Geodesic
Krein Spaces in de Sitter Quantum Theories
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Experimental evidences and theoretical motivations lead to consider the curved space-time relativity based on the de Sitter group $SO_0(1,4)$ or $Sp(2,2)$ as an appealing substitute to the flat space-time Poincaré relativity. Quantum elementary systems are then associated to unitary irreducible representations of that simple Lie group. At the lowest limit of the discrete series lies a remarkable family of scalar representations involving Krein structures and related undecomposable representation cohomology which deserves to be thoroughly studied in view of quantization of the corresponding carrier fields. The purpose of this note is to present the mathematical material needed to examine the problem and to indicate possible extensions of an exemplary case, namely the so-called de Sitterian massless minimally coupled field, i.e. a scalar field in de Sitter space-time which does not couple to the Ricci curvature.
Keywords: de Sitter group; undecomposable representations; Krein spaces; Gupta–Bleuler triplet, cohomology of representations. 
@article{SIGMA_2010_6_a10,
     author = {Jean-Pierre Gazeau and Petr Siegl and Ahmed Youssef},
     title = {Krein {Spaces} in de {Sitter} {Quantum} {Theories}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2010},
     volume = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a10/}
}
TY  - JOUR
AU  - Jean-Pierre Gazeau
AU  - Petr Siegl
AU  - Ahmed Youssef
TI  - Krein Spaces in de Sitter Quantum Theories
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2010
VL  - 6
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a10/
LA  - en
ID  - SIGMA_2010_6_a10
ER  - 
%0 Journal Article
%A Jean-Pierre Gazeau
%A Petr Siegl
%A Ahmed Youssef
%T Krein Spaces in de Sitter Quantum Theories
%J Symmetry, integrability and geometry: methods and applications
%D 2010
%V 6
%U http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a10/
%G en
%F SIGMA_2010_6_a10
Jean-Pierre Gazeau; Petr Siegl; Ahmed Youssef. Krein Spaces in de Sitter Quantum Theories. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a10/

[1] Allen B., “Vacuum states in de Sitter space”, Phys. Rev. D, 32 (1985), 3136–3149 | DOI | MR

[2] Allen B., Folacci A., “Massless minimally coupled scalar field in de Sitter space”, Phys. Rev. D, 35 (1987), 3771–3778 | DOI | MR

[3] Bros J., Gazeau J.-P., Moschella U., “Quantum field theory in the de Sitter universe”, Phys. Rev. Lett., 73 (1994), 1746–1749 | DOI | MR | Zbl

[4] Bros J., Epstein H., Moschella U., “Analyticity properties and thermal effects for general quantum field theory on de Sitter space-time”, Comm. Math. Phys., 196 (1998), 535–570 ; gr-qc/9801099 | DOI | MR | Zbl

[5] Bros J., Moschella U., “Two-point functions and quantum fields in de Sitter universe”, Rev. Math. Phys., 8 (1996), 327–391 ; gr-qc/9511019 | DOI | MR | Zbl

[6] Brunetti R., Guido D., Longo R., “Modular localization and Wigner particles”, Rev. Math. Phys., 14 (2002), 759–785 ; math-ph/0203021 | DOI | MR | Zbl

[7] Caldwell R., Kamionkowski M., “The physics of cosmic acceleration”, Ann. Rev. Nucl. Part. Sci., 59 (2009), 397–429 ; arXiv:0903.0866 | DOI

[8] Chernikov N. A., Tagirov E. A., “Quantum theory of scalar fields in de Sitter space-time”, Ann. Inst. H. Poincaré Sect. A (N.S.), 9 (1968), 109–141 | MR | Zbl

[9] De Bièvre S., Renaud J., “The massless quantum field on the 1+1-dimensional de Sitter space”, Phys. Rev. D, 57 (1998), 6230–6241 | DOI | MR

[10] Dixmier J., “Représentations intégrables du groupe de De Sitter”, Bull. Soc. Math. France, 89 (1961), 9–41 | MR | Zbl

[11] Fulling S. A., Aspects of quantum field theory in curved spacetime, London Mathematical Society Student Texts, 17, Cambridge University Press, Cambridge, 1989 | MR | Zbl

