Abstract
We prove that the GNS-representations of quasifree, Hadamard states on the Weyl-algebra of the quantized Klein-Gordon field propagating in an arbitrary globally hyperbolic spacetime are locally quasiequivalent. We also show that these representations satisfy local primarity and local definiteness if the spacetime is assumed to be ultrastatic. This implies that the local von Neumann algebras associated with these representations are typeIII 1-factors for sufficiently small regions in ultrastatic spacetimes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[Ara] Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of a free Bose field. J. Math. Phys.4, 1343–1362 (1963)
[AY] Araki, H., Yamagami, S.: On quasi-equivalence of quasifree states of the canonical commutation relations. Publ RIMS, Kyoto Univ.18, 283–338 (1982)
[Ban] Bannier, U.: On generally covariant quantum field theory and generalized causal and dynamical structures. Commun. Math. Phys.118, 163–170 (1988)
[Buc] Buchholz, D.: Product states for local algebras. Commun. Math. Phys.36, 287–304 (1974)
[BW] Baumgärtel, H., Wollenberg, M.: Causal nets of operator algebras. Berlin: Akademie Verlag 1992
[CFKS] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer 1987
[Che] Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal.12, 401–414 (1973)
[D] Dieudonné, J.: Treatise on analysis. Vol. III. New York: Academic Press 1972
[Die] Dieckmann, J.: Cauchy-surfaces in globally hyperbolic spacetime. J. Math. Phys.29, 578 (1988)
[Dim] Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys.77 219–228 (1980)
[Dix] Dixmier, J.: LesC *-algèbres et leurs représentations. Paris: Gauthier-Villars, 1964
[FH1] Fredenhagen, K., Haag, R.: Generally covariant quantum field theory and scaling limits. Commun. Math. Phys.108, 91–115 (1987)
[FH2] Fredenhagen, K., Haag, R.: On the derivation of Hawking radiation associated with the formation of a black hole. Commun. Math. Phys.127, 273–284 (1990)
[FNW] Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime. II. Ann. Phys. (N.Y.)136, 243–272 (1981)
[Fri] Friedlander, F.G.: The wave equation on a curved spacetime. Cambridge: Cambridge University Press 1975
[FSW] Fulling, S.A., Sweeny, M., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime. Commun. Math. Phys.63, 257–264 (1978)
[Ful] Fulling, S.A.: Aspects of quantum field theory in curved spacetime. Cambridge: Cambridge University Press 1989
[XXX] Garabedian, P.R.: Partial differential equations. New York: Wiley 1964
[GK] Gonnella, G., Kay, B.S.: Can locally Hadamard quantum states have non-local singularities? Class. Quantum Gravity6, 1445–1454 (1989)
[H1] Haag, R.: Quantum physics and gravitation. In: Velo, G., Wightman, A.S. (eds.): Constructive quantum field theory II. New York: Plenum Press 1990
[H2] Haag, R.: Local quantum physics. Berlin: Springer 1992
[Haw] Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys.43, 199–220 (1975)
[Hei] Heinz, E.: Beiträge zur Störungstheorie der Spektralzerlegung. Math. Ann.123, 415–438 (1951)
[HNS] Haag, R., Narnhofer, H., Stein, U.: On quantum field theory in gravitational background. Commun. Math. Phys.94, 219–238 (1984)
[Hör] Hörmander, L.: Linear partial differential operators. Berlin: Springer 1969
[Kay1] Kay, B.S.: Linear spin-zero quantum fields in external gravitational and scalar fields. Commun. Math. Phys.62, 55–70 (1978)
[Kay2] Kay, B.S.: In: Proc. 10th Inter. Conf. on General Relativity and Gravitation (Padova, 1983), eds. B. Bertotti et al. (Reidel, Dordrecht, 1984), talk (see Workshop Chairman's report by A. Ashtekar, pp. 453–456); see also: Kay, B.S.: Casimir effect in quantum field theory. Phys. Rev. D20, 3052–62 (1979)
[Kay3] Kay, B.S.: The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes. Commun. Math. Phys.100, 57–81 (1985)
[Kay4] Kay, B.S.: A uniqueness result for quasifree KMS states. Helv. Phys. Acta58, 1017–1029 (1985)
[Kay5] Kay, B.S.: Quantum field theory in curved spacetime. In: Bleuler, K., Werner, M. (eds.): Differential geometric methods in theoretical physics. Dordrecht: Kluwer Acadamic Publishers 1988
[KW] Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep.207, 49–136 (1991)
[LR] Lüders, C., Roberts, J.E.: Local quasiequivalence and adiabatic vacuum states. Commun. Math. Phys.134, 29–63 (1990)
[LRT] Leyland, P., Roberts, J.E., Testard, D.: Duality for quantum free fields. Preprint, Marseille (1978)
[MV] Manuceau, J., Verbeure, A.: Quasifree states of the CCR-algebra and Bogoliubov transformations. Commun. Math. Phys.9, 293–302 (1968)
[Rad] Radzikowski, M.J.: The Hadamard condition and Kay's conjecture in (axiomatic) quantum field theory on curved space-time. Doctoral Dissertation, Princeton University, 1992
[RS] Reed, M., Simon, B.: Methods of modern mathematical physics. II. New York: Academic Press 1975
[S] Schwartz, L.: Théorie des distributions. Paris: Hermann 1957
[Wa1] Wald, R.M.: The back reaction effect in particle creation in curved spacetime. Commun. Math. Phys.54, 1–19 (1977)
[Wal2] Wald, R.M.: Trace anomaly of a conformally invariant quantum field in curved spacetime. Phys. Rev. D17, 1477–84 (1978)
[Wal3] Wald, R.M.: General relativity. Chicago: University of Chicago Press 1984
[Wei] Weidmann, J.: Linear operators in Hilbert spaces. Berlin, Heidelberg, New York: Springer 1980
[Wol] Wollenberg, M.: Scaling limits and type of local algebras over curved spacetime. In: Arveson, W.B. et al. (eds.): Operator algebras and topology, Pitman Research Notes in Mathematics Series 270. Harlow: Longman 1992
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Supported by the DFG, SFB 288 “Differentialgeometrie und Quantenphysik”
Rights and permissions
About this article
Cite this article
Verch, R. Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved spacetime. Commun.Math. Phys. 160, 507–536 (1994). https://doi.org/10.1007/BF02173427
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02173427