IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Lokalnie kowariantna kwantowa teoria pola jakopodejscie do kwantowej grawitacji
Katarzyna Rejzner
II. Institute for Theoretical Physics, Hamburg University
Kraków, 08.01.2010
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Outline of the talk
1 Introduction
2 Mathematical preliminariesCategory theory
3 Locally covariant quantum field theoryQFT on curved spacetimeLocal covariance
4 Quantum gravityConservative approachBackground independence
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Problems with quantum gravity
Spacetime is dynamical
"Points" lost their meaning
It is not clear what should be anobservable
"background independance"
Renormalizability
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Objectives
formulate a consistent theory that is valid in a given physicalsituation
answer some interpretational questions
find a relation to experiment
understand better problems of other approaches
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Objectives
formulate a consistent theory that is valid in a given physicalsituation
answer some interpretational questions
find a relation to experiment
understand better problems of other approaches
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Objectives
formulate a consistent theory that is valid in a given physicalsituation
answer some interpretational questions
find a relation to experiment
understand better problems of other approaches
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Objectives
formulate a consistent theory that is valid in a given physicalsituation
answer some interpretational questions
find a relation to experiment
understand better problems of other approaches
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Category
A category C consists ofa class Obj(C) of objects,a class hom(C) of morphisms between the objects. Eachmorphism f has a unique source object a and target object bwhere a, b ∈ Obj(C).Notation: if f : a→ b then we write f ∈ hom(a, b)
for a, b, c ∈ Obj(C), a binary operationhom(a, b)× hom(b, c)→ hom(a, c) called composition ofmorphisms. Notation: f ◦ g.
such that the following axioms hold:associativity if f : a→ b, g : b→ c and h : c→ d thenh ◦ (g ◦ f ) = (h ◦ g) ◦ fidentity for every object c, there exists a morphism idc : c→ c,such that for every hom(a, b) 3 f we have: idb ◦ f = f ◦ ida=f.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Category
A category C consists ofa class Obj(C) of objects,a class hom(C) of morphisms between the objects. Eachmorphism f has a unique source object a and target object bwhere a, b ∈ Obj(C).Notation: if f : a→ b then we write f ∈ hom(a, b)
for a, b, c ∈ Obj(C), a binary operationhom(a, b)× hom(b, c)→ hom(a, c) called composition ofmorphisms. Notation: f ◦ g.
such that the following axioms hold:associativity if f : a→ b, g : b→ c and h : c→ d thenh ◦ (g ◦ f ) = (h ◦ g) ◦ fidentity for every object c, there exists a morphism idc : c→ c,such that for every hom(a, b) 3 f we have: idb ◦ f = f ◦ ida=f.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Category
A category C consists ofa class Obj(C) of objects,a class hom(C) of morphisms between the objects. Eachmorphism f has a unique source object a and target object bwhere a, b ∈ Obj(C).Notation: if f : a→ b then we write f ∈ hom(a, b)
for a, b, c ∈ Obj(C), a binary operationhom(a, b)× hom(b, c)→ hom(a, c) called composition ofmorphisms. Notation: f ◦ g.
such that the following axioms hold:associativity if f : a→ b, g : b→ c and h : c→ d thenh ◦ (g ◦ f ) = (h ◦ g) ◦ fidentity for every object c, there exists a morphism idc : c→ c,such that for every hom(a, b) 3 f we have: idb ◦ f = f ◦ ida=f.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Category
A category C consists ofa class Obj(C) of objects,a class hom(C) of morphisms between the objects. Eachmorphism f has a unique source object a and target object bwhere a, b ∈ Obj(C).Notation: if f : a→ b then we write f ∈ hom(a, b)
for a, b, c ∈ Obj(C), a binary operationhom(a, b)× hom(b, c)→ hom(a, c) called composition ofmorphisms. Notation: f ◦ g.
such that the following axioms hold:associativity if f : a→ b, g : b→ c and h : c→ d thenh ◦ (g ◦ f ) = (h ◦ g) ◦ fidentity for every object c, there exists a morphism idc : c→ c,such that for every hom(a, b) 3 f we have: idb ◦ f = f ◦ ida=f.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Category
A category C consists ofa class Obj(C) of objects,a class hom(C) of morphisms between the objects. Eachmorphism f has a unique source object a and target object bwhere a, b ∈ Obj(C).Notation: if f : a→ b then we write f ∈ hom(a, b)
for a, b, c ∈ Obj(C), a binary operationhom(a, b)× hom(b, c)→ hom(a, c) called composition ofmorphisms. Notation: f ◦ g.
