Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Algebraic field theory on curved manifolds


Author: Martin Olesen
Journal: Trans. Amer. Math. Soc. 347 (1995), 2147-2160
MSC: Primary 81T05; Secondary 46L60, 47D45, 81T20
DOI: https://doi.org/10.1090/S0002-9947-1995-1189546-1
MathSciNet review: 1189546
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [A] H. Araki, A lattice of von Neumann algebras associated with the quantum theory of a free Bose field, J. Math. Phys. 4 (1963), 1343-1362. MR 0158666 (28:1889)
  • [BR] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics $ 1$-$ 2$, Springer, New York, 1967.
  • [BW] J. J. Bisognano and E. H. Wichmann, On the duality condition for a Hermitean scalar field, J. Math. Phys. 12 (1975), 985-1007. MR 0438943 (55:11846)
  • [D] J. Dimock, Algebras of local observables on a manifold, Comm. Math. Phys. 77 (1980), 219-228. MR 594301 (82i:81071)
  • [Da] P. C. W. Davies, Scalar particle production in Schwartzscild and Rindler metrics, J. Phys. A 8 (1975), 609-617.
  • [F] K. Fredenhagen, On the modular structure of local algebras of observables, Comm. Math. Phys. 97 (1985), 79-90. MR 782959 (86f:81069)
  • [Fu] S. A. Fulling, Alternative vacuum states in static spacetimes with horizons, J. Phys. A 10 (1977), 917-951. MR 0469086 (57:8887)
  • [H] R. Haag, Local quantum physics, Springer, New York, 1992. MR 1182152 (94d:81001)
  • [Ha] S. W. Hawking, Particle creation by black holes, Comm. Math. Phys. 43 (1975), 199-220. MR 0381625 (52:2517)
  • [HE] S. W. Hawking and G. E. R. Ellis, The large-scale structure of spacetime, Cambridge University Press, London, 1973.
  • [HHW] R. Haag, N. M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215-236. MR 0219283 (36:2366)
  • [HK] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5 (1964), 848-861. MR 0165864 (29:3144)
  • [Ho] S. S. Horozhy, Introduction to algebraic quantum field theory, Kluwer, Holland, 1990. MR 1063850 (91g:81070)
  • [Ka] B. S. Kay, The double-wedge algebra for quantum fields on Schwartzschild and Minkowski space-times, Comm. Math. Phys. 100 (1985), 57-91. MR 796162 (86i:81101)
  • [KR] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. I, II, Academic Press, 1986. MR 859186 (88d:46106)
  • [KS] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, 1980. MR 567696 (81g:49013)
  • [O] M. Olesen, A type $ {\text{II}}{{\text{I}}_\lambda }$ field theory, Preprint 1993.
  • [PT] G. K. Pedersen and M. Takesaki, The Radon-Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973), 53-87. MR 0412827 (54:948)
  • [R] C. Rigotti, Remarks on the modular operator and local observables, Comm. Math. Phys. 37 (1974), 273-286. MR 0503160 (58:19986)
  • [RvD] M. A. Rieffel and A. van Daele, The commutation theorem for tensor products of von Neumann algebras, London Math. Soc. 7 (1975), 257-260. MR 0383096 (52:3977)
  • [Se] G. L. Sewell, Quantum fields on manifolds: PCT and gravitationally induced thermal states, Ann. Phys. 141 (1982), 201-224. MR 673980 (84j:81100)
  • [St] E. Størmer, Types of von Neumann algebras associated with extremal invariant states, Comm. Math. Phys. 6 (1967), 194-204. MR 0225178 (37:773)
  • [U] W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976), 870-892.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 81T05, 46L60, 47D45, 81T20

Retrieve articles in all journals with MSC: 81T05, 46L60, 47D45, 81T20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1189546-1
Keywords: Quantum field theory, space-time manifolds, von Neumann algebras, modular theory
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society