑
a=1
ψa(x)ψa(x))(
N∑
b=1
ψb(x)ψb(x))
where λ is the coupling constant.2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy Fermionic Model, II
We claim that
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy Fermionic Model, II
We claim that
The perturbation theory for the connected functions of this single slicemodel has a radius of convergence in λ which is uniform in j and N.
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Toy Fermionic Model, II
We claim that
The perturbation theory for the connected functions of this single slicemodel has a radius of convergence in λ which is uniform in j and N.
The J2 model is roughly similar to that model with the role of colorspayed by N = M j angular sectors around the Fermi surface.
3
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Fermionic Tree Expansion
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
p = limΛ→∞
1
|Λ|(
∫
dµC (ψ, ψ)eSΛ(ψa,ψa))
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
p = limΛ→∞
1
|Λ|(
∫
dµC (ψ, ψ)eSΛ(ψa,ψa))
=∞
∑
n=0
(λn/n!)N
∑
a1,...,an,b1,...,bn=1
∑
T
∑
Ω
ε(T ,Ω)(
∏
ℓ∈T
∫ 1
0dwℓ
)
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Fermionic Tree Expansion
Expanding the pressure
p = limΛ→∞
1
|Λ| log Z (Λ)
through the forest formula leads to
p = limΛ→∞
1
|Λ|(
∫
dµC (ψ, ψ)eSΛ(ψa,ψa))
=∞
∑
n=0
(λn/n!)N
∑
a1,...,an,b1,...,bn=1
∑
T
∑
Ω
ε(T ,Ω)(
∏
ℓ∈T
∫ 1
0dwℓ
)
∫
Rnd
dx1...dxnδ(x1 = 0)∏
ℓ∈T
(
Cj ,ab(xℓ, yℓ))
× det[Cj ,ab()]remaining
4
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant
[Cj ,ab()]remaining is the matrix with entries
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant
[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant
[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
5
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant
[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant
[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
Cj(xk , ym) < fj ,k , gj ,m >L2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant
[Cj ,ab()]remaining is the matrix with entries
xTkm(w) · Cj ,ab(xk , ym)
corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .
Suppose we have written
Cj(xk , ym) < fj ,k , gj ,m >L2
(this is realized through fj ,k = fj(xk , ·) and gj ,m = gj(·, ym) if
fj(p).gj(p) = Cj(p)).
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant, II
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant, II
Then we have the Gram inequality:
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant, II
Then we have the Gram inequality:
|det[Cj ,ab()]remaining| ≤∏
anti−fields k
||fj ,k ||∏
fields m
||gj ,m||
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Bounds on the Remaining Determinant, II
Then we have the Gram inequality:
|det[Cj ,ab()]remaining| ≤∏
anti−fields k
||fj ,k ||∏
fields m
||gj ,m||
Proof: The w dependence disappears by extracting the symmetric squareroot v of the positive matrix xT so that xT
km =∑n
k=1 vTknv
Tnm.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in j for the Toy Model
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in j for the Toy Model
There is a factor M−dj/2 per line, or M−dj/4 per field ie entry of theloop determinant. This gives a factor M−dj per vertex
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in j for the Toy Model
There is a factor M−dj/2 per line, or M−dj/4 per field ie entry of theloop determinant. This gives a factor M−dj per vertex
There is a factor M+dj per vertex spatial integration (save one)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in j for the Toy Model
There is a factor M−dj/2 per line, or M−dj/4 per field ie entry of theloop determinant. This gives a factor M−dj per vertex
There is a factor M+dj per vertex spatial integration (save one)
Hence the λ radius of convergence is uniform in j .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in N for the Toy Model
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
There is a factor N per vertex (plus one)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
There is a factor N per vertex (plus one)
The last item is not obvious to prove, because we don’t know all the graph,but only a tree To prove it we organize the sum over the colors from leavesto root of the tree. In this way the pay a factor N at each leaf to know thecolor index which does not go towards the root, then prune the leaf anditerate. The last vertex (the root) is the only special one as it costs two N
factors.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Uniform Radius in N for the Toy Model
There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex
There is a factor N per vertex (plus one)
The last item is not obvious to prove, because we don’t know all the graph,but only a tree To prove it we organize the sum over the colors from leavesto root of the tree. In this way the pay a factor N at each leaf to know thecolor index which does not go towards the root, then prune the leaf anditerate. The last vertex (the root) is the only special one as it costs two N
factors.