such that the following axioms hold:associativity if f : a→ b, g : b→ c and h : c→ d thenh ◦ (g ◦ f ) = (h ◦ g) ◦ fidentity for every object c, there exists a morphism idc : c→ c,such that for every hom(a, b) 3 f we have: idb ◦ f = f ◦ ida=f.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A covariant functor F from C to D is amapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b ∈, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (g) ◦F (f ) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A covariant functor F from C to D is amapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b ∈, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (g) ◦F (f ) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A covariant functor F from C to D is amapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b ∈, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (g) ◦F (f ) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A covariant functor F from C to D is amapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b ∈, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (g) ◦F (f ) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A contravariant functor F from C to D isa mapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (f ) ◦F (g) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A contravariant functor F from C to D isa mapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (f ) ◦F (g) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A contravariant functor F from C to D isa mapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (f ) ◦F (g) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Let C and D be categories. A contravariant functor F from C to D isa mapping that:
associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),
associates to each morphism hom(C) 3 f : a→ b, a morphismhom(D) 3 F (f ) : F (a)→ F (b)
such that the following two conditions hold:
F (idc) = idF (c) for every object c ∈ C.
F (g ◦ f ) = F (f ) ◦F (g) for all morphisms f : a→ b andg : b→ c.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Functor
Covariance:a
f−−−−→ b
F
y yF
F (a)F (f )−−−−→ F (b)
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Category theory
Natural transformation
Let F and G be functors between categories C and D, then a naturaltransformation η from F to G associates to every object a ∈ C amorphism hom(D) 3 ηa : F (a)→ G (a), such that for everymorphism C 3 f : a→ b we have:
ηb ◦F (f ) = G (f ) ◦ ηa
This equation can be expressed by the commutative diagram:
F (a)F (f )−−−−→ F (b)
ηa
y yηb
G (a)G (f )−−−−→ G (b)
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Algebraic formulation
QFT on Minkowski spacetime can be formalized with the use ofHaag-Kastler axioms. Main features of this framework:
Theory formulated in terms of nets ofC∗-algebras (algebras of observables)associated to subsets of Minkowskispacetime: A(O), O ∈ M.
Physical interpretation through states(functionals on observables’ algebras).
There is a special state, that respectssymmetries of M: vacuum state.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Algebraic formulation
QFT on Minkowski spacetime can be formalized with the use ofHaag-Kastler axioms. Main features of this framework:
Theory formulated in terms of nets ofC∗-algebras (algebras of observables)associated to subsets of Minkowskispacetime: A(O), O ∈ M.
Physical interpretation through states(functionals on observables’ algebras).
There is a special state, that respectssymmetries of M: vacuum state.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Algebraic formulation
QFT on Minkowski spacetime can be formalized with the use ofHaag-Kastler axioms. Main features of this framework:
Theory formulated in terms of nets ofC∗-algebras (algebras of observables)associated to subsets of Minkowskispacetime: A(O), O ∈ M.
Physical interpretation through states(functionals on observables’ algebras).
There is a special state, that respectssymmetries of M: vacuum state.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Algebraic formulation
QFT on Minkowski spacetime can be formalized with the use ofHaag-Kastler axioms. Main features of this framework:
Theory formulated in terms of nets ofC∗-algebras (algebras of observables)associated to subsets of Minkowskispacetime: A(O), O ∈ M.
Physical interpretation through states(functionals on observables’ algebras).
There is a special state, that respectssymmetries of M: vacuum state.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Algebraic formulation
QFT on Minkowski spacetime can be formalized with the use ofHaag-Kastler axioms. Main features of this framework:
Theory formulated in terms of nets ofC∗-algebras (algebras of observables)associated to subsets of Minkowskispacetime: A(O), O ∈ M.
Physical interpretation through states(functionals on observables’ algebras).
There is a special state, that respectssymmetries of M: vacuum state.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Algebraic formulation
QFT on Minkowski spacetime can be formalized with the use ofHaag-Kastler axioms. Main features of this framework:
Theory formulated in terms of nets ofC∗-algebras (algebras of observables)associated to subsets of Minkowskispacetime: A(O), O ∈ M.
Physical interpretation through states(functionals on observables’ algebras).
There is a special state, that respectssymmetries of M: vacuum state.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
QFT on curved spacetime
Important insights:
Dimock: Haag-Kastler axioms for globally hyperbolicspacetimes, covariance for isometric diffeomorphisms
Kay: Hadamard condition as a local characterization ofadmissible states
Radzikowski (followed by Köhler): Hadamard conditionformulated in terms of wave-front sets.
Methods of microlocal analysis applied to QFT on curvedspacetime (Fredenhagen, Brunetti, . . . )
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
QFT on curved spacetime
Important insights:
Dimock: Haag-Kastler axioms for globally hyperbolicspacetimes, covariance for isometric diffeomorphisms
Kay: Hadamard condition as a local characterization ofadmissible states
Radzikowski (followed by Köhler): Hadamard conditionformulated in terms of wave-front sets.
Methods of microlocal analysis applied to QFT on curvedspacetime (Fredenhagen, Brunetti, . . . )
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
QFT on curved spacetime
Important insights:
Dimock: Haag-Kastler axioms for globally hyperbolicspacetimes, covariance for isometric diffeomorphisms
Kay: Hadamard condition as a local characterization ofadmissible states
Radzikowski (followed by Köhler): Hadamard conditionformulated in terms of wave-front sets.
Methods of microlocal analysis applied to QFT on curvedspacetime (Fredenhagen, Brunetti, . . . )
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
QFT on curved spacetime
Important insights:
Dimock: Haag-Kastler axioms for globally hyperbolicspacetimes, covariance for isometric diffeomorphisms
Kay: Hadamard condition as a local characterization ofadmissible states
Radzikowski (followed by Köhler): Hadamard conditionformulated in terms of wave-front sets.
Methods of microlocal analysis applied to QFT on curvedspacetime (Fredenhagen, Brunetti, . . . )
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Local covariance
Ideas developed recently by: Brunetti-Fredenhagen-Verch,Hollands-Wald.
Application of category theory provides a formulation whichdoesn’t fix the background
One constructs the theory simultaneously on all spacetimes (of agiven class) in a coherent way
The theory is fixed by a covariant functor between certaincategories
Fields are natural transformations
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Local covariance
Ideas developed recently by: Brunetti-Fredenhagen-Verch,Hollands-Wald.
Application of category theory provides a formulation whichdoesn’t fix the background
One constructs the theory simultaneously on all spacetimes (of agiven class) in a coherent way
The theory is fixed by a covariant functor between certaincategories
Fields are natural transformations
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Local covariance
Ideas developed recently by: Brunetti-Fredenhagen-Verch,Hollands-Wald.
Application of category theory provides a formulation whichdoesn’t fix the background
One constructs the theory simultaneously on all spacetimes (of agiven class) in a coherent way
The theory is fixed by a covariant functor between certaincategories
Fields are natural transformations
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Local covariance
Ideas developed recently by: Brunetti-Fredenhagen-Verch,Hollands-Wald.
Application of category theory provides a formulation whichdoesn’t fix the background
One constructs the theory simultaneously on all spacetimes (of agiven class) in a coherent way
The theory is fixed by a covariant functor between certaincategories
Fields are natural transformations
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Local covariance
Ideas developed recently by: Brunetti-Fredenhagen-Verch,Hollands-Wald.
Application of category theory provides a formulation whichdoesn’t fix the background
One constructs the theory simultaneously on all spacetimes (of agiven class) in a coherent way
The theory is fixed by a covariant functor between certaincategories
Fields are natural transformations
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Categories
Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).
• Preserving orientation and time-orientation of the embeddedspacetime.
Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Locally covariant quantum field theory
A locally covariant quantum field theory is defined as a covariantfunctor A between Man and Alg. This means that the followingdiagram commutes for all morphismsψ ∈ homMan((M1, g1), (M2, g2)) and all objects of Man:
(M1, g)ψ−−−−→ (M2, g′)
A
y yA
A (M1, g)A (ψ)−−−−→ A (M2, g′)
Denoting αψ ≡ A (ψ), the covariance property reads:
αψ′ ◦ αψ = αψ′◦ψ , αidM = idA (M,g) ,
for all morphisms ψ′ from homMan((M2, g2), (M3, g3)),ψ ∈ homMan((M1, g1), (M2, g2)) and all objects of Man.
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IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Further axioms
One can also include two further axioms which are important in QFT:causality and time-slice axiom.
Causality: If there exist morphismsψj ∈ homMan((Mj, gj), (M, g)), j = 1, 2, such that the setsψ1(M1) and ψ2(M2) are causally separated in (M, g), then:
[αψ1(A (M1, g1)), αψ2(A (M2, g2))] = {0},
where [., .] is the commutator of given C∗ algebras.
Time-slice axiom: If the morphismψ ∈ homMan((M, g), (M′, g′)) is such that ψ(M) contains aCauchy-surface in (M′, g′), then αψ is an isomorphism.
