The 𝒫(𝜑)₂ Model on de Sitter Space

Barata, João; Jäkel, Christian; Mund, Jens

doi:10.1090/memo/1389

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

The $\mathscr {P}(\varphi )_2$ Model on de Sitter Space

About this Title

João C. A. Barata, Christian D. Jäkel and Jens Mund

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 281, Number 1389
ISBNs: 978-1-4704-5548-4 (print); 978-1-4704-7322-8 (online)
DOI: https://doi.org/10.1090/memo/1389
Published electronically: January 3, 2023
Keywords: De Sitter space, unitary irreducible representations, Fourier–Helgason transformation, (constructive) quantum field theory

PDF View full volume as PDF

8. The interacting vacuum
9. The interacting representation of $SO(1,2)$
10. Local algebras for the interacting field
11. The equations of motion and the stress-energy tensor
12. Summary
A. A local flat tube theorem
B. One particle structures
C. Sobolev spaces on the circle and on the sphere
D. Some identities involving Legendre functions

Abstract

In 1975 Figari, Høegh-Krohn and Nappi constructed the ${\mathscr P}(\varphi )_2$ model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular,

[$i.)$] we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs of $SO_0(1,2)$ for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables.

[$ii.)$] we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations of $SO(3)$ on the Euclidean sphere, and discuss how the ${\mathscr P}(\varphi )_2$ interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physical infrared problems are absent on de Sitter space. As shown in Figari, Høegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag–Kastler axioms for the ${\mathscr P}(\varphi )_2$ model and we explain how the generators of the boosts and the rotations for the interacting quantum field theory arise from the stress-energy tensor. Finally, we show that the interacting quantum fields satisfy the equations of motion in their covariant form.

In summary, we argue that the de Sitter ${\mathscr P}(\varphi )_2$ model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation.

