×

Classical wave operators and asymptotic quantum field operators on curved space-times. (English) Zbl 0539.35063

Summary: We consider a class of Lorentzian metrics on \({\mathbb{R}}^ 4\) which are stationary at time-like infinity and Minkowskian either at space-like or time-like infinity. For these metrics, we construct wave operators for the classical Klein-Gordon equation. For certain transient cases we also construct inverse wave operators. Given the classical wave operators, we show how to construct in and out field operators for the quantum Klein- Gordon equation and a particle interpretation for this theory. Finally, we give several results of a general nature on the construction of classical and quantum scattering operators paying special attention to the stationary case. The methods and results of the paper could be generalized to other external field problems (not just gravitational).

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35P25 Scattering theory for PDEs
81T08 Constructive quantum field theory
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] N. Bogolubov , A. Logunov , and I. Todorov , Introduction to Axiomatic Quantum Field Theory , Benjamin , Reading, Mass ., 1975 . · Zbl 1114.81300
[2] P. Bongaarts , The electron-positron field coupled to external electromagnetic potentials as an elementary C*-algebra theory , Ann. Phys. (N. Y.) , t. 56 , 1970 , p. 108 . MR 260291
[3] P. Bongaarts , S. Ruijsenaars , The Klein paradox as a many particle problem , Annals of Physics , t. 101 , 1976 , p. 289 . MR 443657
[4] O. Bratteli and D. Robinson , Operator Algebras and Quantum Statistical Mechanics II , Berlin - Heidelberg - New York , Springer , 1981 . MR 611508 | Zbl 0463.46052 · Zbl 0463.46052
[5] Y. Choquet-Bruhat , Hyperbolic differential equations on a manifold , in Battelle Rencontres , DeWitt and Wheeler, eds. Benjamin , N. Y ., 1968 . MR 239299 | Zbl 0169.43202 · Zbl 0169.43202
[6] Y. Choquet-Bruhat , D. Christodoulou , and M. Francaviglia , On the wave equation in curved space-time , Ann. Inst. Henri Poincaré , t. A 21 , 1979 , p. 339 . Numdam | MR 574143 | Zbl 0454.58016 · Zbl 0454.58016
[7] P. Cotta-Ramusino , W. Krüger , R. Schrader , Quantum scattering by external metrics and Yang-Mills potentials , Ann. Inst. Henri Poincaré , t. A 21 , 1979 , p. 43 . Numdam | MR 557051 | Zbl 0441.35055 · Zbl 0441.35055
[8] J. Dimock , Scalar quantum field in an external gauge field , J. Math. Phys. , t. 20 , 1979 , p. 1791 . MR 543917 | Zbl 0425.35075 · Zbl 0425.35075
[9] J. Dimock , Scalar quantum field in an external gravitational field , J. Math. Phys. , t. 20 , 1979 , p. 2549 . MR 553516 | Zbl 0455.35105 · Zbl 0455.35105
[10] J. Dimock , Algebras of local observables on a manifold , Commun. Math. Phys. , t. 77 , 1980 , p. 219 . Article | MR 594301 | Zbl 0455.58030 · Zbl 0455.58030
[11] E. Furlani , Suny at Buffalo thesis ( 1982 ).
[12] S. Hawking and G. Ellis , The Large Scale Structure of Space-time , Cambridge University Press , Cambridge , 1973 . MR 424186 | Zbl 0265.53054 · Zbl 0265.53054
[13] Articles by C. Isham and P. Hajicek in: Differential Geometric Methods in Mathematical Physics II , Bleuler, Petry, Reetz (eds). Springer Lecture Notes in Mathematics , t. 676 , Springer-Verlag , Berlin - Heidelberg - N. Y ., 1978 . MR 519605
[14] B. Kay , Linear spin zero quantum fields in external gravitational and scalar . fields I , Commun. Math. Phys. , t. 62 , 1978 , p. 55 . Article | MR 506366
[15] B. Kay , Linear spin zero quantum fields in external gravitational and scalar fields II , Commun. Math. , Phys. , t. 71 , 1980 , p. 29 . Article | MR 556899
[16] B. Kay , A Uniqueness result in the Segal-Weinless approach to linear Bose fields , J. Math. Phys. , t. 20 , 1979 , p. 1712 . MR 543906
[17] B. Kay , Quantum fields in curved space-times and scattering theory , in: Differential Geometric Methods in Mathematical Physics. Proceedings , 1980 . Andersson, Doebner, Petry (eds). Springer Lecture Notes in Mathematics , t. 905 , Springer-Verlag . Berlin - Heidelberg - N. Y ., 1982 . Zbl 0544.35078 · Zbl 0544.35078
[18] J. Leray , Hyperbolic Differential Equations , Princeton Lectures Notes , 1953 (unpublished). MR 80849 · JFM 58.0415.01
[19] L. Lundberg , Relativistic quantum theory for charged spinless particles in external vector fields , Commun. Math. Phys. , t. 31 , 1973 , p. 295 . Article | MR 378634 · Zbl 1125.81311
[20] S. Nelson , L2 asymptotes for the Klein-Gordon equation , Proc. Am. Math. Soc. , t. 27 , 1971 , p. 110 . MR 271561 | Zbl 0214.10201 · Zbl 0214.10201
[21] G. Nenciu , Strong external fields in Q. E. D. rigorous results . Proceedings of the International Summer School in Heavy Ion Physics , Predeal, Roumania , 1978 .
[22] S. Paneitz and I. Segal , Quantization of wave equations and hermitian structures in partial differential varieties , Proc. Nat. Acad. Sci. U. S. A. , t. 12 , 1980 , p. 6943 . MR 603063 | Zbl 0469.35065 · Zbl 0469.35065
[23] M. Reed , B. Simon , Methods of Modern Mathematical Physics III , Academic Press , N. Y ., 1979 . MR 529429 | Zbl 0405.47007 · Zbl 0405.47007
[24] S. Ruijsenaars , Gauge invariance and implementability of the S operator for spin 0 and spin 1/2 in time dependent external fields . Journal of Functional Analysis , t. 33 , 1979 , p. 47 . MR 545384
[25] I. Segal , in Applications of Mathematics to Problems in Theoretical Physics , F. Lurçat (ed.), Gordon and Breach , N. Y ., 1967 .
[26] R. Seiler , Particles with spin S \geq 1 in an external field, in Invariant Wave Equations , Lecture Notes in Physics , t. 73 , Springer-Verlag , Berlin - Heidelberg - New York , 1978 . MR 495993
[27] D. Shale , Linear symmetries of free Bose fields . Trans. Am. Math. Soc. , t. 103 , 1962 , p. 149 . MR 137504 | Zbl 0171.46901 · Zbl 0171.46901
[28] J. Slawny , On factor representations and the C*-algebra of the canonical commutation relations , Commun. Math. Phys. , t. 24 , 1972 , p. 151 . Article | MR 293942 | Zbl 0225.46068 · Zbl 0225.46068
[29] R. Wald , Existence of the S-matrix in quantum field theory in curved space-time , Ann. Phys. (N. Y.) , t. 118 , 1979 , p. 490 . MR 533774
[30] M. Weinless , Existence and uniqueness of the vacuum for linear quantized fields , J. Funct. Anal. , t. 4 , 1969 , p. 350 . MR 253687 | Zbl 0205.57504 · Zbl 0205.57504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.