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Further axioms
One can also include two further axioms which are important in QFT:causality and time-slice axiom.
Causality: If there exist morphismsψj ∈ homMan((Mj, gj), (M, g)), j = 1, 2, such that the setsψ1(M1) and ψ2(M2) are causally separated in (M, g), then:
[αψ1(A (M1, g1)), αψ2(A (M2, g2))] = {0},
where [., .] is the commutator of given C∗ algebras.
Time-slice axiom: If the morphismψ ∈ homMan((M, g), (M′, g′)) is such that ψ(M) contains aCauchy-surface in (M′, g′), then αψ is an isomorphism.
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IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
QFT on curved spacetimeLocal covariance
Fields
Let D be a functor that associates to each spacetimeM a space of testfunctions D(M) ∈ Obj(Test). A field Φ is defined as a naturaltransformation between D and A . To any object (M, g) ∈Man itassociates a morphism Φ(M,g) : D(M, g)→ A (M, g) in such a way,that for each given isometric embedding χ : (M1, g1) −→ (M2, g2)following diagram commutes
D(M1, g1)Φ(M1,g1)−−−−−→ A (M1, g1)
χ∗
y yαχD(M2, g2)
Φ(M2,g2)−−−−−→ A (M2, g2)
where χ∗ is the push forward under D . This means:
αχ ◦ Φ(M1,g1) = Φ(M2,g2) ◦ χ∗ .
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Perturbative quantum gravity
splitting a metric gµν into backgroundmetric ηµν and fluctuation hµν :
gµν = ηµν + hµν
The fluctuation metric is to be quantized
The renormalization scheme:Epstein-Glaser renormalization (valid infinite order).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Perturbative quantum gravity
splitting a metric gµν into backgroundmetric ηµν and fluctuation hµν :
gµν = ηµν + hµν
The fluctuation metric is to be quantized
The renormalization scheme:Epstein-Glaser renormalization (valid infinite order).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Perturbative quantum gravity
splitting a metric gµν into backgroundmetric ηµν and fluctuation hµν :
gµν = ηµν + hµν
The fluctuation metric is to be quantized
The renormalization scheme:Epstein-Glaser renormalization (valid infinite order).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Perturbative quantum gravity
splitting a metric gµν into backgroundmetric ηµν and fluctuation hµν :
gµν = ηµν + hµν
The fluctuation metric is to be quantized
The renormalization scheme:Epstein-Glaser renormalization (valid infinite order).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Perturbative quantum gravity
splitting a metric gµν into backgroundmetric ηµν and fluctuation hµν :
gµν = ηµν + hµν
The fluctuation metric is to be quantized
The renormalization scheme:Epstein-Glaser renormalization (valid infinite order).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Perturbative quantum gravity
splitting a metric gµν into backgroundmetric ηµν and fluctuation hµν :
gµν = ηµν + hµν
The fluctuation metric is to be quantized
The renormalization scheme:Epstein-Glaser renormalization (valid infinite order).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Relative Cauchy evolution
Let N+ and N− be two spacetimes thatembed into two other spacetimes M1 andM2 around Cauchy surfaces, via causalembeddings given by χk,±, k = 1, 2.
Then β = αχ1+α−1χ2+
αχ2−α−1χ1− is an
automorphism of A (M1).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Relative Cauchy evolution
Let N+ and N− be two spacetimes thatembed into two other spacetimes M1 andM2 around Cauchy surfaces, via causalembeddings given by χk,±, k = 1, 2.
Then β = αχ1+α−1χ2+
αχ2−α−1χ1− is an
automorphism of A (M1).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Relative Cauchy evolution
Let N+ and N− be two spacetimes thatembed into two other spacetimes M1 andM2 around Cauchy surfaces, via causalembeddings given by χk,±, k = 1, 2.
Then β = αχ1+α−1χ2+
αχ2−α−1χ1− is an
automorphism of A (M1).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Relative Cauchy evolution
Let N+ and N− be two spacetimes thatembed into two other spacetimes M1 andM2 around Cauchy surfaces, via causalembeddings given by χk,±, k = 1, 2.
Then β = αχ1+α−1χ2+
αχ2−α−1χ1− is an
automorphism of A (M1).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Relative Cauchy evolution
Let N+ and N− be two spacetimes thatembed into two other spacetimes M1 andM2 around Cauchy surfaces, via causalembeddings given by χk,±, k = 1, 2.
Then β = αχ1+α−1χ2+
αχ2−α−1χ1− is an
automorphism of A (M1).