Hence the λ radius of convergence is uniform in N.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Model in a RG Slice
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
(using Gevrey cutoffs fj to get fractional exponential decay).
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
(using Gevrey cutoffs fj to get fractional exponential decay).
This is much worse than the factor M−3j/2 that would be needed.
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Model in a RG Slice
We claim that this model is roughly similar to the Toy Model, withdimension d = 3.
The naive estimate on the slice propagator is (using integration by parts)
|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2
(using Gevrey cutoffs fj to get fractional exponential decay).
This is much worse than the factor M−3j/2 that would be needed.
But the situation improves if we cut the Fermi slice into smaller pieces(called sectors).
9
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Sectors
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
(using again Gevrey cutoffs fja for fractional power decay).
10
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
(using again Gevrey cutoffs fja for fractional power decay).But since N = M j
M−2j =M−3j/2
√N
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
J2 Sectors
Suppose we divide the j-th slice into M j sectors, each of size roughly M−j
in all three directions.A sector propagator C j ,a has now prefactor M−2j and
|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2
(using again Gevrey cutoffs fja for fractional power decay).But since N = M j
M−2j =M−3j/2
√N
so that the bound is identical to that of the toy model.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
Hence an approximate rhombus should be an approximate parallelogram.
11
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
Hence an approximate rhombus should be an approximate parallelogram.
Momentum conservation δ(p1 + p2 + p3 + p4) at each vertex follows fromtranslation invariance of J2. Hence p1, p2, p3, p4 form a quadrilateral. For j
large we have |pk | ≃√
2Mµ hence the quadrilateral is an approximaterhombus. Hence the four sectors to which p1, p2, p3 and p4 should beroughly equal two by two (parallelogram condition).
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.
Hence an approximate rhombus should be an approximate parallelogram.
Momentum conservation δ(p1 + p2 + p3 + p4) at each vertex follows fromtranslation invariance of J2. Hence p1, p2, p3, p4 form a quadrilateral. For j
large we have |pk | ≃√
2Mµ hence the quadrilateral is an approximaterhombus. Hence the four sectors to which p1, p2, p3 and p4 should beroughly equal two by two (parallelogram condition).
It means that the interaction is roughly of the color (or Gross-Neveu) typewith respect to these angular sectors:
(
∑
a
ψaψa
)(
∑
b
ψbψb
)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Anisotropic Sectors
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
12
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
This is the source of painful technical complications (anisotropic angularsectors) which were developed by Feldman, Magnen, Trubowitz and myself.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
This is the source of painful technical complications (anisotropic angularsectors) which were developed by Feldman, Magnen, Trubowitz and myself.
One should use in fact M j/2 longer sectors in the tangential direction (oflength M−j/2). The corresponding propagators have dual decay because thesectors are still aprroximately flat.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Anisotropic Sectors
The rhombus rule is not fully correct for almost degenerate rhombuses.
This is the source of painful technical complications (anisotropic angularsectors) which were developed by Feldman, Magnen, Trubowitz and myself.
One should use in fact M j/2 longer sectors in the tangential direction (oflength M−j/2). The corresponding propagators have dual decay because thesectors are still aprroximately flat.
Ultimately the conclusion is unchanged: the radius of convergence of J2 ina slice is independent of the slice index j .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Anisotropic Sectors
In order for sectors defined in momentum space to correspond topropagators with dual decay in direct space, it is essential that their lengthin the tangential direction is not too big, otherwise the curvature is toostrong for the stationary phase method to apply.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Results on 2d Interacting Fermi Liquid
14
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
14
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
In 2002 Benfatto, Giuliani and Mastropietro extended our analysis tothe case of non-rotation invariant curves close to the jellium case.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
In 2002 Benfatto, Giuliani and Mastropietro extended our analysis tothe case of non-rotation invariant curves close to the jellium case.
Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.
14
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Results on 2d Interacting Fermi Liquid
This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).
In 2002 Benfatto, Giuliani and Mastropietro extended our analysis tothe case of non-rotation invariant curves close to the jellium case.
Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.Following this road, Feldman, Knorrer and Trubowitz built in greatdetail a 2D Fermi liquids with non-parity invariant surfaces in animpressive series of papers completed around 2003-2004.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The Hubbard model at half-filling in 2 dimensions
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The Hubbard model at half-filling in 2 dimensions
There are not many sectors for H2; it is not a Fermi liquid in the sense ofSalmhofer but a “Luttinger liquid” with logarithmic corrections.(Afchain-Magnen-R., 2004).
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The Hubbard model at half-filling in 2 dimensions
There are not many sectors for H2; it is not a Fermi liquid in the sense ofSalmhofer but a “Luttinger liquid” with logarithmic corrections.(Afchain-Magnen-R., 2004).
As the Hubbard filling factor moves from zero to half-filling, there is acrossover between Fermi and Luttinger behavior (Benfatto, Giuliani,Mastropietro)
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors
Consider the curvature radius of the curve (cos k1 + cos k2)2 = M−2j , which
is
R =(sin2 k1 + sin2 k2)
3/2
| cos k1 sin2 k2 + cos k2 sin2 k1|.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors
Consider the curvature radius of the curve (cos k1 + cos k2)2 = M−2j , which
is
R =(sin2 k1 + sin2 k2)
3/2
| cos k1 sin2 k2 + cos k2 sin2 k1|.
Consider also the distance d(k1) to the Fermi curve cos k1 + cos k2 = 0,and the width w(k1) of the band M−j ≤ | cos k1 + cos k2| ≤
√2M.M−j .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors
Consider the curvature radius of the curve (cos k1 + cos k2)2 = M−2j , which
is
R =(sin2 k1 + sin2 k2)
3/2
| cos k1 sin2 k2 + cos k2 sin2 k1|.
Consider also the distance d(k1) to the Fermi curve cos k1 + cos k2 = 0,and the width w(k1) of the band M−j ≤ | cos k1 + cos k2| ≤
√2M.M−j .
One finds
d(k1) ≃ w(k1) ≃M−j
M−j/2 + k1,
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors
Consider the curvature radius of the curve (cos k1 + cos k2)2 = M−2j , which
is
R =(sin2 k1 + sin2 k2)
3/2
| cos k1 sin2 k2 + cos k2 sin2 k1|.
Consider also the distance d(k1) to the Fermi curve cos k1 + cos k2 = 0,and the width w(k1) of the band M−j ≤ | cos k1 + cos k2| ≤
√2M.M−j .
One finds
d(k1) ≃ w(k1) ≃M−j
M−j/2 + k1,
R(k1) ≃k31 + M−3j/2
M−j,
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors, II
( ( ( ((
(
(
(
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors, IIRemember the condition that the sector length should not be bigger thanthat anisotropic length.
( ( ( ((
(
(
(
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors, IIRemember the condition that the sector length should not be bigger thanthat anisotropic length.
(! (" (# ($ % $ # " !(!
("
(#
($
%
$
#
"
!
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors, IIRemember the condition that the sector length should not be bigger thanthat anisotropic length.
(! (" (# ($ % $ # " !(!
("
(#
($
%
$
#
"
!
This leads to slice |k1| or |k2| according to a geometric progression from 1to M−j/2 to form the angular sectors in this model.
17
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors, IIRemember the condition that the sector length should not be bigger thanthat anisotropic length.