References

R. Abraham and J. L. Marsden, Foundations of Mechanics (Second Edition), Addison-Wesley Publishing Company, lnc., (1987).
Gian Fabrizio De Angelis, Diego de Falco, and Glauco Di Genova, Random fields on Riemannian manifolds: a constructive approach, Comm. Math. Phys. 103 (1986), no. 2, 297–303. MR 826866
Huzihiro Araki, Mathematical theory of quantum fields, International Series of Monographs on Physics, vol. 101, Oxford University Press, New York, 1999. Translated from the 1993 Japanese original by Ursula Carow-Watamura. MR 1799198
Huzihiro Araki, A lattice of von Neumann algebras associated with the quantum theory of a free Bose field, J. Mathematical Phys. 4 (1963), 1343–1362. MR 158666, DOI 10.1063/1.1703912
Huzihiro Araki, von Neumann algebras of local observables for free scalar field, J. Mathematical Phys. 5 (1964), 1–13. MR 160487, DOI 10.1063/1.1704063
Huzihiro Araki, Relative Hamiltonian for faithful normal states of a von Neumann algebra, Publ. Res. Inst. Math. Sci. 9 (1973/74), 165–209. MR 342080, DOI 10.2977/prims/1195192744
H. Araki, Positive cone, Radon-Nikodým theorems, relative Hamiltonian and the Gibbs condition in statistical mechanics. An application of the Tomita-Takesaki theory, $C^*$-algebras and their applications to statistical mechanics and quantum field theory (Proc. Internat. School of Physics “Enrico Fermi”, Course LX, Varenna, 1973) Proc. Internat. School Phys. Enrico Fermi, LX, North-Holland, Amsterdam-New York-Oxford, 1976, pp. 64–100. MR 675642
Huzihiro Araki, Golden-Thompson and Peierls-Bogolubov inequalities for a general von Neumann algebra, Comm. Math. Phys. 34 (1973), 167–178. MR 341114
Huzihiro Araki, Multiple time analyticity of a quantum statistical state satisfying the $\textrm {KMS}$ boundary condition, Publ. Res. Inst. Math. Sci. Ser. A 4 (1968/69), 361–371. MR 252003, DOI 10.2977/prims/1195194880
Huzihiro Araki and Tetsuya Masuda, Positive cones and $L_{p}$-spaces for von Neumann algebras, Publ. Res. Inst. Math. Sci. 18 (1982), no. 2, 759–831 (339–411). MR 677270, DOI 10.2977/prims/1195183577
H. Araki and E. J. Woods, Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas, J. Mathematical Phys. 4 (1963), 637–662. MR 152295, DOI 10.1063/1.1704002
William B. Arveson, Operator algebras and measure preserving automorphisms, Acta Math. 118 (1967), 95–109. MR 210866, DOI 10.1007/BF02392478
Kendall Atkinson and Weimin Han, Spherical harmonics and approximations on the unit sphere: an introduction, Lecture Notes in Mathematics, vol. 2044, Springer, Heidelberg, 2012. MR 2934227, DOI 10.1007/978-3-642-25983-8
Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave equations on Lorentzian manifolds and quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2007. MR 2298021, DOI 10.4171/037
C. Bär and K. Fredenhagen (ed.), Quantum field theory on curved spacetimes, Lecture Notes Phys. 786 (2009) 1–155.
V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568–640. MR 21942, DOI 10.2307/1969129
A. O. Barut and C. Fronsdal, On non-compact groups. II. Representations of the $2+1$ Lorentz group, Proc. Roy. Soc. London Ser. A 287 (1965), 532–548. MR 189615, DOI 10.1098/rspa.1965.0195
Asim O. Barut and Ryszard Rączka, Theory of group representations and applications, 2nd ed., World Scientific Publishing Co., Singapore, 1986. MR 889252, DOI 10.1142/0352
C. Bellaiche, Le comportement asymptotique de $p(t, x, y)$ quand $t \to 0$, Soc. Math. France, Asterisque 84–85 (1981) 151–187.
G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicolò, E. Olivieri, E. Presutti, and E. Scacciatelli, Ultraviolet stability in Euclidean scalar field theories, Comm. Math. Phys. 71 (1980), no. 2, 95–130. MR 560344
Antonio N. Bernal and Miguel Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Phys. 257 (2005), no. 1, 43–50. MR 2163568, DOI 10.1007/s00220-005-1346-1
J. Bertrand, P. Bertrand and J.P. Ovarlez, The Mellin Transform, The Transforms and Applications Handbook, Chapter 12, Ed. A.D. Poularikas, CRC Press Inc., 1995.
S. De Bièvre and M. Merkli, The Unruh effect revisited, Classical Quantum Gravity 23 (2006), no. 22, 6525–6541. MR 2272019, DOI 10.1088/0264-9381/23/22/026
Lothar Birke and Jürg Fröhlich, KMS, etc, Rev. Math. Phys. 14 (2002), no. 7-8, 829–871. Dedicated to Professor Huzihiro Araki on the occasion of his 70th birthday. MR 1932668, DOI 10.1142/S0129055X02001442
Joseph J. Bisognano and Eyvind H. Wichmann, On the duality condition for a Hermitian scalar field, J. Mathematical Phys. 16 (1975), 985–1007. MR 438943, DOI 10.1063/1.522605
Joseph J. Bisognano and Eyvind H. Wichmann, On the duality condition for quantum fields, J. Mathematical Phys. 17 (1976), no. 3, 303–321. MR 438944, DOI 10.1063/1.522898
H. J. Borchers, On revolutionizing quantum field theory with Tomita’s modular theory, J. Math. Phys. 41 (2000), no. 6, 3604–3673. MR 1768633, DOI 10.1063/1.533323
H. J. Borchers and D. Buchholz, Global properties of vacuum states in de Sitter space, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), no. 1, 23–40 (English, with English and French summaries). MR 1671209
Armand Borel, Groupes linéaires algébriques, Ann. of Math. (2) 64 (1956), 20–82 (French). MR 93006, DOI 10.2307/1969949
M. Born, W. Heisenberg and P. Jordan, Zur Quantenmechanik. II., Zeitschrift f. Physik 35(8-9) (1926) 557–615.
Nicolas Bourbaki, Integration. II. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2004. Translated from the 1963 and 1969 French originals by Sterling K. Berberian. MR 2098271
O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II, Springer-Verlag, New York-Heidelberg-Berlin, 1981.
Jacques Bros, Complexified de Sitter space: analytic causal kernels and Källen-Lehmann-type representation, Nuclear Phys. B Proc. Suppl. 18B (1990), 22–28 (1991). Recent advances in field theory (Annecy-le-Vieux, 1990). MR 1128126, DOI 10.1016/0920-5632(91)90119-Y
Jacques Bros and Detlev Buchholz, Towards a relativistic KMS-condition, Nuclear Phys. B 429 (1994), no. 2, 291–318. MR 1299069, DOI 10.1016/0550-3213(94)00298-3
J. Bros, H. Epstein, V. Glaser, and R. Stora, Quelques aspects globaux des problèmes d’edge-of-the-wedge, Hyperfunctions and theoretical physics (Rencontre, Nice, 1973; dédié à la mémoire de A. Martineau), Lecture Notes in Math., Vol. 449, Springer, Berlin-New York, 1975, pp. 185–218 (French). MR 419831
