Application of linear hyperbolic PDE to linear quantum fields in curved spacetimes : especially black holes, time machines and a new semi-local vacuum concept
Journées équations aux dérivées partielles (2000), article no. 9, 19 p.

Several situations of physical importance may be modelled by linear quantum fields propagating in fixed spacetime-dependent classical background fields. For example, the quantum Dirac field in a strong and/or time-dependent external electromagnetic field accounts for the creation of electron-positron pairs out of the vacuum. Also, the theory of linear quantum fields propagating on a given background curved spacetime is the appropriate framework for the derivation of black-hole evaporation (Hawking effect) and for studying the question whether or not it is possible in principle to manufacture a time-machine. It is a well-established metatheorem that any question concerning such a linear quantum field may be reduced to a definite question concerning the corresponding classical field theory (i.e. linear hyperbolic PDE with non-constant coefficients describing the background in question) - albeit not necessarily a question which would have arisen naturally in a purely classical context. The focus in this talk will be on the covariant Klein-Gordon equation in a fixed curved background, although we shall draw on analogies with other background field problems and with the time-dependent harmonic oscillator. The aim is to give a sketch-impression of the whole subject of Quantum Field Theory in Curved Spacetime, focussing on work with which the author has been personally involved, and also to mention some ideas and work-in-progress by the author and collaborators towards a new “semi-local” vacuum construction for this subject. A further aim is to introduce, and set into context, some recent advances in our understanding of the general structure of quantum fields in curved spacetimes which rely on classical results from microlocal analysis.

     author = {Kay, Bernard},
     title = {Application of linear hyperbolic {PDE} to linear quantum fields in curved spacetimes : especially black holes, time machines and a new semi-local vacuum concept},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {9},
     publisher = {Universit\'e de Nantes},
     year = {2000},
     zbl = {01808699},
     mrnumber = {2001h:83042},
     language = {en},
     url = {}
AU  - Kay, Bernard
TI  - Application of linear hyperbolic PDE to linear quantum fields in curved spacetimes : especially black holes, time machines and a new semi-local vacuum concept
JO  - Journées équations aux dérivées partielles
PY  - 2000
DA  - 2000///
PB  - Université de Nantes
UR  -
UR  -
UR  -
LA  - en
ID  - JEDP_2000____A9_0
ER  - 
Kay, Bernard. Application of linear hyperbolic PDE to linear quantum fields in curved spacetimes : especially black holes, time machines and a new semi-local vacuum concept. Journées équations aux dérivées partielles (2000), article  no. 9, 19 p.

[1] S. W. Hawking. Particle creation by black holes. Commun. Math. Phys., 43, (1975), pp. 199-220. | MR 52 #2517

[2] R. Haag. Local Quantum Physics. Springer-Verlag, TMP, 1992. | MR 94d:81001 | Zbl 0777.46037

[3] B. S. Kay, R. M. Wald. Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep., 207 No. 2, (1991), pp. 49-136. | MR 93b:81189 | Zbl 0861.53074

[4] M. J. Radzikowski. The Hadamard Condition and Kay's Conjecture in (Axiomatic) Quantum Field Theory on Curved Space-time. Ph.D. dissertation. Princeton University, 1992. Available through University Microfilms International, 300 N. Zeeb Road, Ann Arbor, Michigan 48106 U.S.A.

[5] J. J. Duistermaat, L. Hörmander. Fourier integral operators I. Acta Mathematica, 127, (1971), pp. 79-183. | MR 52 #9299 | Zbl 0212.46601

[6] J. J. Duistermaat, L. Hörmander. Fourier integral operators II. Acta Mathematica, 128, (1972), pp. 183-269. | MR 52 #9300 | Zbl 0232.47055

[7] L. Hörmander. The Analysis of Linear Partial Differential Operators I, II-IV. Springer-Verlag, 1990, 1983-85. | Zbl 0521.35002

[8] R. Brunetti, K. Fredenhagen and M. Köhler. The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys., 180, (1996), pp. 633-652. | MR 98b:81153 | Zbl 0923.58052

[9] W. Junker. Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved space-time. Rev. Math. Phys., 8, (1996), pp. 1091-1159 and Erratum (to appear). | MR 98b:81158 | Zbl 0869.53053

[10] W. Junker. Application of microlocal analysis to the theory of quantum fields interacting with a gravitational field. (preprint) hep-th/9701039. | Zbl 0882.35103

[11] R. Verch. Wavefront sets in algebraic quantum field theory. Commun. Math. Phys., 205, (1999), pp. 337-367 | MR 2000j:81147 | Zbl 01379895

[12] R. Brunetti, K. Fredenhagen. Microlocal analysis and interacting quantum field theories : Renormalization on physical backgrounds. Commun. Math. Phys., 208, (2000), pp. 623-661. | MR 2001g:81176 | Zbl 1040.81067

