Algebraic field theory on curved manifolds

Olesen, Martin

doi:10.1090/S0002-9947-1995-1189546-1

Algebraic field theory on curved manifolds
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by Martin Olesen PDF

Trans. Amer. Math. Soc. 347 (1995), 2147-2160 Request permission

Abstract:

In this paper we set up an algebraic framework for the study of quantum field theory in a class of manifolds, which includes Minkowski space and the Kruskal spacetime. The formalism provides a unifying framework for studying problems of Bisognano-Wichmann type, e.g., Hawking radiation in black hole geometries. Analogously to flat spacetime, we establish a correspondence between isometries of certain wedge domains of spacetime and the modular structure of the local algebras. Under an ergodic hypothesis, the wedge algebras are shown to be type III factors as expected, and we derive a result concerning factorization of the equilibrium state. This result generalizes a similar one obtained by Sewell in [Ann. Phys. 141 (1982), 201-224]. Finally an example of a quantum field theory satisfying the basic axioms is constructed. The local algebras are field algebras of bosonic free field solutions to the Klein-Gordon equation twisted through a PCT-like conjugation, and we show that this model realizes the abstract properties developed on the axiomatic basis.

References

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

Operator algebras and quantum statistical mechanics

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
J. Dimock, Algebras of local observables on a manifold, Comm. Math. Phys. 77 (1980), no. 3, 219–228. MR 594301, DOI 10.1007/BF01269921

Scalar particle production in Schwartzscild and Rindler metrics

8

Klaus Fredenhagen, On the modular structure of local algebras of observables, Comm. Math. Phys. 97 (1985), no. 1-2, 79–89. MR 782959, DOI 10.1007/BF01206179
S. A. Fulling, Alternative vacuum states in static space-times with horizons, J. Phys. A 10 (1977), no. 6, 917–951. MR 469086, DOI 10.1088/0305-4470/10/6/014
Rudolf Haag, Local quantum physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. Fields, particles, algebras. MR 1182152, DOI 10.1007/978-3-642-97306-2
S. W. Hawking, Particle creation by black holes, Comm. Math. Phys. 43 (1975), no. 3, 199–220. MR 381625, DOI 10.1007/BF02345020

The large-scale structure of spacetime

R. Haag, N. M. Hugenholtz, and M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215–236. MR 219283, DOI 10.1007/BF01646342
Rudolf Haag and Daniel Kastler, An algebraic approach to quantum field theory, J. Mathematical Phys. 5 (1964), 848–861. MR 165864, DOI 10.1063/1.1704187
S. S. Horuzhy, Introduction to algebraic quantum field theory, Mathematics and its Applications (Soviet Series), vol. 19, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian by K. M. Cook. MR 1063850
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, DOI 10.1007/BF01212687
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186, DOI 10.1016/S0079-8169(08)60611-X
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696

A type

field theory

Gert K. Pedersen and Masamichi Takesaki, The Radon-Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973), 53–87. MR 412827, DOI 10.1007/BF02392262
Carlo Rigotti, Remarks on the modular operator and local observables, Comm. Math. Phys. 61 (1978), no. 3, 267–273. MR 503160, DOI 10.1007/BF01940769
Marc A. Rieffel and Alfons Van Daele, The commutation theorem for tensor products of von Neumann algebras, Bull. London Math. Soc. 7 (1975), no. 3, 257–260. MR 383096, DOI 10.1112/blms/7.3.257
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
Erling Størmer, Types of von Neumann algebras associated with extremal invariant states, Comm. Math. Phys. 6 (1967), 192–204. MR 225178, DOI 10.1007/BF01659976

Notes on black hole evaporation

14