[12] Garidi T., Huguet E., Renaud J., “de Sitter waves and the zero curvature limit”, Phys. Rev. D, 67 (2003), 124028, 5 pp., ages ; gr-qc/0304031 | DOI | MR

[13] Gazeau J.-P., “An introduction to quantum field theory in de Sitter space-time”, Cosmology and Gravitation: XIIth Brazilian School of Cosmology and Gravitation, AIP Conf. Proc., 910, Amer. Inst. Phys., Melville, NY, 2007, 218–269 | MR | Zbl

[14] Gazeau J.-P., Novello M., “The question of mass in (anti-) de Sitter spacetimes”, J. Phys. A: Math. Theor., 41 (2008), 304008, 14 pp., ages | DOI | MR | Zbl

[15] Gazeau J.-P., Renaud J., Takook M. V., “Gupta–Bleuler quantization for minimally coupled scalar fields in de Sitter space”, Classical Quantum Gravity, 17 (2000), 1415–1434 ; gr-qc/9904023 | DOI | MR | Zbl

[16] Gazeau J.-P., Youssef A., “A discrete nonetheless remarkable brick in de Sitter: the “massless minimally coupled field””, Proceedings of the XXVII$^{\rm th}$ International Colloquium on Group Theoretical Methods in Physics (Yerevan, 2008) (to appear) , Phys. Atomic Nuclei

[17] Hua L. K., Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963 | MR | Zbl

[18] Isham C. J., “Quantum field theory in curved space-times, a general mathematical framework”, Differential Geometrical Methods in Mathematical Physics II (Proc. Conf., Univ. Bonn, Bonn, 1977), Lecture Notes in Math., 676, eds. K. Bleuler et al., Springer, Berlin, 1978, 459–512 | MR

[19] Jakobczyk L., Strocchi F., “Krein structures for Wightman and Schwinger functions”, J. Math. Phys., 29 (1988), 1231–1235 | DOI | MR | Zbl

[20] Kirsten K., Garriga J., “Massless minimally coupled fields in de Sitter space: $O(4)$-symmetric states versus de Sitter-invariant vacuum”, Phys. Rev. D, 48 (1993), 567–577 ; gr-qc/9305013 | DOI | MR

[21] Linder E. V., “Resource letter DEAU-1: dark energy and the accelerating universe”, Amer. J. Phys., 76 (2008), 197–204 ; arXiv:0705.4102 | DOI

[22] Magnus W., Oberhettinger F., Soni R. P., Formulas and theorems for the special functions of mathematical physics, Springer-Verlag, New York, 1966 | MR

[23] Mickelsson J., Niederle J., “Contractions of representations of de Sitter groups”, Comm. Math. Phys., 27 (1972), 167–180 | DOI | MR | Zbl

[24] Morchio G., Pierotti D., Strocchi F., “Infrared and vacuum structure in two-dimensional local quantum field theory models. The massless scalar field”, J. Math. Phys., 31 (1990), 1467–1477 | DOI | MR | Zbl

[25] Newton T. D., “A note on the representations of the de Sitter group”, Ann. of Math. (2), 51 (1950), 730–733 | DOI | MR | Zbl

[26] Newton T. D., Wigner E. P., “Localized states for elementary systems”, Rev. Modern Phys., 21 (1949), 400–406 | DOI | Zbl

[27] Pinczon G., Simon J., “Extensions of representations and cohomology”, Rep. Math. Phys., 16 (1979), 49–77 | DOI | MR | Zbl

[28] Schmidt H. J., “On the de Sitter space-time – the geometric foundation of inflationary cosmology”, Fortschr. Phys., 41:3 (1993), 179–199 | MR

[29] Takahashi B., “Sur les représentations unitaires des groupes de Lorentz généralisés”, Bull. Soc. Math. France, 91 (1963), 289–433 | MR | Zbl

[30] Thomas L. H., “On unitary representations of the group of de Sitter space”, Ann. of Math. (2), 42 (1941), 113–126 | DOI | MR

[31] Wald R. M., Quantum fields theory in curved spacetime and black hole thermodynamics Chicago Lectures in Physics, University of Chicago Press, Chicago, IL, 1994 | MR