[13] H. Sahlmann and R. Verch, Passivity and microlocal spectrum condition. (preprint) math-ph/0002021. | Zbl 1010.81046

[14] C. J. Fewster. A general worldline quantum inequality. Class. Quantum Grav., 17, (2000), pp. 1897-1911. | MR 2001k:81186 | Zbl 1079.81555

[15] N. D. Birrell, P. C. W. Davies. Quantum Fields in Curved Space. Cambridge University Press, 1982. | MR 83h:81061 | Zbl 0476.53017

[16] R. M. Wald. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago University Press, 1994. | MR 95i:81176 | Zbl 0842.53052

[17] N. Dencker. On the propagation of polarization sets for systems of real principal type. J. Funct. Anal., 46, (1982), pp. 351-372. | MR 84c:58081 | Zbl 0487.58028

[18] K. Kratzert. Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime. (preprint) math-ph/0003015. | Zbl 0958.81050

[19] S. Hollands. The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes. (preprint) gr-qc/9906076. | Zbl 0976.58023

[20] A. Bachelot. Creation of fermions at the charged black hole horizon. prépublication de l'Unité CNRS 5466, 1999.

[21] A. Bachelot. Gravitational scattering of electromagnetic field by Schwarzschild black-hole. Ann. Inst. Henri-Poincaré - Physique théorique, 54, (1991), pp. 261-320. | Numdam | MR 92k:83030 | Zbl 0743.53037

[22] J.-P. Nicolas. Scattering of linear Dirac fields by a spherically symmetric black hole. Ann. Inst. Henri Poincaré - Physique théorique, 62, (1995), pp. 145-179. | Numdam | MR 96c:81237 | Zbl 0826.53072

[23] F. Melnyk. Wave operators for the massive charged linear Dirac field on the Reissner-Nordström metric. Class. Quantum Grav., 17, (2000), pp. 2281-2296. | MR 2001j:83015 | Zbl 0949.83032

[24] J. Leray. Hyperbolic partial differential equations. Princeton Lecture Notes (mimeographed) Princeton University, 1952.

[25] R. Geroch. Domain of dependence. J. Math. Phys., 11, (1970), pp. 437-449. | MR 42 #5585 | Zbl 0189.27602

[26] S. W. Hawking, G. F. R. Ellis. The Large Scale Structure of Space-time. Cambridge University Press, 1973. | MR 54 #12154 | Zbl 0265.53054

[27] J. Dieckmann. Cauchy surfaces in a globally hyperbolic space-time. J. Math. Phys. 29, (1988), pp. 578-579. | MR 89e:83009 | Zbl 0644.53061

[28] A. Lichnérowicz. Propagateurs et commutateurs en relativité générale. Publ. I.H.E.S., 10, (1961), pp. 293-344. | Numdam | Zbl 0098.42607

[29] Y. Choquet-Bruhat. Hyperbolic partial differential equations on a manifold. In : Battelle Rencontres (eds. C. deWitt-Morette, J. A. Wheeler) Benjamin (New-York) 1967.

[30] J. Dimock. Algebras of local observables on a manifold. Commun. Math. Phys., 77, (1980), pp. 219-228. | MR 82i:81071 | Zbl 0455.58030

[31] B. S. Kay. Talk at 10th International Conference on General Relativity and Gravitation (Padova, 1983) (see Workshop Chairman's Report by A. Ashtekar. in the proceedings eds. B. Bertotti et al. Reidel (Dordrecht), 1984., pp. 453-456.).

[32] B. S. Kay. Quantum field theory in curved spacetime. In : Differential Geometrical Methods in Theoretical Physics. eds. K. Bleuler, M. Werner. Reidel (Dordrecht), 1988, pp. 373-393. | MR 89k:81104 | Zbl 0872.53052

[33] R. Verch. Local definiteness, primarity and quasi equivalence of quasifree Hadamard states in curved spacetime. Commun. Math. Phys., 160, (1994), pp. 507-536. | MR 95a:81159 | Zbl 0790.53077

[34] R. M. Wald. On the trace anomaly of a conformally invariant quantum field on curved spacetime. Phys. Rev., D17, (1978), pp. 1477-1484. | MR 57 #2334

[35] M. J. Radzikowski. Micro-local approach to the Hadamard condition in quantum field theory on curved spacetime. Commun. Math. Phys., 179, (1996), pp. 529-553. | MR 97f:81107 | Zbl 0858.53055

[36] M. J. Radzikowski. A local-to-global singularity theorem for quantum field theory on curved space-time. Commun. Math. Phys., 180, (1996), pp. 1-22. (With an Appendix by R. Verch.). | MR 97f:81108 | Zbl 0874.58079

[37] A. S. Wightman. Introduction to some aspects of the relativistic dynamics of quantum fields. In : 1964 Cargèse Lectures in Theoretical Physics : High Energy Electromagnetic Interactions and Field Theory. ed. M. Lévy. Gordon and Breach (New York), 1967.