Jacques Bros, Henri Epstein, and Ugo Moschella, Towards a general theory of quantized fields on the anti-de Sitter space-time, Comm. Math. Phys. 231 (2002), no. 3, 481–528. MR 1946447, DOI 10.1007/s00220-002-0726-z
Jacques Bros, Henri Epstein, and Ugo Moschella, Analyticity properties and thermal effects for general quantum field theory on de Sitter space-time, Comm. Math. Phys. 196 (1998), no. 3, 535–570. MR 1645200, DOI 10.1007/s002200050435
Bros, J., Epstein, H., and Moschella, U., The lifetime of a massive particle in a de Sitter universe, J. Cosmology & Astroparticle Physics 02 (2008) 003.
Jacques Bros and Ugo Moschella, Two-point functions and quantum fields in de Sitter universe, Rev. Math. Phys. 8 (1996), no. 3, 327–391. MR 1388256, DOI 10.1142/S0129055X96000123
Jacques Bros and Ugo Moschella, Fourier analysis and holomorphic decomposition on the one-sheeted hyperboloid, Géométrie complexe. II. Aspects contemporains dans les mathématiques et la physique, Hermann Éd. Sci. Arts, Paris, 2004, pp. 27–58. MR 2493569
François Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97–205 (French). MR 84713
R. Brunetti, K. Fredenhagen, and M. Köhler, The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes, Comm. Math. Phys. 180 (1996), no. 3, 633–652. MR 1408521
Romeo Brunetti and Klaus Fredenhagen, Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds, Comm. Math. Phys. 208 (2000), no. 3, 623–661. MR 1736329, DOI 10.1007/s002200050004
R. Brunetti, D. Guido, and R. Longo, Modular localization and Wigner particles, Rev. Math. Phys. 14 (2002), no. 7-8, 759–785. Dedicated to Professor Huzihiro Araki on the occasion of his 70th birthday. MR 1932665, DOI 10.1142/S0129055X02001387
Romeo Brunetti, Klaus Fredenhagen, and Rainer Verch, The generally covariant locality principle—a new paradigm for local quantum field theory, Comm. Math. Phys. 237 (2003), no. 1-2, 31–68. Dedicated to Rudolf Haag. MR 2007173, DOI 10.1007/s00220-003-0815-7
D. Buchholz, On the manifestations of particles, published in Mathematical Physics Towards the 21st Century, Proceedings Beer-Sheva 1993, Ben Gurion University Press, 1994.
D. Buchholz, C. D’Antoni, and K. Fredenhagen, The universal structure of local algebras, Comm. Math. Phys. 111 (1987), no. 1, 123–135. MR 896763
Detlev Buchholz, Jens Mund, and Stephen J. Summers, Covariant and quasi-covariant quantum dynamics in Robertson-Walker spacetimes, Classical Quantum Gravity 19 (2002), no. 24, 6417–6434. MR 1956312, DOI 10.1088/0264-9381/19/24/310
Detlev Buchholz and Stephen J. Summers, Stable quantum systems in anti-de Sitter space: causality, independence, and spectral properties, J. Math. Phys. 45 (2004), no. 12, 4810–4831. MR 2105223, DOI 10.1063/1.1804230
Claude Chevalley, Theory of Lie Groups. I, Princeton Mathematical Series, vol. 8, Princeton University Press, Princeton, NJ, 1946. MR 15396
Bruno Chilian and Klaus Fredenhagen, The time slice axiom in perturbative quantum field theory on globally hyperbolic spacetimes, Comm. Math. Phys. 287 (2009), no. 2, 513–522. MR 2481748, DOI 10.1007/s00220-008-0670-7
Y. Choquet-Bruhat, Hyperbolic partial differential equations on a manifold, Battelle Rencontres. 1967 Lectures in Mathematics and Physics, W. A. Benjamin, Inc., New York-Amsterdam, 1968, pp. 84–106. MR 239299
Claudio D’Antoni and László Zsidó, Analytic extension of vector valued functions, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 171–202. MR 1902801
Claudio D’Antoni and László Zsidó, The flat tube theorem for vector valued functions, Mathematical physics in mathematics and physics (Siena, 2000) Fields Inst. Commun., vol. 30, Amer. Math. Soc., Providence, RI, 2001, pp. 115–138. MR 1867550
Jean-Marc Delort, L’équation de Klein-Gordon à données petites faiblement décroissantes, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, École Polytech., Palaiseau, 1997, pp. Exp. No. V, 14 (French). MR 1482811
Jean-Marc Delort, Sur le temps d’existence pour l’équation de Klein-Gordon semi-linéaire en dimension 1, Bull. Soc. Math. France 125 (1997), no. 2, 269–311 (French, with English and French summaries). MR 1478033
J.-M. Delort, Minoration du temps d’existence pour l’équation de Klein-Gordon non-linéaire en dimension 1 d’espace, Ann. Inst. H. Poincaré C Anal. Non Linéaire 16 (1999), no. 5, 563–591 (French, with English and French summaries). MR 1712572, DOI 10.1016/S0294-1449(99)80028-6
Jean-Marc Delort, Existence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 1, 1–61 (French, with English and French summaries). MR 1833089, DOI 10.1016/S0012-9593(00)01059-4
Jean-Marc Delort and Jérémie Szeftel, Almost global solutions for non Hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces, Journées “Équations aux Dérivées Partielles”, École Polytech., Palaiseau, 2005, pp. Exp. No. XV, 13. MR 2352782
J. Dereziński and C. Gérard, Spectral scattering theory of spatially cut-off $P(\phi )_2$ Hamiltonians, Comm. Math. Phys. 213 (2000), no. 1, 39–125. MR 1782144, DOI 10.1007/s002200000233
J. Dereziński, V. Jakšić, and C.-A. Pillet, Perturbation theory of $W^*$-dynamics, Liouvilleans and KMS-states, Rev. Math. Phys. 15 (2003), no. 5, 447–489. MR 1993788, DOI 10.1142/S0129055X03001679
J. Dimock, The non-relativistic limit of $P(\phi )_{2}$ quantum field theories: two-particle phenomena, Comm. Math. Phys. 57 (1977), no. 1, 51–66. MR 475455
J. Dimock, Algebras of local observables on a manifold, Comm. Math. Phys. 77 (1980), no. 3, 219–228. MR 594301
J. Dimock, $P(\varphi )_{2}$ models with variable coefficients, Ann. Physics 154 (1984), no. 2, 283–307. MR 747204, DOI 10.1016/0003-4916(84)90152-0
J. Dimock, Markov quantum fields on a manifold, Rev. Math. Phys. 16 (2004), no. 2, 243–255. MR 2048318, DOI 10.1142/S0129055X04001947
H. Dym and H. P. McKean, Fourier series and integrals, Probability and Mathematical Statistics, No. 14, Academic Press, New York-London, 1972. MR 442564
Jean-Pierre Eckmann and Konrad Osterwalder, An application of Tomita’s theory of modular Hilbert algebras: duality for free Bose Fields, J. Functional Analysis 13 (1973), 1–12. MR 345548, DOI 10.1016/0022-1236(73)90062-1
