Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 060, 22 pages      arXiv:1708.00538

Wavepackets on de Sitter Spacetime

João C.A. Barata a and Marcos Brum ab
a) Instituto de Física, Universidade de São Paulo, Rua do Matão 1371, São Paulo, 05508-090, Brasil
b) Departamento de Matemática, Universidade Federal do Rio de Janeiro, Campus Duque de Caxias, Rodovia Washington Luiz Km 104,5, Duque de Caxias, 25265-970, Brazil

Received April 16, 2019, in final form August 12, 2019; Published online August 15, 2019

We construct wavepackets on de Sitter spacetime, with masses consistently defined from the eigenvalues of an irreducible representation of a Casimir element in the universal enveloping algebra of the Lorentz algebra and analyse their asymptotic behaviour. Furthermore, we show that, in the limit as the de Sitter radius tends to infinity, the wavepackets tend to the wavepackets of Minkowski spacetime and the plane waves arising after contraction have support sharply located on the mass shell.

Key words: quantum field theory on de Sitter spacetime; Haag-Ruelle scattering theory; theory of group representations; algebraic quantum field theory.

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  1. Akhmedov E.T., Lecture notes on interacting quantum fields in de Sitter space, Internat. J. Modern Phys. D 23 (2014), 1430001, 61 pages, arXiv:1309.2557.
  2. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  3. Angelopoulos E., Flato M., Fronsdal C., Sternheimer D., Massless particles, conformal group, and de Sitter universe, Phys. Rev. D 23 (1981), 1278-1289.
  4. Araki H., Haag R., Collision cross sections in terms of local observables, Comm. Math. Phys. 4 (1967), 77-91.
  5. Bargmann V., Irreducible unitary representations of the Lorentz group, Ann. of Math. 48 (1947), 568-640.
  6. Barut A.O., Böhm A., Reduction of a class of ${\rm O}(4,2)$ representations with respect to ${\rm SO}(4,1)$ and ${\rm SO}(3,2)$, J. Math. Phys. 11 (1970), 2938-2945.
  7. Bros J., Epstein H., Gaudin M., Moschella U., Pasquier V., Triangular invariants, three-point functions and particle stability on the de Sitter universe, Comm. Math. Phys. 295 (2010), 261-288, arXiv:0901.4223.
  8. 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, arXiv:gr-qc/9801099.
  9. Bros J., Epstein H., Moschella U., The lifetime of a massive particle in a de Sitter universe, J. Cosmol. Astropart. Phys. 2008 (2008), no. 2, 003, 8 pages, arXiv:hep-th/0612184.
  10. Bros J., Epstein H., Moschella U., Particle decays and stability on the de Sitter universe, Ann. Henri Poincaré 11 (2010), 611-658, arXiv:0812.3513.
  11. Bros J., Gazeau J.P., Moschella U., Quantum field theory in the de Sitter universe, Phys. Rev. Lett. 73 (1994), 1746-1749.
  12. Bros J., Moschella U., Two-point functions and quantum fields in de Sitter universe, Rev. Math. Phys. 8 (1996), 327-391, arXiv:gr-qc/9511019.
  13. Bros J., Moschella U., Fourier analysis and holomorphic decomposition on the one-sheeted hyperboloid, in Géométrie complexe. II. Aspects contemporains dans les mathématiques et la physique, Hermann Éd. Sci. Arts, Paris, 2004, 27-58, arXiv:math-ph/0311052.
  14. Dixmier J., Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9-41.
  15. Dobrev V.K., Mack G., Petkova V.B., Petrova S.G., Todorov I.T., Harmonic analysis on the $n$-dimensional Lorentz group and its application to conformal quantum field theory, Lecture Notes in Phys., Vol. 63, Springer-Verlag, Berlin - Heidelberg - New York, 1977.
  16. Dooley A.H., Rice J.W., On contractions of semisimple Lie groups, Trans. Amer. Math. Soc. 289 (1985), 185-202.
  17. Dybalski W., From Faddeev-Kulish to LSZ. Towards a non-perturbative description of colliding electrons, Nuclear Phys. B 925 (2017), 455-469, arXiv:1706.09057.
  18. Dybalski W., Gérard C., A criterion for asymptotic completeness in local relativistic QFT, Comm. Math. Phys. 332 (2014), 1167-1202, arXiv:1308.5187.
  19. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York, 1954, available at
  20. Folland G.B., A course in abstract harmonic analysis, 2nd ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2016.
  21. Gangolli R., On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. 93 (1971), 150-165.
  22. Garidi T., What is mass in de Sitterian physics?, arXiv:hep-th/0309104.
  23. Garidi T., Huguet E., Renaud J., de Sitter waves and the zero curvature limit, Phys. Rev. D 67 (2003), 124028, 5 pages, arXiv:gr-qc/0304031.
  24. Gazeau J.P., Novello M., The question of mass in (anti-) de Sitter spacetimes, J. Phys. A: Math. Theor. 41 (2008), 304008, 14 pages.
  25. Gel'fand I.M., Graev M.I., Vilenkin N.Ya., Generalized functions, Vol. 5, Integral geometry and representation theory, Academic Press, New York - London, 1966.
