Abstract
The quantum theory of both linear, and interacting fields on curved space-times is discussed. It is argued that generic curved space-time situations force the adoption of the algebraic approach to quantum field theory: and a suitable formalism is presented for handling an arbitrary quasi-free state in an arbitrary globally hyperbolic space-time.
For the interacting case, these quasi-free states are taken as suitable starting points, in terms of which expectation values of field operator products may be calculated to arbitrary order in perturbation theory. The formal treatment of interacting fields in perturbation theory is reduced to a treatment of “free” quantum fields interacting with external sources.
Central to the approach is the so-called two-current operator, which characterises the effect of external sources in terms of purely algebraic (i.e. representation free) properties of the source-free theory.
The paper ends with a set of “Feynman rules” which seems particularly appropriate to curved space-times in that it takes care of those aspects of the problem which are specific to curved space-times (and independent of interaction). Heuristically, the scheme calculates “in-in” rather than “in-out” matrix elements. Renormalization problems are discussed but not treated.
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Birrell, N. D., Taylor, J. G.: Analysis of interacting quantum fields in curved spacetime. J. Math. Phys. (to appear)
Kay, B. S.: Linear spin-zero quantum fields in external gravitational and scalar fields I. A one-particle structure for the stationary case. Commun. Math. Phys.62, 55–70 (1978)
Hajicek, P.: A new generating functional for expectation values of field operator products. Preprint, Berne (Oct. 1978)
DeWitt, B. S.: Quantum theory in curved space-time. Phys. Lett.19C, 295–357 (1975)
Parker, L.: The production of elementary particles by strong gravitational fields. In: Proceedings of the Symposium on Asymptotic Properties of Space-Time. New York: Plenum 1977
Parker, L.: Aspects of quantum field theory in curved space-time: Effective action and energy-momentum tensor. In: Proceedings of the NATO Advanced Study Institute on Gravitation: Recent Developments, (eds. M. Levy, S. Deser). New York: Plenum (in press) (Milwaukee preprint UWM-4867-78-9)
Isham, C. J.: Quantum field theory in curved space-time, an overview, In: 8th Texas Symposium on Relativistic Astrophysics. Imperial College preprint ICTP/76/5 (Jan. 1977)
Davies, P. C. W.: Quantum fields in curved space. In: GRG Finstein Centennial Volume. New York: Plenum (to be published) King's College, London Preprint
Hawking, S. W.: Particle creation by black holes. Commun. Math. Phys.43, 199–220 (1975)
Fulling, S. A.: Nonuniqueness of canonical field quantization in Riemannian space-time. Phys. Rev.D7, 2850–2862 (1973)
Emch, G. G.: Algebraic methods in statistical mechanics and quantum field theory. New York-London-Sydney-Toronto: Wiley 1972
Haag, R., Kastler, D.: An algebraic approach to quantum field theory: J. Math. Phys.5, 848–861 (1964)
Streater, R. F. (ed.): Mathematics of contemporary physics: New York-London: Academic 1972
Segal, I. E.: Mathematical problems of relativistic physics. Providence: Am. Math. Soc. 1963
Hajicek, P.: Observables for quantum fields on curved backgrounds. In: Differential geometric methods in mathematical physics II. Proceedings, Bonn, 1977 (eds. K. Bleuler, H. R. Petry, A. Reetz). Berlin, Heidelberg, New York: Springer 1978
Isham, C. J.: Quantum field theory in curved space-time—a general mathematical framework. In: Differential geometric methods in mathematical physics II. Proceedings, Bonn 1977 (eds. K. Bleuler, H. R. Petry, A. Reetz). Berlin, Heidelberg, New York: Springer 1978
Ashtekar, A., Magnon, A.: Quantum fields in curved space-time: Proc. Roy. Soc. Lond.A346, 375–394 (1975)
Kay, B. S.: Linear spin-zero quantum fields in external gravitational and scalar fields II: The description of dynamics in the generic case: Trieste preprint IC/77/143 (unpublished) (a superseded “first edition” of Sects. 1, 2 of present paper)
Boulware, D.: To appear in: Proceedings of NATO advanced study institute on gravitation: Recent developments, (eds. M. Levy, S. Deser). New York: Plenum (in press)
Schwinger, J.: Particles, sources and fields. Vols. I, II. Reading, Mass.: Addison-Wesley 1970, 1973
Schwinger J.: In: 1960 Brandeis Lectures: mimeographed lecture notes, Brandeis University
Hawking, S. W., Ellis, G. F. R.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973
Geroch, R.: Domain of dependence: J. Math. Phys.11, 437–449 (1970)
Seifert, H. J.: Kausal Lorentzräume. Doctoral Dissertation, Hamburg University 1968
Misner, C., Thorne, K., Wheeler, J. A.: Gravitation. San Francisco-London: Freeman 1973
Choquet-Bruhat, Y.: Hyperbolic partial differential equations on a manifold. In: Battelle Rencontres (eds. B. S. deWitt, J. A. Wheeler) New York: Benjamin 1967
Leray, J.: Hyperbolic partial differential equations. Princeton lecture notes. Princeton: Princeton University 1952 (mimeographed)
Lichnerowicz, A.: Propagateurs et commutateurs en relativité générale. Publ. I.H.E.S.10 (1961)
Segal, I. E.: Representations of the canonical commutation relations. In: Cargèse lectures on theoretical physics. New York: Gordon and Breach 1967
Kuchar, K.: Geometry of hyperspace I. J. Math. Phys.17, 777–791 (1976)
Slawny, J.: On factor representations and the C*-algebra of the canonical commutation relations: Commun. Math. Phys.24, 151–170 (1972)
Manuceau, J.: C*-algebra de relations de commutation. Ann. Inst. H. Poincaré8, 139 (1968)
Bogolubov, Logunov, Todorov: Introduction to axiomatic quantum field theory. Reading, Mass.: Benjamin 1975
Robinson, D. W.: The ground state of the Bose gas. Commun. Math. Phys.1, 159–174 (1965)
Robinson, D. W.: A theorem concerning the positive metric. Commun. Math. Phys.1, 89–94 (1965)
Manuceau, J., Verbeure, A.: Quasi-free states of the C.C.R. algebra and Bogolubov transformations. Commun. Math. Phys.9, 293–302 (1968)
Abers, E. S., Lee, B. W.: Gauge Theories. Phys. Lett.9C, 1–141 (1973)
Faddeev, L. D.: Article in: Methods in field theory. (eds. R. Balian, J. Zinn-Justin). Amsterdam-New York-Oxford: North Holland 1976
't Hooft, G., Veltman, M.: One-loop divergencies in the theory of gravitation. Ann. Inst. H. Poincaré,20, 69–94 (1974)
't Hooft, G.: Quantum Gravity: A fundamental problem and some radical ideas. Talk given at 8th GRG conference, Waterloo 1977 (to be published). Utrecht Preprint (May 1978)
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Communicated by R. Haag
Work partly supported by the Schweizerische Nationalfonds
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Kay, B.S. Linear spin-zero quantum fields in external gravitational and scalar fields. Commun.Math. Phys. 71, 29–46 (1980). https://doi.org/10.1007/BF01230084
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DOI: https://doi.org/10.1007/BF01230084