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Background independence
Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces
Θµν(x).=
δβh
δhµν(x)|h=0 is a derivation
valued distribution which is covariantlyconserved
background independance condition:Θµν = 0
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Background independence
Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces
Θµν(x).=
δβh
δhµν(x)|h=0 is a derivation
valued distribution which is covariantlyconserved
background independance condition:Θµν = 0
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Background independence
Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces
Θµν(x).=
δβh
δhµν(x)|h=0 is a derivation
valued distribution which is covariantlyconserved
background independance condition:Θµν = 0
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Background independence
Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces
Θµν(x).=
δβh
δhµν(x)|h=0 is a derivation
valued distribution which is covariantlyconserved
background independance condition:Θµν = 0
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Background independence
Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces
Θµν(x).=
δβh
δhµν(x)|h=0 is a derivation
valued distribution which is covariantlyconserved
background independance condition:Θµν = 0
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Background independence
Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces
Θµν(x).=
δβh
δhµν(x)|h=0 is a derivation
valued distribution which is covariantlyconserved
background independance condition:Θµν = 0
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Background independence
Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces
Θµν(x).=
δβh
δhµν(x)|h=0 is a derivation
valued distribution which is covariantlyconserved
background independance condition:Θµν = 0
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Problems
Nonrenormalizability: In every order new counter terms. Onehas to show that theory can be applied when QG effects aresmall.
Constraints have to be imposed. Best developped withinperturbation theory: BRST
BRST cohomology has to be formulated for global quantities:Fields (natural transformations)
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Problems
Nonrenormalizability: In every order new counter terms. Onehas to show that theory can be applied when QG effects aresmall.
Constraints have to be imposed. Best developped withinperturbation theory: BRST
BRST cohomology has to be formulated for global quantities:Fields (natural transformations)
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conservative approachBackground independence
Problems
Nonrenormalizability: In every order new counter terms. Onehas to show that theory can be applied when QG effects aresmall.
Constraints have to be imposed. Best developped withinperturbation theory: BRST
BRST cohomology has to be formulated for global quantities:Fields (natural transformations)
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conclusions
Locally covariant quantum field theory turned out to besuccessful in describing QFT on CS.
New mathematical tools of LCQFT can be applied toperturbative quantum gravity.
Main mathematical difficulties: BRST cohomology,renormalization procedure.
Relations to other approaches: TQFT, perturbative QG, loops,strings . . .
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conclusions
Locally covariant quantum field theory turned out to besuccessful in describing QFT on CS.
New mathematical tools of LCQFT can be applied toperturbative quantum gravity.
Main mathematical difficulties: BRST cohomology,renormalization procedure.
Relations to other approaches: TQFT, perturbative QG, loops,strings . . .
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conclusions
Locally covariant quantum field theory turned out to besuccessful in describing QFT on CS.
New mathematical tools of LCQFT can be applied toperturbative quantum gravity.
Main mathematical difficulties: BRST cohomology,renormalization procedure.
Relations to other approaches: TQFT, perturbative QG, loops,strings . . .
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
IntroductionMathematical preliminaries
Locally covariant quantum field theoryQuantum gravity
Conclusions
Conclusions
Locally covariant quantum field theory turned out to besuccessful in describing QFT on CS.
New mathematical tools of LCQFT can be applied toperturbative quantum gravity.
Main mathematical difficulties: BRST cohomology,renormalization procedure.
Relations to other approaches: TQFT, perturbative QG, loops,strings . . .
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
Appendix References
References I
K. Fredenhagen, R. Brunetti, Towards a Background IndependentFormulation of Perturbative Quantum Gravity,arXiv:gr-qc/0603079v3
R. Brunetti, K. Fredenhagen, R. Verch, The generally covariantlocality principle - A new paradigm for local quantum fieldtheory, Commun. Math. Phys. 237 (2003) 31-68
Haag, R., Local Quantum Physics, 2nd ed. Springer-Verlag,Berlin, Heidelberg, New York, 1996
Radzikowski, M.J., Micro-local approach to the Hadamardcondition in quantum field theory in curved spacetime, Commun.Math. Phys. 179, 529 (1996)
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
Appendix References
References II
R.Brunetti, K.Fredenhagen, Microlocal analysis and interactingquantum field theories: Renormalization on physicalbackgrounds, Commun. Math. Phys. 208, 623 (2000)
Segal G., Two-dimensional conformal field theory and modularfunctors,Proc. IXth Intern. Congr. Math. Phys. (Bristol,Philadelphia). Eds. B.Simon, A.Truman and I.M.Davies, IOPPubl. Ltd, 1989, 22-37
Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola
Appendix References
Thank you for your attention
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