(! (" (# ($ % $ # " !(!
("
(#
($
%
$
#
"
!
This leads to slice |k1| or |k2| according to a geometric progression from 1to M−j/2 to form the angular sectors in this model.
Sincecos k1 + cos k2 = 2 cos(πk+/2) cos(πk−/2) .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
H2 Sectors, IIRemember the condition that the sector length should not be bigger thanthat anisotropic length.
(! (" (# ($ % $ # " !(!
("
(#
($
%
$
#
"
!
This leads to slice |k1| or |k2| according to a geometric progression from 1to M−j/2 to form the angular sectors in this model.
Sincecos k1 + cos k2 = 2 cos(πk+/2) cos(πk−/2) .
it is better to slice cos(πk±/2) with indices (s+, s−)
| cos(πk±/2)| ≃ M−s±
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation Rule, Power Counting
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation Rule, Power Counting
The two smallest indices among sj ,+ for j = 1, 2, 3, 4 differ by at most oneunit, and the two smallest indices among sj ,− for j = 1, 2, 3, 4 differ by atmost one unit.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Momentum Conservation Rule, Power Counting
The two smallest indices among sj ,+ for j = 1, 2, 3, 4 differ by at most oneunit, and the two smallest indices among sj ,− for j = 1, 2, 3, 4 differ by atmost one unit.
Because the total number of sectors is growing logarithmically, notpower-like, the scaling of the model resembles more the one-dimensionalmodel (Luttinger liqud) than to the jellium models in 2 or 3 dimensions.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).
Fix m ∈ Zd+1 . The number of 4-tuples S1, · · · S4 of sectors for whichthere exist ki ∈ R
d , i = 1, · · · , 4 satisfying
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).
Fix m ∈ Zd+1 . The number of 4-tuples S1, · · · S4 of sectors for whichthere exist ki ∈ R
d , i = 1, · · · , 4 satisfying
k ′i ∈ Si , |ki − k ′
i | ≤ const M−j , i = 1, · · · , 4
and
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).
Fix m ∈ Zd+1 . The number of 4-tuples S1, · · · S4 of sectors for whichthere exist ki ∈ R
d , i = 1, · · · , 4 satisfying
k ′i ∈ Si , |ki − k ′
i | ≤ const M−j , i = 1, · · · , 4
and|k1 + · · · + k4| ≤ const (1 + |m|) M−j
19
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).
Fix m ∈ Zd+1 . The number of 4-tuples S1, · · · S4 of sectors for whichthere exist ki ∈ R
d , i = 1, · · · , 4 satisfying
k ′i ∈ Si , |ki − k ′
i | ≤ const M−j , i = 1, · · · , 4
and|k1 + · · · + k4| ≤ const (1 + |m|) M−j
is bounded byconst(1 + |m|)dM(3d−4)j 1 + jδd ,2 .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).
Fix m ∈ Zd+1 . The number of 4-tuples S1, · · · S4 of sectors for whichthere exist ki ∈ R
d , i = 1, · · · , 4 satisfying
k ′i ∈ Si , |ki − k ′
i | ≤ const M−j , i = 1, · · · , 4
and|k1 + · · · + k4| ≤ const (1 + |m|) M−j
is bounded byconst(1 + |m|)dM(3d−4)j 1 + jδd ,2 .
Here, k ′ = k|k| denotes the projection of k onto the Fermi surface.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
So for d = 3 we find M5j 4-tuples which correspond to the choice of twosectors (M2j × M2j) and of one angular twist M j .
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
So for d = 3 we find M5j 4-tuples which correspond to the choice of twosectors (M2j × M2j) and of one angular twist M j .
Therefore the jellium model interaction is not of the vector type.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
The jellium model in 3 dimensions
So for d = 3 we find M5j 4-tuples which correspond to the choice of twosectors (M2j × M2j) and of one angular twist M j .
Therefore the jellium model interaction is not of the vector type.