Volker Enss, Characterization of particles by means of local observables, Comm. Math. Phys. 45 (1975), no. 1, 35–52. MR 400967
Jacques Faraut, Noyaux sphériques sur un hyperboloïde à une nappe, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973–1975) Lecture Notes in Math., Vol. 497, Springer, Berlin-New York, 1975, pp. 172–210 (French). MR 420156
J. Feldman and F. P. Greenleaf, Existence of Borel transversals in groups, Pacific J. Math. 25 (1968), 455–461. MR 230837
Joel S. Feldman and Ryszard Rączka, The relativistic field equation of the $\lambda F^{4}_{3}$ quantum field theory, Ann. Physics 108 (1977), no. 1, 212–229. MR 452279, DOI 10.1016/0003-4916(77)90357-8
J. M. G. Fell and R. S. Doran, Representations of $^*$-algebras, locally compact groups, and Banach $^*$-algebraic bundles. Vol. 1, Pure and Applied Mathematics, vol. 125, Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups and algebras. MR 936628
C. J. Fewster, Bisolutions to the Klein-Gordon equation and quantum field theory on two-dimensional cylinder spacetimes, Classical Quantum Gravity 16 (1999), no. 3, 769–788. MR 1682550, DOI 10.1088/0264-9381/16/3/010
C. J. Fewster, A unique continuation result for Klein-Gordon bisolutions on a two-dimensional cylinder, Classical Quantum Gravity 16 (1999), no. 3, 789–796. MR 1682546, DOI 10.1088/0264-9381/16/3/011
Rodolfo Figari, Raphael Hoegh-Krohn, and Chiara R. Nappi, Interacting relativistic boson fields in the de Sitter universe with two space-time dimensions, Comm. Math. Phys. 44 (1975), no. 3, 265–278. MR 384010
Franca Figliolini and Daniele Guido, The Tomita operator for the free scalar field, Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), no. 4, 419–435 (English, with French summary). MR 1034596
Franca Figliolini and Daniele Guido, On the type of second quantization factors, J. Operator Theory 31 (1994), no. 2, 229–252. MR 1331775
M. Flato, J. Simon, H. Snellman, and D. Sternheimer, Simple facts about analytic vectors and integrability, Ann. Sci. École Norm. Sup. (4) 5 (1972), 423–434. MR 376960
V. A. Fock, On the representation of an arbitrary function by an integral involving Legendre’s functions with a complex index, C. R. (Doklady) Acad. Sci. URSS (N.S.) 39 (1943), 253–256. MR 9665
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
T.T. Frankel, The geometry of physics, an introduction, Cambridge University Press, 1997; see also Third Edition 2012.
J. Fröhlich, Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint, Adv. in Appl. Math. 1 (1980), no. 3, 237–256. MR 603131, DOI 10.1016/0196-8858(80)90012-3
J. Fröhlich, K. Osterwalder, and E. Seiler, On virtual representations of symmetric spaces and their analytic continuation, Ann. of Math. (2) 118 (1983), no. 3, 461–489. MR 727701, DOI 10.2307/2006979
T. Garidi, What is mass in desitterian physics? arXiv:hep-th/0309104v1.
I. M. Gel′fand and M. A. Naĭmark, Unitary representations of the Lorentz group, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 411–504 (Russian). MR 24440
I. M. Gel′fand and M. I. Graev, Analogue of the Plancherel formula for the classical groups, Trudy Moskov. Mat. Obšč. 4 (1955), 375–404 (Russian). MR 71714
I. M. Gel′fand and M. I. Graev, Unitary representations of the real unimodular group (principal nondegenerate series), Amer. Math. Soc. Transl. (2) 2 (1956), 147–205. MR 76291
I. M. Gel′fand and M. I. Graev, Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry. I, Trudy Moskov. Mat. Obšč. 8 (1959), 321–390; addendum: 9 (1959), 562 (Russian). MR 126719
I.M. Gelfand and M.I. Graev, Admissible complexes of lines in ${\mathbf {CP}}^n$, Funct. Anal. Appl. 3 (1968) 39–52.
I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5: Integral geometry and representation theory, Academic Press, New York-London, 1966. Translated from the Russian by Eugene Saletan. MR 207913
C. Gérard, On the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. Henri Poincaré 1 (2000), no. 3, 443–459. MR 1777307, DOI 10.1007/s000230050002
Christian Gérard and Christian D. Jäkel, Thermal quantum fields with spatially cutoff interactions in $1+1$ space-time dimensions, J. Funct. Anal. 220 (2005), no. 1, 157–213. MR 2114702, DOI 10.1016/j.jfa.2004.08.003
Christian Gérard and Christian D. Jäkel, Thermal quantum fields without cut-offs in $1+1$ space-time dimensions, Rev. Math. Phys. 17 (2005), no. 2, 113–173. MR 2133630, DOI 10.1142/S0129055X05002303
Christian Gérard and Annalisa Panati, Spectral and scattering theory for space-cutoff $P(\phi )_2$ models with variable metric, Ann. Henri Poincaré 9 (2008), no. 8, 1575–1629. MR 2465734, DOI 10.1007/s00023-008-0396-2
C. Gérard and M. Wrochna, Construction of Hadamard states by pseudo-differential calculus, Comm. Math. Phys. 325 (2014), no. 2, 713–755. MR 3148100, DOI 10.1007/s00220-013-1824-9
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), no. 1, 15–35 (English, with French summary). MR 984146
J. Glimm, Boson fields with nonlinear self-interaction in two dimensions, Comm. Math. Phys. 8 (1968) 12–25.
James Glimm and Arthur Jaffe, A $\lambda \phi ^{4}$ quantum field without cutoffs. I, Phys. Rev. (2) 176 (1968), 1945–1951. MR 247845
James Glimm and Arthur Jaffe, The $\lambda (\Pi ^{4})_{2}$ quantum field theory without cutoffs. II. The field operators and the approximate vacuum, Ann. of Math. (2) 91 (1970), 362–401. MR 256677, DOI 10.2307/1970582
James Glimm and Arthur Jaffe, The $\lambda (\phi ^{4})_{2}$ quantum field theory without cutoffs. III. The physical vacuum, Acta Math. 125 (1970), 203–267. MR 269234, DOI 10.1007/BF02392335
James Glimm and Arthur Jaffe, The $\lambda \phi _{2}^{4}$ quantum field theory without cutoffs. IV. Perturbations of the Hamiltonian, J. Mathematical Phys. 13 (1972), 1568–1584. MR 317683, DOI 10.1063/1.1665879
J. Glimm, A. Jaffe, and T. Spencer, The particle structure of the weakly coupled $P(\phi )_2$ model and other applications of the high temperature expansions, Part II: The cluster expansion, In G. Velo and A.S. Wightman (ed.), Constructive Quantum Field Theory (Erice), Lecture Notes in Physics, Vol. 25 (1973) 199–242.
James Glimm, Arthur Jaffe, and Thomas Spencer, The Wightman axioms and particle structure in the ${\scr P}(\phi )_{2}$ quantum field model, Ann. of Math. (2) 100 (1974), 585–632. MR 363256, DOI 10.2307/1970959
