zbMATH — the first resource for mathematics

Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions. (English) Zbl 1335.81103
There exist many different approaches to the subject of Feynman path integrals, often called functional integrals by mathematicians. Very often the sequential approach, which is used by the authors, defines a functional integral as the limit of finite dimensional integrals when the dimension tends to infinity. However, it has been noticed in the past that the path integrals are very sensitive to our approximation choice, and so the question arises whether the quantization of a classical system might be non-unique. Berezin has posed this problem in an article. The present paper tries to provide an answer Berezin’s problem using what the authors call the method of Feynman formulae. By this method one obtains representations of semigroups solving evolution equations by limits of \(n\)-fold iterated integrals over the phase space of a physical system. Some representations are based on integral operators. Feynman formulae appear to be suitable for numerical computations. Different quantizations now correspond to path integrals with different (pseudo)measures, though they have the same integrand. This answers Berezin’s problem. Finally, so-called symbols of Lévy-Khintchine type are considered with respect to models of one particle interacting with some scalar potential and a magnetic field.

81S40 Path integrals in quantum mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
47D08 Schrödinger and Feynman-Kac semigroups
81S05 Commutation relations and statistics as related to quantum mechanics (general)
PDF BibTeX Cite
Full Text: DOI arXiv
[1] Albeverio, S.; Guatteri, G.; Mazzucchi, S., Phase space Feynman path integrals, J. Math. Phys., 43, 2847-2857, (2002) · Zbl 1059.81107
[2] Albeverio, S.; Høegh-Krohn, R.; Mazzucchi, S., Mathematical Theory of Feynman Path Integrals: An Introduction, 523, (2008), Springer Verlag: Springer Verlag, Berlin · Zbl 1222.46001
[3] Berezin, F. A., Non-Wiener functional integrals, Theor. Math. Phys., 6, N2, 141-155, (1971)
[4] Berezin, F. A., Feynman path integrals in a phase space, Sov. Phys. Usp., 23, 763-788, (1980)
[5] Bock, W.; Grothaus, M., White noise approach to phase space Feynman path integrals, Theory Probab. Math. Stat., 85, 7-21, (2012) · Zbl 1285.60071
[6] Bock, W.; Grothaus, M.; Jung, S., The Feynman integrand for the charged particle in a constant magnetic field as white noise distribution, Commun. Stochastic Anal., 6, N4, 649-668, (2012) · Zbl 1331.81115
[7] Böttcher, B.; Butko, Ya. A.; Schilling, R. L.; Smolyanov, O. G., Feynman formulae and path integrals for some evolutionary semigroups related to τ-quantization, Russ. J. Math. Phys., 18, N4, 387-399, (2011) · Zbl 1311.58006
[8] Bouchaud, J.-P.; Georges, A., Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195, N4-5, 127-293, (1990)
[9] Bouchemla, N.; Chetouani, L., Path integral solution for a particle with position dependent mass, Acta Phys. Pol., B, 40, N10, 2711-2723, (2009)
[10] Butko, Ya. A., Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold, Math. Notes, 83, 301-316, (2008) · Zbl 1155.58302
[11] Butko, Ya. A., Function integrals corresponding to a solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a Riemannian manifold, J. Math. Sci., 151, 2629-2638, (2008) · Zbl 1151.35375
[12] Butko, Ya.; Grothaus, M.; Smolyanov, O. G., Feynman formula for a class of second-order parabolic equations in a bounded domain, Dokl. Math., 78, 590-595, (2008) · Zbl 1218.81037
[13] Butko, Ya.; Grothaus, M.; Smolyanov, O. G., Lagrangian Feynman formulae for second order parabolic equations in bounded and unbounded domains, Infinite Dimens. Anal., Quantum Probab. Relat. Top., 13, 377-392, (2010) · Zbl 1204.47096
[14] Butko, Ya.; Schilling, R. L.; Smolyanov, O. G., Feynman formulae for Feller semigroups, Dokl. Math., 82, 679-683, (2010) · Zbl 1213.47047
[15] Butko, Ya. A.; Schilling, R. L.; Smolyanov, O. G., Hamiltonian Feynman-Kac and Feynman formulae for dynamics of particles with position-dependent mass, Int. J. Theor. Phys., 50, 2009-2018, (2011) · Zbl 1226.81114
[16] Butko, Ya. A.; Schilling, R. L.; Smolyanov, O. G., Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations, Infinite Dimens. Anal., Quantum Probab. Relat. Top., 15, N3, 1250015, (2012) · Zbl 1273.47070
[17] Cartier, P.; De Witt-Morette, C., Functional Integration: Action and Symmetries, (2006), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1122.81004
[18] Chernoff, P., Product formulas, nonlinear semigroups and addition of unbounded operators, Mem. Am. Math. Soc., 140, (1974) · Zbl 0283.47041
[19] Daubechies, I.; Klauder, J. R., Quantum-mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II, J. Math. Phys., 26, 2239-2256, (1985) · Zbl 0979.81517
[20] De Witt-Morette, C.; Maheshwari, A.; Nelson, B., Path integration in non-relativistic quantum mechanics, Phys. Rep., 50, 255-372, (1979)
[21] Elworthy, D.; Truman, A., Feynman maps, Cameron-Martin formulae and anharmonic oscillators, Ann. Inst. Henri Poincare, 41, N2, 115-142, (1984) · Zbl 0578.28013
[22] Feynman, R. P., Space-time approach to nonrelativistic quantum mechanics, Rev. Mod. Phys., 20, 367-387, (1948) · Zbl 1371.81126
[23] Feynman, R. P., An operational calculus having application in quantum electrodynamics, Phys. Rev., 84, 108-128, (1951) · Zbl 0044.23304
[24] Freidlin, M., Functional Integration and Partial Differential Equations, (1985), Princeton University Press: Princeton University Press, Princeton, New Jersey · Zbl 0568.60057
[25] Gadella, M.; Kuru, Ş.; Negro, J., Self-adjoint Hamiltonians with a mass jump: General matching conditions, Phys. Lett. A, 362, 265-268, (2007) · Zbl 1197.81119
[26] Gadella, M.; Smolyanov, O. G., Feynman formulas for particles with position-dependent mass, Dokl. Math., 77, 120-123, (2007) · Zbl 1159.35426
[27] Gangulu, A.; Kuru, Ş.; Negro, J.; Nieto, L. M., A study of the bound states for square potential wells with position-dependent mass, Phys. Lett. A, 360, 228-233, (2006) · Zbl 1236.81181
[28] Garrod, C., Hamiltonian path integral methods, Rev. Mod. Phys., 38, 483-494, (1966) · Zbl 0192.30403
[29] Grosche, C.; Steiner, F., Handbook of Feynman Path Integrals, (1998), Springer · Zbl 1029.81045
[30] Ichinose, W., The phase space Feynman path integral with gauge invariance and its convergence, Rev. Math. Phys., 12, 1451-1463, (2000) · Zbl 0966.81039
[31] Jacob, N., Pseudo-Differential Operators and Markov Processes, 1-3, (2005), Imperial College Press: Imperial College Press, London · Zbl 1142.26003
[32] Johnson, G. W.; Lapidus, M. L., The Feynman Integral and Feynman’s Operational Calculus, (2000), Clarendon Press: Clarendon Press, Oxford · Zbl 0952.46044
[33] Kitada, H.; Kumano-go, H., A family of Fourier integral operators and the fundamental solution for a Schrödinger equation, Osaka J. Math., 18, 291-360, (1981) · Zbl 0472.35034
[34] Kleinert, H., Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, (2006), World Scientific Publishing Co.: World Scientific Publishing Co., Singapore · Zbl 0942.81038
[35] Kumano-go, N., A Hamiltonian path integral for a degenerate parabolic pseudo-differential operators, J. Math. Sci. Univ. Tokyo, 3, 57-72, (1996) · Zbl 0870.35134
[36] Kumano-go, N.; Fujiwara, D., Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations, Bull. Sci. Math., 132, 313-357, (2008) · Zbl 1141.81018
[37] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 298-305, (2000) · Zbl 0948.81595
[38] Laskin, N., Lévy flights over quantum paths, Commun. Nonlinear Sci. Numer. Simul., 12, 2-18, (2007) · Zbl 1101.81079
