igor salom and Đorđe Šijački
DESCRIPTION
Generalization of the Gell-Mann decontraction formula for sl(n,R) and its applications in affine gravity. Igor Salom and Đorđe Šijački. Generalization of the Gell-Mann formula for sl(n,R) and applications in affine gravity - Talk outline -. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/1.jpg)
Generalization of the Gell-Mann decontraction formula
for sl(n,R) and its applications in affine gravity
Igor Salom and Đorđe Šijački
![Page 2: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/2.jpg)
Generalization of the Gell-Mann formula for sl(n,R) and applications in affine gravity
- Talk outline -
• sl(n,R) algebra in theory of gravity – what is specific?
• What is the Gell-Mann decontraction formula an why is it important in this context?
• Validity domain and need for generalization• Generalization of the Gell-Mann formula• Illustration: application of the formula in affine
theory of gravity
![Page 3: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/3.jpg)
sl(n,R) algebra in gravity and HEP
• Affine models of gravity in n space-time dimensions (gauging Rn Λ GL(n,R) symmetry)
• “World spinors” in n space-time dimensions
• Algebra of M-theory is often extended to r528 Λ gl(32,R)
• Systems with conserved n-dimensional volume (strings, pD-branes...)
• Effective QCD in terms of Regge trajectories
![Page 4: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/4.jpg)
In these context we need to know how to represent SL(n,R) generators…
…in some simple, “easy to use” form if possible,
…in SO(n) (or SO(1,n-1)) subgroup basis,
…for infinite-dimensional unitary representations,
…and, in particular, for infinite-dimensional spinorial representations: SL(n,R) is double cover of SL(n,R)!
![Page 5: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/5.jpg)
How to find SL(n,R) generators?
• Induction from parabolic subgroups• Construct generators as differential operators in
the space of group parameters• Analytical continuation of complexified SU(n)
representations• ...• Using the Gell-Mann decontraction formula
![Page 6: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/6.jpg)
Now, what is the Gell-Mann decontraction formula?
Loosely speaking: it is formula inverse to the Inönü-Wigner contraction.
![Page 7: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/7.jpg)
The Gell-Mann decontraction formula
Gell-Mann formula
(as named by R. Hermann)
Gell-Mann formula?
Inönü-Wigner contraction
![Page 8: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/8.jpg)
Example: Poincare to de Sitter
• Define function of Poincare generators:
• Check:
• …unfortunately, this works so nicely only for so(m,n) cases. Not for sl(n,R).
![Page 9: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/9.jpg)
SL(n,R) group
• Definition: group of unimodular n x n real matrices (with matrix multiplication)
• Algebra relations:so(n)
irrep. of traceless symmetric matrices
![Page 10: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/10.jpg)
Rn(n+1)/2-1 Λ Spin(n)SL(n,R)
Representations of this group are easy to find
Inönü-Wigner contraction of SL(n,R) Find representations of the contracted semidirect
product and apply Gell-Mann formula to get sl(n,R) representations.
![Page 11: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/11.jpg)
Space of square integrable functions over Spin(n) manifold
• Space of square integrable functions is rich enough to contain representatives from all equivalence classes of irreps. of both SL(n,R) and Tn(n+1)/2-1 Λ Spin(n) groups (Haris Chandra).
• As a basis we choose Wigner D functions:
k indices label SL(n,R) SО(n)
multiplicity
![Page 12: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/12.jpg)
Contracted algebra representations
• Contracted abelian operators U represent as multiplicative Wigner D functions:
• Action of spin(n) subalgebra is “natural” one:
Matrix elements are simply products of Spin(n) CG
coefficients
![Page 13: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/13.jpg)
Try to use Gell-Mann formula
• Take and plug it in the Gell-Mann formula, i.e.:
and then check commutation relations.• works only in spaces over SO(n)/(SO(p)×SO(q)), q+p=n • no spinorial representations here• no representations with multiplicity w.r.t. Spin(n) → Insufficient for most of physical applications!(“Conditions for Validity of the Gell-Mann Formula in the
Case of sl(n,R) and/or su(n) Algebras”, Igor Salom and Djordje Šijački, in Lie theory and its applications in physics, American Institute of Physics Conference Proceedings, 1243 (2010) 191-198.)