James Glimm and Arthur Jaffe, Quantum physics, Springer-Verlag, New York-Berlin, 1981. A functional integral point of view. MR 628000
J. Glimm and A. Jaffe, Collected Papers, Vol. I: Quantum Field Theory and Statistical Mechanics; Expositions; Vol. II: Constructive Quantum Field Theory, Selected Papers, Birkhäuser, Boston, 1985.
I.S. Gradstein and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1966; see also Academic Press, New York, 2007.
Daniele Guido and Roberto Longo, A converse Hawking-Unruh effect and $\textrm {dS}^2$/CFT correspondence, Ann. Henri Poincaré 4 (2003), no. 6, 1169–1218. MR 2031163, DOI 10.1007/s00023-003-0159-z
F. Guerra, Uniqueness of the vacuum energy density and van Hove phenomenon in the infinite volume limit for two-dimensional self-coupled Bose fields, Phys. Rev. Lett. 28 (1972) 1213–1215.
Francesco Guerra, Quantum field theory and probability theory. Outlook on new possible developments, Trends and developments in the eighties (Bielefeld, 1982/1983) World Sci. Publishing, Singapore, 1985, pp. 214–243. MR 853750
F. Guerra, L. Rosen, and B. Simon, The $\textbf {P}(\phi )_{2}$ Euclidean quantum field theory as classical statistical mechanics. I, II, Ann. of Math. (2) 101 (1975), 111–189; ibid. (2) 101 (1975), 191–259. MR 378670, DOI 10.2307/1970988
Rudolf Haag, Local quantum physics, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996. Fields, particles, algebras. MR 1405610, DOI 10.1007/978-3-642-61458-3
Klaus Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265–277. MR 332046
K. C. Hannabuss, The localizability of particles in de Sitter space, Proc. Cambridge Philos. Soc. 70 (1971), 283–302. MR 299103, DOI 10.1017/s0305004100049896
Harish-Chandra, Infinite irreducible representations of the Lorentz group, Proc. Roy. Soc. London Ser. A 189 (1947), 372–401. MR 21941, DOI 10.1098/rspa.1947.0047
Harish-Chandra, On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. Soc. 70 (1951), 28–96. MR 44515, DOI 10.1090/S0002-9947-1951-0044515-0
Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc. 75 (1953), 185–243. MR 56610, DOI 10.1090/S0002-9947-1953-0056610-2
Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26–65. MR 58604, DOI 10.1090/S0002-9947-1954-0058604-0
Harish-Chandra, Representations of semisimple Lie groups. V, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 1076–1077. MR 64779, DOI 10.1073/pnas.40.11.1076
Harish-Chandra, Plancherel formula for the $2\times 2$ real unimodular group, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 337–342. MR 47055, DOI 10.1073/pnas.38.4.337
Harish-Chandra, The Plancherel formula for complex semisimple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485–528. MR 63376, DOI 10.1090/S0002-9947-1954-0063376-X
Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math. 80 (1958) 241–310; Spherical functions on a semisimple Lie group II, Amer. J. Math. 80 (1958) 553–613.
Harish-Chandra, Harmonic analysis on real reductive groups I, J. Funct. Anal. 19 (1975) 104–204; Harmonic analysis on real reductive groups II, Invent. Math. 36 (1976) 1–55; Harmonic analysis on real reductive groups III, Ann. Math. 104 (1976) 117–201.
S. W. Hawking, Particle creation by black holes, Comm. Math. Phys. 43 (1975), no. 3, 199–220. MR 381625
S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 424186
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 145455
Sigurdur Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR 1280714, DOI 10.1090/surv/039
Sigurdur Helgason, Integral geometry and Radon transforms, Springer, New York, 2011. MR 2743116, DOI 10.1007/978-1-4419-6055-9
Sigurdur Helgason, Support theorems for horocycles on hyperbolic spaces, Pure Appl. Math. Q. 8 (2012), no. 4, 921–927. MR 2959914, DOI 10.4310/PAMQ.2012.v8.n4.a4
Stefan Hollands and Robert M. Wald, Local Wick polynomials and time ordered products of quantum fields in curved spacetime, Comm. Math. Phys. 223 (2001), no. 2, 289–326. MR 1864435, DOI 10.1007/s002200100540
Stefan Hollands and Robert M. Wald, Existence of local covariant time ordered products of quantum field in curved spacetime, Comm. Math. Phys. 231 (2002), no. 2, 309–345. MR 1946335, DOI 10.1007/s00220-002-0719-y
Stefan Hollands and Robert M. Wald, On the renormalization group in curved spacetime, Comm. Math. Phys. 237 (2003), no. 1-2, 123–160. Dedicated to Rudolf Haag. MR 2007177, DOI 10.1007/s00220-003-0837-1
Stefan Hollands, Massless interacting scalar quantum fields in deSitter spacetime, Ann. Henri Poincaré 13 (2012), no. 4, 1039–1081. MR 2913629, DOI 10.1007/s00023-011-0140-1
Stefan Hollands, Correlators, Feynman diagrams, and quantum no-hair in deSitter spacetime, Comm. Math. Phys. 319 (2013), no. 1, 1–68. MR 3034025, DOI 10.1007/s00220-012-1653-2
R. Hoole, A. Jaffe and C. Jäkel, Quantization Domains, see arXiv:1304.6138.
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
Lars Hörmander, The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR 1996773, DOI 10.1007/978-3-642-61497-2
Roger Howe and Eng-Chye Tan, Nonabelian harmonic analysis, Universitext, Springer-Verlag, New York, 1992. Applications of $\textrm {SL}(2,\textbf {R})$. MR 1151617, DOI 10.1007/978-1-4613-9200-2
Kenkichi Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558. MR 29911, DOI 10.2307/1969548
Arthur Jaffe, Divergence of perturbation theory for bosons, Comm. Math. Phys. 1 (1965), 127–149. MR 193979
Arthur Jaffe and Gordon Ritter, Quantum field theory on curved backgrounds. I. The Euclidean functional integral, Comm. Math. Phys. 270 (2007), no. 2, 545–572. MR 2276455, DOI 10.1007/s00220-006-0166-2
A. Jaffe and G. Ritter, Quantum field theory on curved backgrounds. II. Spacetime symmetries, see arXiv:0704.0052.
Arthur Jaffe and Gordon Ritter, Reflection positivity and monotonicity, J. Math. Phys. 49 (2008), no. 5, 052301, 10. MR 2421900, DOI 10.1063/1.2907660
Arthur Jaffe, Christian D. Jäkel, and Roberto E. Martinez II, Complex classical fields: an example, J. Funct. Anal. 266 (2014), no. 3, 1833–1881. MR 3146837, DOI 10.1016/j.jfa.2013.08.033
Christian D. Jäkel and Jens Mund, The Haag-Kastler axioms for the $\mathcal P(\varphi )_2$ model on the de Sitter space, Ann. Henri Poincaré 19 (2018), no. 3, 959–977. MR 3769251, DOI 10.1007/s00023-018-0647-9
C.D. Jäkel and U.A. Wiedemann, Local observables for massless particles on the two-dimensional de Sitter space, in preparation.
Palle E. T. Jorgensen and Gestur Ólafsson, Unitary representations and Osterwalder-Schrader duality, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) Proc. Sympos. Pure Math., vol. 68, Amer. Math. Soc., Providence, RI, 2000, pp. 333–401. MR 1767902, DOI 10.1090/pspum/068/1767902
Jay Jorgenson and Serge Lang, Spherical inversion on SL${_n}(\textbf {R})$, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001. MR 1834111, DOI 10.1007/978-1-4684-9302-3
Res Jost, The general theory of quantized fields, Lectures in Applied Mathematics, American Mathematical Society, Providence, RI, 1965. Mark Kac, editor. MR 177667
R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. I-IV, American Mathematical Society, 1997-1998.
Eberhard Kaniuth and Keith F. Taylor, Induced representations of locally compact groups, Cambridge Tracts in Mathematics, vol. 197, Cambridge University Press, Cambridge, 2013. MR 3012851
Bernard S. Kay, A uniqueness result in the Segal-Weinless approach to linear Bose fields, J. Math. Phys. 20 (1979), no. 8, 1712–1713. MR 543906, DOI 10.1063/1.524253
Bernard S. Kay, A uniqueness result for quasifree KMS states, Helv. Phys. Acta 58 (1985), no. 6, 1017–1029. MR 821118
Bernard S. Kay, Purification of KMS states, Helv. Phys. Acta 58 (1985), no. 6, 1030–1040. MR 821119
Bernard S. Kay, The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes, Comm. Math. Phys. 100 (1985), no. 1, 57–81. MR 796162
Bernard S. Kay, Sufficient conditions for quasifree states and an improved uniqueness theorem for quantum fields on space-times with horizons, J. Math. Phys. 34 (1993), no. 10, 4519–4539. MR 1235954, DOI 10.1063/1.530354
Bernard S. Kay and Robert M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon, Phys. Rep. 207 (1991), no. 2, 49–136. MR 1133130, DOI 10.1016/0370-1573(91)90015-E
A.A. Kirillov, Introduction to the Theory of Representations and Noncommutative Harmonic Analysis I, Fundamental Concepts. Representations of Virasoro and Affine Algebras, Springer, 1994.
A.A. Kirillov, Introduction to the Theory of Representations and Noncommutative Harmonic Analysis II. Springer, 1994.
A. A. Kirillov, Merits and demerits of the orbit method, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 433–488. MR 1701415, DOI 10.1090/S0273-0979-99-00849-6
Abel Klein and Lawrence J. Landau, Stochastic processes associated with KMS states, J. Functional Analysis 42 (1981), no. 3, 368–428. MR 626451, DOI 10.1016/0022-1236(81)90096-3
Abel Klein and Lawrence J. Landau, Construction of a unique selfadjoint generator for a symmetric local semigroup, J. Functional Analysis 44 (1981), no. 2, 121–137. MR 642913, DOI 10.1016/0022-1236(81)90007-0
Abel Klein and Lawrence J. Landau, From the Euclidean group to the Poincaré group via Osterwalder-Schrader positivity, Comm. Math. Phys. 87 (1982/83), no. 4, 469–484. MR 691039
Abel Klein and Lawrence J. Landau, Singular perturbations of positivity preserving semigroups via path space techniques, J. Functional Analysis 20 (1975), no. 1, 44–82. MR 381580, DOI 10.1016/0022-1236(75)90053-1
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15, Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 238225
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups. II, Invent. Math. 60 (1980), no. 1, 9–84. MR 582703, DOI 10.1007/BF01389898
Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the $2\times 2$ real unimodular group, Amer. J. Math. 82 (1960), 1–62. MR 163988, DOI 10.2307/2372876
Serge Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, vol. 191, Springer-Verlag, New York, 1999. MR 1666820, DOI 10.1007/978-1-4612-0541-8
G. Lebeau, Équations des ondes semi-linéaires. II. Contrôle des singularités et caustiques non linéaires, Invent. Math. 95 (1989), no. 2, 277–323 (French, with English summary). MR 974905, DOI 10.1007/BF01393899
N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 350075
G. Lemaître, Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extragalactiques, Annals of the Scientific Society of Brussels 47A (1927) 49–59; translated in: A Homogeneous Universe of Constant Mass and Growing Radius Accounting for the Radial Velocity of Extragalactic Nebulae, Monthly Notices of the Royal Astronomical Society 91 (1931) 483–490.
Jean Leray, Hyperbolic differential equations, Institute for Advanced Study (IAS), Princeton, NJ, 1953. MR 63548
P. Leyland, J. Roberts and D. Testard, Duality for free quantum fields, Marseille preprint (1978) unpublished.
Roberto Longo, Algebraic and modular structure of von Neumann algebras of physics, Operator algebras and applications, Part 2 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, RI, 1982, pp. 551–566. MR 679537
André Lichnerowicz, Propagateurs et commutateurs en relativité générale, Cahiers de Phys. 15 (1961), 189–199 (French). MR 158726
Ronald L. Lipsman, Group representations, Lecture Notes in Mathematics, Vol. 388, Springer-Verlag, Berlin-New York, 1974. A survey of some current topics. MR 372116
George W. Mackey, Imprimitivity for representations of locally compact groups. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537–545. MR 31489, DOI 10.1073/pnas.35.9.537
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
George W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. MR 56611, DOI 10.2307/1969786
George W. Mackey, Infinite-dimensional group representations, Bull. Amer. Math. Soc. 69 (1963), 628–686. MR 153784, DOI 10.1090/S0002-9904-1963-10973-8
George W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. MR 89999, DOI 10.1090/S0002-9947-1957-0089999-2
George W. Mackey, Induced representations of groups and quantum mechanics, W. A. Benjamin, Inc., New York-Amsterdam; Editore Boringhieri, Turin, 1968. MR 507212
N. W. Macfadyen, A Laplace transform on the Lorentz groups. II. The general case, Comm. Math. Phys. 34 (1973), 297–314. MR 330930
N. W. Macfadyen, The horospheric approach to harmonic analysis on a semi-simple Lie group, Rep. Mathematical Phys. 6 (1974), no. 2, 265–288. MR 376957, DOI 10.1016/0034-4877(74)90009-3