[38] S. A. Fulling, S. N. M. Ruijsenaars. Temperature, periodicity and horizons. Phys. Rep., 152, (1987), pp. 135-176. | MR 89a:81091

[39] G. Gonnella, B. S. Kay. Can locally Hadamard quantum states have non-local singularities ? Class. Quantum Grav., 6, (1989), pp. 1445-1454. | MR 90k:81156 | Zbl 0678.53082

[40] W. G. Unruh. Notes on black hole evaporation. Phys. Rev., D14, (1976), pp. 870-892.

[41] J. J. Bisognano, E. H. Wichmann. On the duality condition for a Hermitian scalar field. J. Math. Phys., 16, (1975), pp. 985-1007, and J. J. Bisognano, E. H. Wichmann. On the duality condition for quantum fields. J. Math. Phys., 17, (1976), pp. 303-321. | MR 55 #11846 | Zbl 0316.46062

[42] R. Haag, H. Narnhofer and U. Stein. On quantum field theory in gravitational background. Commun. Math. Phys., 94, (1984), pp. 219-238. | MR 86c:81069

[43] J. Dimock, B. S. Kay. Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric I. Ann. Phys. (NY), 175, (1987), pp. 366-426. | MR 88h:83043 | Zbl 0628.53080

[44] K. Fredenhagen, R. Haag. On the derivation of Hawking radiation associated with the formation of a black hole. Commun. Math. Phys. 127, (1990), pp. 273-284. | MR 90m:83057 | Zbl 0692.53040

[45] A. Bachelot. Scattering of scalar fields by spherical gravitational collapse. J. Math. Pures Appl., 76 (1997), pp. 155-210. | MR 98a:83068 | Zbl 0872.53066

[46] A. Bachelot. Quantum vacuum polarization at the black hole horizon. Ann. Inst. Henri-Poincaré - Physique théorique, 67, (1997), pp. 181-222. | Numdam | MR 98i:83046 | Zbl 0897.53064

[47] A. Bachelot. The Hawking effect. Ann. Inst. Henri Poincaré - Physique théorique, 70, (1999), 41-99. | Numdam | MR 2000b:83041 | Zbl 0919.53034

[48] B. S. Kay. Sufficient conditions for quasifree states and an improved uniqueness theorem for quantum field theory on spacetimes with horizons. J. Math. Phys., 34, (1993), pp. 4519-4539. | MR 94k:81191 | Zbl 0809.53076

[49] W. Rindler. Kruskal space and the uniformly accelerated frame. Am. J. Phys., 34, (1966) pp. 1174-1178.

[50] B. S. Kay. The principle of locality and quantum field theory on (non-globally hyperbolic) curved spacetimes. Rev. Math. Phys. Special Issue Dedicated to the 70th Birthday of R. Haag, (1992), pp. 167-195. | MR 93k:81157 | Zbl 0779.53052

[51] S. W. Hawking. The chronology protection conjecture. Phys. Rev., D46, (1992), pp. 603-611. | MR 93c:83086

[52] P. T. Chrusciel. A remark on differentiability of Cauchy horizons. Class. Quant. Grav., 15, (1998), pp. 3845-3848. | MR 2000d:83094 | Zbl 0933.83004

[53] M. S. Morris, K. S. Thorne and U. Yurtsever. Wormholes, time machines, and the weak energy condition. Phys. Rev. Lett., 61, (1988), 1446-1449.

[54] B. S. Kay, M. J. Radzikowski and R. M. Wald. Quantum field theory on space-tims with a compactly generated Cauchy horizon. Commun. Math. Phys., 183, (1997), pp. 533-556. | MR 98i:81170 | Zbl 0883.53057

[55] C. R. Cramer and B. S. Kay. Stress energy must be singular on the Misner space horizon even for automorphic fields. Class. Quant. Grav., 13, (1996), pp. L143-L149. | MR 98b:81154 | Zbl 0865.53068

[56] C. R. Cramer and B. S. Kay. The thermal and two-particle stress-energy must be ill-defined on the 2-d Misner space chronology horizon. Phys. Rev., D57, (1998), pp. 1052-1056. | MR 99c:83068

[57] M. Visser. The reliability horizon. (preprint) gr-qc/9710020. (To be published in the proceedings of 8th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 8), Jerusalem, Israel, 22-27 June 1997).

[58] P. Hajicek. On quantum field theory in curved space-time. Nuovo Cimento, B33, (1976), 597-612, and P. Hajicek. Theory of particle detection in curved spacetimes. Phys. Rev., D15, (1977), 2757-2774.

[59] B. S. Kay, A. R. Borrott, C. R. Cramer. Geometrical foundation for the trace anomaly of the 2-D quantum wave equation and the question of a local vacuum for quantum field theory in curved spacetime. (in preparation).

[60] S. Hollands, B. S. Kay. A semi-local vacuum concept for quantum field theory in curved spacetime. (in preparation)