  26. Haag R., Quantum field theories with composite particles and asymptotic conditions, Phys. Rev. 112 (1958), 669-673.
  27. Hannabuss K.C., The localizability of particles in de Sitter space, Proc. Cambridge Philos. Soc. 70 (1971), 283-302.
  28. Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241-310.
  29. Harish-Chandra, Spherical functions on a semisimple Lie group. II, Amer. J. Math. 80 (1958), 553-613.
  30. Hawking S.W., Particle creation by black holes, Comm. Math. Phys. 43 (1975), 199-220.
  31. Helgason S., An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297-308.
  32. Helgason S., Lie groups and symmetric spaces, in Battelle Rencontres, 1967 Lectures in Mathematics and Physics, Editors C.M. DeWitt, J.A. Wheeler, W.A. Benjamin, Benjamin, New York, 1968, 1-71.
  33. Helgason S., Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, Vol. 83, Amer. Math. Soc., Providence, RI, 2000.
  34. Hepp K., On the connection between the LSZ and Wightman quantum field theory, Comm. Math. Phys. 1 (1965), 95-111.
  35. Hepp K., On the connection between Wightman and LSZ quantum field theory, in Proceedings of the 8th Brandeis Summer Institute in Theoretical Physics, Lecture in Theoretical Physics: Axiomatic Field Theory,Gordon and Breach, New York, 1965, 135-240.
  36. Hilgert J., Neeb K.H., Structure and geometry of Lie groups, Springer Monographs in Mathematics, Springer, New York, 2012.
  37. Inonu E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA 39 (1953), 510-524.
  38. Joung E., Mourad J., Parentani R., Group theoretical approach to quantum fields in de Sitter space. I. The principal series, J. High Energy Phys. 2006 (2006), no. 8, 082, 36 pages, arXiv:hep-th/0606119.
  39. Joung E., Mourad J., Parentani R., Group theoretical approach to quantum fields in de Sitter space. II. The complementary and discrete series, J. High Energy Phys. 2007 (2007), no. 9, 030, 40 pages, arXiv:0707.2907.
  40. Limić N., Niederle J., Rączka R., Continuous degenerate representations of noncompact rotation groups. II, J. Math. Phys. 7 (1966), 2026-2035.
  41. Limić N., Niederle J., Rączka R., Eigenfunction expansions associated with the second-order invariant operator on hyperboloids and cones. III, J. Math. Phys. 8 (1967), 1079-1093.
  42. Marolf D., Morrison I.A., Srednicki M., Perturbative $S$-matrix for massive scalar fields in global de Sitter space, Classical Quantum Gravity 30 (2013), 155023, 42 pages, arXiv:1209.6039.
  43. Mickelsson J., Niederle J., Contractions of representations of de Sitter groups, Comm. Math. Phys. 27 (1972), 167-180.
  44. Mizony M., Semi-groupes de causalité et formalisme hilbertien de la mécanique quantique, Publ. Dép. Math. (Lyon) (1984), no. 3B, 47-64.
  45. Molchanov V.F., Harmonic analysis on a hyperboloid of one sheet, Sov. Math. Dokl. 7 (1966), 1553-1556.
  46. O'Neill B., Semi-Riemannian geometry with applications to relativity, Pure and Applied Mathematics, Vol. 103, Academic Press, Inc., New York, 1983.
  47. Primet G., Contractions de groupes de Lie semi-simples sur le groupe de Poincaré généralisé, Publ. Dép. Math. (Lyon) (1983), no. 6D, 1-69.
  48. Rączka R., Limić N., Niederle J., Discrete degenerate representations of noncompact rotation groups. I, J. Math. Phys. 7 (1966), 1861-1876.
  49. Ruelle D., On the asymptotic condition in quantum field theory, Helv. Phys. Acta 35 (1962), 147-163.
  50. Schrödinger E., Expanding universes, Cambridge University Press, Cambridge, 1956.
  51. Stein E.M., Shakarchi R., Functional analysis: introduction to further topics in analysis, Princeton Lectures in Analysis, Vol. 4, Princeton University Press, Princeton, NJ, 2011.
  52. Stein E.M., Weiss G., Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, Vol. 32, Princeton University Press, Princeton, N.J., 1971.
  53. Strichartz R.S., Harmonic analysis on hyperboloids, J. Funct. Anal. 12 (1973), 341-383.
  54. Takahashi R., Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289-433.
  55. Thieleker E.A., The unitary representations of the generalized Lorentz groups, Trans. Amer. Math. Soc. 199 (1974), 327-367.
  56. Unruh W.G., Notes on black-hole evaporation, Phys. Rev. D 14 (1976), 870-892.
  57. Vilenkin N.Ja., Klimyk A.U., Representation of Lie groups and special functions, Vol. 2, Class I representations, special functions, and integral transforms, Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1993.
  58. Warner G., Harmonic analysis on semi-simple Lie groups. II, Die Grundlehren der mathematischen Wissenschaften, Vol. 189, Springer-Verlag, New York - Heidelberg, 1972.
  59. Wigner E., On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. 40 (1939), 149-204.

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