N. W. Macfadyen, A new approach to harmonic analysis on the group $\textrm {SL}(2,\,R)$, Rep. Mathematical Phys. 6 (1974), no. 2, 237–248. MR 379753, DOI 10.1016/0034-4877(74)90006-8
Kyûya Masuda, Anti-locality of the one-half power of elliptic differential operators, Publ. Res. Inst. Math. Sci. 8 (1972), 207–210. MR 331120, DOI 10.2977/prims/1195193233
J. Mickelsson and J. Niederle, Contractions of representations of de Sitter groups, Comm. Math. Phys. 27 (1972), 167–180. MR 309502
R. A. Minlos, Generalized random processes and their extension in measure, Trudy Moskov. Mat. Obšč. 8 (1959), 497–518 (Russian). MR 108851
V. F. Molčanov, Harmonic analysis on a hyperboloid of one sheet, Dokl. Akad. Nauk SSSR 171 (1966), 794–797 (Russian). MR 206158
V. F. Molčanov, An analog of Plancherel’s formula for hyperboloids, Dokl. Akad. Nauk SSSR 183 (1968), 288–291 (Russian). MR 236310
Jens Mund, Bert Schroer, and Jakob Yngvason, String-localized quantum fields and modular localization, Comm. Math. Phys. 268 (2006), no. 3, 621–672. MR 2259209, DOI 10.1007/s00220-006-0067-4
M. A. Naimark, Linear representations of the Lorentz group, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by Ann Swinfen and O. J. Marstrand; translation edited by H. K. Farahat. MR 170977
Tadao Nakano, Quantum field theory in terms of Euclidean parameters, Progr. Theoret. Phys. 21 (1959), 241–259. MR 101779, DOI 10.1143/PTP.21.241
Heide Narnhofer, Quantum Anosov systems, Rigorous quantum field theory, Progr. Math., vol. 251, Birkhäuser, Basel, 2007, pp. 213–223. MR 2279219, DOI 10.1007/978-3-7643-7434-1_{1}5
H. Narnhofer, I. Peter, and W. Thirring, How hot is the de Sitter space?, Internat. J. Modern Phys. B 10 (1996), no. 13-14, 1507–1520. Memorial issue for H. Umezawa. MR 1405180, DOI 10.1142/S0217979296000611
Edward Nelson, A quartic interaction in two dimensions, Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) MIT Press, Cambridge, Mass.-London, 1966, pp. 69–73. MR 210416
Edward Nelson, Quantum fields and Markoff fields, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Proc. Sympos. Pure Math., Vol. XXIII, Amer. Math. Soc., Providence, RI, 1973, pp. 413–420. MR 337206
Edward Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis 12 (1973), 97–112. MR 343815, DOI 10.1016/0022-1236(73)90091-8
Edward Nelson, The free Markoff field, J. Functional Analysis 12 (1973), 211–227. MR 343816, DOI 10.1016/0022-1236(73)90025-6
Yu. A. Neretin, Restriction of functions holomorphic in a domain to curves lying on the boundary of the domain, and discrete $\textrm {SL}_2(\textbf {R})$-spectra, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 3, 67–86 (Russian, with Russian summary); English transl., Izv. Math. 62 (1998), no. 3, 493–513. MR 1642160, DOI 10.1070/im1998v062n03ABEH000202
Yurii A. Neretin, Some remarks on quasi-invariant actions of loop groups and the group of diffeomorphisms of the circle, Comm. Math. Phys. 164 (1994), no. 3, 599–626. MR 1291246
C. Neumann, Ueber die Mehler’schen Kegelfunctionen und deren Anwendung auf elektrostatische Probleme, Math. Ann. 18 (1881), no. 2, 195–236 (German). MR 1510099, DOI 10.1007/BF01445848
Raymond O. Wells Jr., Differential analysis on complex manifolds, 3rd ed., Graduate Texts in Mathematics, vol. 65, Springer, New York, 2008. With a new appendix by Oscar Garcia-Prada. MR 2359489, DOI 10.1007/978-0-387-73892-5
Fritz Oberhettinger, Tables of Mellin transforms, Springer-Verlag, New York-Heidelberg, 1974. MR 352890
Masanori Ohya and Dénes Petz, Quantum entropy and its use, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993. MR 1230389, DOI 10.1007/978-3-642-57997-4
Konrad Osterwalder and Robert Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys. 31 (1973), 83–112. MR 329492
Konrad Osterwalder and Robert Schrader, Axioms for Euclidean Green’s functions. II, Comm. Math. Phys. 42 (1975), 281–305. With an appendix by Stephen Summers. MR 376002
I. Peter, Quantum Field Theory in Curved Space-Time with an Application to the Reduced Model of the de Sitter Universe, Dissertation, Universität Wien (1995).
L. S. Pontryagin, Topological groups, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR 201557
L. Pukánszky, The Plancherel formula for the universal covering group of $\textrm {SL}(R,\,2)$, Math. Ann. 156 (1964), 96–143. MR 170981, DOI 10.1007/BF01359927
John C. Quigg, On the irreducibility of an induced representation, Pacific J. Math. 93 (1981), no. 1, 163–179. MR 621605
Marek Jan Radzikowski, The Hadamard condition and Kay’s conjecture in (axiomatic) quantum field theory on curved space-time, ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.)–Princeton University. MR 2688333
Marek J. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Comm. Math. Phys. 179 (1996), no. 3, 529–553. MR 1400751
M. Reed, and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, 1972; Vol. II: Fourier Analysis, Self-adjointness, Academic Press, 1975.
A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009–1038.
David Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 289084
Paul J. Sally Jr., Intertwining operators and the representations of $\textrm {SL}(2,\,\textbf {R})$, J. Functional Analysis 6 (1970), 441–453. MR 299727, DOI 10.1016/0022-1236(70)90071-6
Paul J. Sally Jr., Analytic continuation of the irreducible unitary representations of the universal covering group of $\textrm {SL}(2,\,R)$, Memoirs of the American Mathematical Society, No. 69, American Mathematical Society, Providence, RI, 1967. MR 235068
Robert Schrader, Local operator products and field equations in $P(\varphi )_{2}$ theories, Fortschr. Physik 22 (1974), 611–631. MR 674158, DOI 10.1002/prop.19740221102
E. Schrödinger, Expanding Universe, Cambridge University Press, 2011.
Schulmann et al., The Einstein-De Sitter-Weyl-Klein Debate, in Vol. 8 of The Collected Papers of Albert Einstein, covering correspondence during period 1914–1918 (1998), 351–357.
Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 209834
Julian Schwinger, On the Euclidean structure of relativistic field theory, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 956–965. MR 97250, DOI 10.1073/pnas.44.9.956