![Page 14: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/14.jpg)
• All irreducible representations of SL(3,R) and SL(4,R) are known – Dj. Šijački found using different approach
• Matrix elements of SL(3,R) representations with multiplicity indicate an expression of the form:
• This is a correct, “generalized” formula!• Similarly in SL(4,R) case.
Learning from the solved cases
!Additional label,
overall 2, matching the group rank!
![Page 15: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/15.jpg)
Spin(n) left action generators
![Page 16: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/16.jpg)
Generalized formula in SL(5,R) case
new terms
Not easy even to check that this is correct (i.e. closes algebra relations).
4 labels, matching the group rank.
Generalization of the Gell-Mann formula for sl(5,R) and su(5) algebras, Igor Salom and Djordje Šijački, International Journal of Geometric Methods in Modern Physics, 7 (2010) 455-470.
![Page 17: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/17.jpg)
Can we find the generalized formula for arbitrary n?
• Idea: rewrite all generalized formulas (n=3,4,5) in Cartesian coordinates.
• All formulas fit into a general expression, now valid for arbitratry n:
• Using a D-functions identity:
direct calculation shows that the expression satisfies algebra relations.
Overall n-1 parameters, matching the group rank! They
determine Casimir values.
![Page 18: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/18.jpg)
• Matrix elements:
• All required properties met: simple expression in Spin(n) basis valid for arbitrary representation (including infinite
dimensional ones, and spinorial ones, and with nontrivial multiplicity)!
Matrix elements for arbitrary SL(n,R) irreducible representation
![Page 19: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/19.jpg)
Collateral result for su(n)
• Multiplying shear generators T → iT turns algebra into su(n)
• All results applicable to su(n): su(n) matrices in so(n) basis – a nontrivial result.
![Page 20: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/20.jpg)
• A generic affine theory Lagrangian in n space-time dimensions :
• A symmetry breaking mechanism is required.
Application – affine theory of gravity
What kind of fields are these?
![Page 21: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/21.jpg)
sl(n,R) matrix elements appear in vertices
![Page 22: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/22.jpg)
Example: n=5, multiplicity free
• Vector component of infinite-component bosonic multifield, transforming as a multiplicity free SL(5,R) representation labelled by
• Similarly for the term:
![Page 23: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/23.jpg)
Example: n=5, nontrivial multiplicity
• Due to multiplicity, there are , a priori, 5 different 5-dimensional vector components, i.e. Lorentz subfields, of the infinite-component bosonic multifield – one vector field for each valid combination of left indices k.
From the form of the generalized Gell-Mann formula we deduce
that all component can not belong to the same irreducible
representation
![Page 24: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/24.jpg)
Example: n=5, nontrivial multiplicity
• Sheer connection transforms these fields one into another. Interaction terms are:
![Page 25: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/25.jpg)
Conclusion
• Not much use in gravity for the original Gell-Mann formula (for sl(n,R) case), but we generalized it, in the case of arbitrary n. New formula is of a simple form and applicable to all irreducible representations.
• If you ever need expressions for SL(n,R) or SU(n) generators in SO(n) basis, you can find them in Generalization of the Gell-Mann decontraction formula for sl(n,R) and su(n) algebras, Igor Salom and Djordje Šijački, International Journal of Geometric Methods in Modern Physics, 8 (2011), 395-410.
![Page 26: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/26.jpg)
![Page 27: Igor Salom and Đorđe Šijački](https://reader036.vdocuments.site/reader036/viewer/2022062500/56815b13550346895dc8bf0b/html5/thumbnails/27.jpg)
Relation to the max weight labels
• Labels are weights of max weight vector :
Using:
one obtains: