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Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field. (English) Zbl 0606.60060
A rigorous path integral representation of the solution of the Cauchy problem for the pure-imaginary-time Schrödinger equation \(\partial_ 1\psi (t,x)=-[H-mc^ 2]\psi (t,x)\) is established. H is the quantum Hamiltonian associated, via the Weyl correspondence, with the classical Hamiltonian \([(cp-eA(x))^ 2+m^ 2c^ 4]^{1/2}+e\Phi (x)\) of a relativistic spinless particle in an electromagnetic field. The problem is connected with a time homogeneous Lévy process.

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
81P20 Stochastic mechanics (including stochastic electrodynamics)
82B10 Quantum equilibrium statistical mechanics (general)
83C50 Electromagnetic fields in general relativity and gravitational theory
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[1] Berezin, F.A., ?ubin, M. A.: Symbols of operators and quantization. Colloq. Math. Soc. Janos Bolyai 5. Hilbert Space Operators, 21-52, Tihany 1970
[2] Calderón, A. P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nat. Acad. Sci. USA69, 1185-1187 (1972) · Zbl 0244.35074 · doi:10.1073/pnas.69.5.1185
[3] Daubechies, I., Lieb, E.H.: one-electron relativistic molecules with Coulomb interaction. Commun. Math. Phys.90, 497-510 (1983) · Zbl 0946.81522 · doi:10.1007/BF01216181
[4] Daubechies, I.: One electron molecules with relativistic kinetic energy: Properties of the discrete spectrum. Commun. Math. Phys.94, 523-535 (1984) · doi:10.1007/BF01403885
[5] Erdélyi, A.: Higher transcendental functions, Vol. 2. New York: McGraw-Hill 1953 · Zbl 0051.30303
[6] Erdélyi, A.: Tables of integral transforms, Vol. 2, New York: McGraw-Hill 1954 · Zbl 0055.36401
[7] Garrod, C.: Hamiltonian path-integral methods. Rev. Mod. Phys.38, 483-494 (1966) · Zbl 0192.30403 · doi:10.1103/RevModPhys.38.483
[8] Gaveau, B.: Representation formulas of the Cauchy problem for hyperbolic systems generalizing Dirac system. J. Funct. Anal.58, 310-319 (1984) · Zbl 0562.35055 · doi:10.1016/0022-1236(84)90045-4
[9] Gaveau, B., Jacobson, T., Kac, M., Schulman, L. S.: Relativistic extension of the analogy between quantum mechanics and Brownian motion. Phys. Rev. Lett.53 (5), 419-422 (1984) · doi:10.1103/PhysRevLett.53.419
[10] Grossmann, A., Loupias, G., Stein, E. M.: An algebra of pseudodifferential operators and quantum mechanics in phase space. Ann. Inst. Fourier18, 343-368 (1968) · Zbl 0176.45102
[11] Herbst, I. W.: Spectral theory of the operator (p 2+m 2)1/2?Ze 2/r. Commun. Math. Phys.53, 285-294 (1977) · Zbl 0375.35047 · doi:10.1007/BF01609852
[12] Hörmander, L.: The analysis of linear partial differential operators III. Berlin, Heidelberg, New York, Tokyo: 1985 · Zbl 0601.35001
[13] Ichinose, T., Tamura, H.: Propagation of a Dirac particle. A path integral approach. J. Math. Phys.25, 1810-1819 (1984) · Zbl 0557.35101 · doi:10.1063/1.526360
[14] Ichinose, T.: Path integral for a hyperbolic system of the first order. Duke Math. J.51, 1-36 (1984) · Zbl 0542.35047 · doi:10.1215/S0012-7094-84-05101-9
[15] Ichinose, T.: Path integral formulation of the propagator for a two-dimensional Dirac particle. Physica124A, 419-426 (1984) · Zbl 0598.35104
[16] Ikeda N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam, Tokyo: North-Holland/Kodansha 1981 · Zbl 0495.60005
[17] Itô, K.: Stochastic processes. Lecture Notes Series, 16, Aårhus University 1969
[18] Kumano-go, H.: Pseudo-differential operators of multiple symbol and the Calderón-Vaillancourt theorem. J. Math. Soc. Jpn27, 113-120 (1975) · Zbl 0294.35068 · doi:10.2969/jmsj/02710113
[19] Kumano-go, H.: Pseudo-differential operators. Cambridge, Massachusetts: The MIT Press 1981 · Zbl 0472.35034
[20] Landau, L. D., Lifschitz, E. M.: Course of theoretical physics, Vol. 2. The classical theory of fields, 4th revised English ed. Oxford: Pergamon Press 1975
[21] Mizrahi, M. M.: The Weyl correspondence and path integrals. J. Math. Phys.16, 2201-2206 (1975) · doi:10.1063/1.522468
[22] Mizrahi, M. M.: Phase space path integrals, without limiting procedure. J. Math. Phys.19, 298-307 (1978); Erratum, ibid.21, 1965 (1980) · doi:10.1063/1.523504
[23] Muramatu, T., Nagase, M.: On sufficient conditions for the boundedness of pseudo-differential operators. Proc. Jpn Acad.55 (A), 293-296 (1979) · Zbl 0454.35089 · doi:10.3792/pjaa.55.293
[24] Nagase, M.: TheL p -boundedness of pseudo-differential operators with non-regular symbols. Commun. Partial. Differ. Equations.,2, 1045-1061 (1977) · Zbl 0397.35071 · doi:10.1080/03605307708820054
[25] Reed, M., Simon, B.: Methods of modern mathematical physics, IV: Analysis of operators. New York: Academic Press 1978 · Zbl 0401.47001
[26] Shubin, M. A.: Essential self-adjointness of uniformly hypoelliptic operators. Vestn. Mosk. Univ. Ser. I.30, 91-94 (1975); English transl. Mosc. Univ. Math. Bull.30, 147-150 (1975) · Zbl 0295.47054
[27] Simon, B.: Functional integration and quantum physics. New York: Academic Press 1979 · Zbl 0434.28013
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