The power counting corresponds to M−3j per sector propagator. Twopropagators pay for one vertex integration (M4j) and one sector choice(M2j) but there is nothing to pay for the angular twist.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Scaling in direct space
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Scaling in direct space
In direct space the problem is different.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Scaling in direct space
In direct space the problem is different.
The propagator in x-space satisfies a naive bound
Cj(x , y) ≤ M−jeM−j |x−y |
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Scaling in direct space
In direct space the problem is different.
The propagator in x-space satisfies a naive bound
Cj(x , y) ≤ M−jeM−j |x−y |
But integrating over angles leads to an additional 1/|x − y | decay, because
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Scaling in direct space
In direct space the problem is different.
The propagator in x-space satisfies a naive bound
Cj(x , y) ≤ M−jeM−j |x−y |
But integrating over angles leads to an additional 1/|x − y | decay, because
∫ π
0sin θdθdφe i cos θ|x−y | = sin |x − y |/|x − y |
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Scaling in direct space
In direct space the problem is different.
The propagator in x-space satisfies a naive bound
Cj(x , y) ≤ M−jeM−j |x−y |
But integrating over angles leads to an additional 1/|x − y | decay, because
∫ π
0sin θdθdφe i cos θ|x−y | = sin |x − y |/|x − y |
Hence for typical distances |x − y | ≃ M j the propagator obeys an improvedestimate
Cj(x , y) ≤ M−2jeM−j |x−y |
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Scaling in direct space
In direct space the problem is different.
The propagator in x-space satisfies a naive bound
Cj(x , y) ≤ M−jeM−j |x−y |
But integrating over angles leads to an additional 1/|x − y | decay, because
∫ π
0sin θdθdφe i cos θ|x−y | = sin |x − y |/|x − y |
Hence for typical distances |x − y | ≃ M j the propagator obeys an improvedestimate
Cj(x , y) ≤ M−2jeM−j |x−y |
The problem is that this bound is wrong at small distances.21
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Solution: Hadamard bound.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Solution: Hadamard bound.
Hadamard bound for a 2n by 2n determinant yields
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Solution: Hadamard bound.
Hadamard bound for a 2n by 2n determinant yields
|detaij | ≤∏
i
√
∑
j
|aij |2 ≤ (2n)n| supij
|aij |2n
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Solution: Hadamard bound.
Hadamard bound for a 2n by 2n determinant yields
|detaij | ≤∏
i
√
∑
j
|aij |2 ≤ (2n)n| supij
|aij |2n
This bound used after a cluster expansion between cubes solves theproblem for the main part of the propagator.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Solution: Hadamard bound.
Hadamard bound for a 2n by 2n determinant yields
|detaij | ≤∏
i
√
∑
j
|aij |2 ≤ (2n)n| supij
|aij |2n
This bound used after a cluster expansion between cubes solves theproblem for the main part of the propagator.
Correct just renormalizable power counting is recovered for the mainpart of the theory
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Solution: Hadamard bound.
Hadamard bound for a 2n by 2n determinant yields
|detaij | ≤∏
i
√
∑
j
|aij |2 ≤ (2n)n| supij
|aij |2n
This bound used after a cluster expansion between cubes solves theproblem for the main part of the propagator.
Correct just renormalizable power counting is recovered for the mainpart of the theory
A factor n! is lost in the bound at order n but this is what is allowedby the 1/n! symmetry factor.
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Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay
Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model
Solution: Hadamard bound.
Hadamard bound for a 2n by 2n determinant yields
|detaij | ≤∏
i
√
∑
j
|aij |2 ≤ (2n)n| supij
|aij |2n
This bound used after a cluster expansion between cubes solves theproblem for the main part of the propagator.
Correct just renormalizable power counting is recovered for the mainpart of the theory
A factor n! is lost in the bound at order n but this is what is allowedby the 1/n! symmetry factor.
An auxiliary (superrenormalizable) expansion is needed to treat the smalldistance part.
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