Irving Segal, Construction of non-linear local quantum processes. I, Ann. of Math. (2) 92 (1970), 462–481. MR 272306, DOI 10.2307/1970628
Geoffrey Sewell, Quantum fields on manifolds: PCT and gravitationally induced thermal states, Ann. Physics 141 (1982), no. 2, 201–224. MR 673980, DOI 10.1016/0003-4916(82)90285-8
Geoffrey L. Sewell, Relativity of temperature and the Hawking effect, Phys. Lett. A 79 (1980), no. 1, 23–24. MR 597629, DOI 10.1016/0375-9601(80)90306-0
Barry Simon, Real analysis, A Comprehensive Course in Analysis, Part 1, American Mathematical Society, Providence, RI, 2015. With a 68 page companion booklet. MR 3408971, DOI 10.1090/simon/001
Barry Simon, The $P(\phi )_{2}$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1974. MR 489552
Barry Simon and Raphael Høegh-Krohn, Hypercontractive semigroups and two dimensional self-coupled Bose fields, J. Functional Analysis 9 (1972), 121–180. MR 293451, DOI 10.1016/0022-1236(72)90008-0
W. de Sitter, On the relativity of inertia: Remarks concerning Einstein’s latest hypothesis, Proc. Kon. Ned. Acad. Wet. 19 (1917), 1217–1225.
W. de Sitter, On the curvature of space, Proc. Kon. Ned. Acad. Wet. 20 (1917) 229–243.
Chester Snow, Hypergeometric and Legendre functions with applications to integral equations of potential theory, National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, DC, 1952. MR 48145
Thomas Spencer and Francesco Zirilli, Scattering states and bound states in $\lambda {\cal P}(\phi )_{2}$, Comm. Math. Phys. 49 (1976), no. 1, 1–16. MR 413888
R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that, W. A. Benjamin, Inc., New York-Amsterdam, 1964. MR 161603
Alexander Strohmaier, Rainer Verch, and Manfred Wollenberg, Microlocal analysis of quantum fields on curved space-times: analytic wave front sets and Reeh-Schlieder theorems, J. Math. Phys. 43 (2002), no. 11, 5514–5530. MR 1936535, DOI 10.1063/1.1506381
K. Symanzik, Euclidean quantum field theory. I. Equations for a scalar model, J. Mathematical Phys. 7 (1966), 510–525. MR 187686, DOI 10.1063/1.1704960
K. Symanzik, A modified model of Euclidean quantum field theory, Courant Institute of Mathematical Sciences Report IMM-NYU 327, June 1964, and
K. Symanzik, Euclidean quantum field theory, in Local Quantum Theory, Varenna Lectures 1968, R. Jost (Editor), Academic Press, New York (1969) 152–226.
Reiji Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289–433 (French). MR 179296
M. Takesaki, Theory of Operator Algebras I, II, III, Series: Encyclopaedia of Mathematical Sciences (Book 124, 125, 127) Springer, 2001-2003.
W. Thirring, Lehrbuch der Mathematischen Physik, Band 2, Klassische Feldtheorie, Springer, 1990.
W. Thirring, Lehrbuch der Mathematischen Physik, Band 4, Quantenmechanik grosser Systeme, Springer, 1990.
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D14 (1976) 870–892.
V. S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups, Cambridge Studies in Advanced Mathematics, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1071183
Rainer Verch, Antilocality and a Reeh-Schlieder theorem on manifolds, Lett. Math. Phys. 28 (1993), no. 2, 143–154. MR 1229897, DOI 10.1007/BF00750307
Vasiliĭ Sergeevič Vladimirov, Methods of the theory of functions of many complex variables, The M.I.T. Press, Cambridge, Mass.-London, 1966. Translated from the Russian by Scripta Technica, Inc; Translation edited by Leon Ehrenpreis. MR 201669
N.Ya. Vilenken, and A.U. Klimyk, Representations of Lie Groups and Special Functions Vol. 1–3, Kluwer Academic Publishers, Dordrecht, 1991–1993.
A. M. Vershik and M. I. Graev, The structure of complementary series and special representations of the groups $\textrm {O}(n,1)$ and $\textrm {U}(n,1)$, Uspekhi Mat. Nauk 61 (2006), no. 5(371), 3–88 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 5, 799–884. MR 2328257, DOI 10.1070/RM2006v061n05ABEH004356
A. M. Veršik, I. M. Gel′fand, and M. I. Graev, Representations of the group $\textrm {SL}(2,\,\textbf {R})$, where $\textbf {R}$ is a ring of functions, Uspehi Mat. Nauk 28 (1973), no. 5(173), 83–128 (Russian). MR 393337
N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions. Vol. 1, Mathematics and its Applications (Soviet Series), vol. 72, Kluwer Academic Publishers Group, Dordrecht, 1991. Simplest Lie groups, special functions and integral transforms; Translated from the Russian by V. A. Groza and A. A. Groza. MR 1143783, DOI 10.1007/978-94-011-3538-2
N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions. Vol. 2, Mathematics and its Applications (Soviet Series), vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1993. Class I representations, special functions, and integral transforms; Translated from the Russian by V. A. Groza and A. A. Groza. MR 1220225, DOI 10.1007/978-94-017-2883-6
Robert M. Wald, On the Euclidean approach to quantum field theory in curved spacetime, Comm. Math. Phys. 70 (1979), no. 3, 221–242. MR 554493
A. G. Walker, Note on a distance invariant and the calculation of Ruse’s invariant, Proc. Edinburgh Math. Soc. (2) 7 (1942), 16–26. MR 7638, DOI 10.1017/S0013091500024251
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. (2) 40 (1939), no. 1, 149–204. MR 1503456, DOI 10.2307/1968551
Eugene P. Wigner, Group theory and its application to the quantum mechanics of atomic spectra, Pure and Applied Physics, Vol. 5, Academic Press, New York-London, 1959. Expanded and improved ed. Translated from the German by J. J. Griffin. MR 106711
Karen Yagdjian, Huygens’ principle for the Klein-Gordon equation in the de Sitter spacetime, J. Math. Phys. 54 (2013), no. 9, 091503, 18. MR 3135584, DOI 10.1063/1.4821115
Karen Yagdjian, Semilinear hyperbolic equations in curved spacetime, Fourier analysis, Trends Math., Birkhäuser/Springer, Cham, 2014, pp. 391–415. MR 3362031

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

The $\mathscr {P}(\varphi )_2$ Model on de Sitter Space

About this Title

Table of Contents

Abstract

References [Enhancements On Off] (What's this?)

memo_has_moved_text();The $\mathscr {P}(\varphi )_2$ Model on de Sitter Space

About this Title

Table of Contents

Abstract

References [Enhancements On Off] (What's this?)

The $\mathscr {P}(\varphi )_2$ Model on de Sitter Space