[39] Lejay, A., A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE, Stochastic Processes Appl., 110, 145-176, (2004) · Zbl 1075.60070
[40] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and Their Applications, 16, (1995), Birkhäuser: Birkhäuser, Basel, Boston, Berlin · Zbl 0816.35001
[41] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1-77, (2000) · Zbl 0984.82032
[42] Nelson, E., Feynman integrals and the Schrödinger equation, J. Math. Phys., 3, 332-343, (1964) · Zbl 0133.22905
[43] Obrezkov, O. O., The proof of the Feynman-Kac formula for heat equation on a compact Riemannian manifold, Infinite Dimens. Anal., Quantum Probab. Relat. Top., 6, 311-320, (2003) · Zbl 1067.58030
[44] Obrezkov, O.; Smolyanov, O. G.; Truman, A., The generalized Chernoff theorem and randomized Feynman formula, Dokl. Math., 71, 105-110, (2005)
[45] Orlov, Yu. N.; Sakbaev, V. Zh.; Smolyanov, O. G., Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian, Theor. Math. Phys., 172, N1, 987-1000, (2012) · Zbl 1280.81046
[46] Plyashechnik, A. S., Feynman Formula for Schrödinger-type equations with time- and space-dependent coefficients, Russ. J. Math. Phys., 19, N3, 340-359, (2012) · Zbl 1260.81082
[47] Plyashechnik, A. S., Feynman Formulas for second-order parabolic equations with variable coefficients, Russ. J. Math. Phys., 20, N3, 3, (2013) · Zbl 1285.81030
[48] Reed, M.; Simon, B., Methods of Modern Mathematical Physics: Functional Analysis, I, (1980), Academic Press
[49] Sakbaev, V. G.; Smolyanov, O. G., Dynamics of a quantum particle with discontinuous position-dependent mass, Dokl. Math., 82, 630-634, (2010) · Zbl 1200.81054
[50] Simon, B., Functional Integration and Quantum Physics, (2004), AMS Chelsea Publishing
[51] Smolyanov, O. G.; Shamarov, N. N., Feynman and Feynman-Kac formulae for evolution equations with Vladimirov operator, Dokl. Math., 77, 345-349, (2008) · Zbl 1185.35351
[52] Smolyanov, O. G.; Shamarov, N. N., Hamiltonian Feynman integrals for equations with the Vladimirov operator, Dokl. Math., 81, 209-214, (2010) · Zbl 1210.46061
[53] Smolyanov, O. G.; Shavgulidze, E. T., Kontinualnye Integraly, (1990), Moscow State University Press: Moscow State University Press, Moscow
[54] Smolyanov, O. G.; Shavgulidze, E. T., The support of symplectic Feynman measure and uncertainty principle, Dokl. Acad. Nauk USSR, 323, 1038-1042, (1992) · Zbl 0797.28009
[55] Smolyanov, O. G.; Tokarev, A. G.; Truman, A., Hamiltonian Feynman path integrals via the Chernoff formula, J. Math. Phys., 43, 5161-5171, (2002) · Zbl 1060.58009
[56] Smolyanov, O. G.; Weizsäcker, H. v.; Wittich, O., Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions, Stochastic Proceses, Physics and Geometry: New Interplays. II: A Volume in Honor of Sergio Albeverio, 589-602, (2000), American Mathematical Society: American Mathematical Society, Providence · Zbl 0978.58015
[57] Smolyanov, O. G.; Weizsäcker, H. v.; Wittich, O., Diffusion on compact Riemannian manifolds, and surface measures, Dokl. Math., 61, 230-234, (2000) · Zbl 1047.58006
[58] Smolyanov, O. G.; Weizsäcker, H. v.; Wittich, O., Chernoff’s theorem and the construction of semigroups, 349-358, (2003), Birkhäuser: Birkhäuser, Basel · Zbl 1057.47044
[59] Smolyanov, O. G.; Weizsäcker, H. v.; Wittich, O., Chernoff’s theorem and discrete time approximations of Brownian motion on manifolds, Potential Anal., 26, 1-29, (2007) · Zbl 1107.58014
[60] Smolyanov, O. G.; Weizsäcker, H. v.; Wittich, O., Surface measures and initial boundary value problems generated by diffusions with drift, Dokl. Math., 76, 606-610, (2007) · Zbl 1156.58014
[61] Stannat, W., (Nonsymmetric) Dirichlet operators on L1: Existence, uniqueness and associated Markov processes, Ann. della Scuola Normale Superiore di Pisa, Cl. Sci. 4e série, 28, N1, 99-140, (1999) · Zbl 0946.31003
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.