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De Witt-Morette, C.; Elworthy, K. D.; Nelson, B. L.; Sammelman, G. S.

A stochastic scheme for constructing solutions of the Schrödinger equations. (English) Zbl 0446.60045

Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 32, 327-341 (1980).
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Cited in 1 Review
Cited in 8 Documents

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
58D30 Applications of manifolds of mappings to the sciences
58J65 Diffusion processes and stochastic analysis on manifolds
60H05 Stochastic integrals
81S40 Path integrals in quantum mechanics
60J60 Diffusion processes
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory

Keywords:

curved space; external gauge field; multiply connected configuration space; Feynman path integration; historical discussion
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References:

[1] H. Airault , Subordination de processus dans Ie fibré tangent et formes harmoniques , C. R. Acad. Sc. Paris , t. 282 , Sér. A , 1976 , p. 1311 - 1314 . MR 420740 | Zbl 0336.53029 · Zbl 0336.53029
[2] S.J. Avis and C.J. Isham , Lorentz gauge invariant vacuum functionals for quantized spinor fields in non-simply connected space-times . Preprint Imperial College, London SW7 2BZ, 1979 . MR 542970
[3] J.M.C. Clark , An introduction to stochastic differential equations on manifolds , Geometric Methods in Systems Theory (D. Q. Mayne, R. W. Brockett, eds.), Reidel , Holland, 1973 . Zbl 0296.60035 · Zbl 0296.60035
[4] B.S. De Witt , Dynamical theory in curved spaces. I . A review of the classical and quantum action principles , Rev. Mod. Phys. , t. 29 , 1957 , p. 337 - 397 . MR 95691 | Zbl 0118.23301 · Zbl 0118.23301 · doi:10.1103/RevModPhys.29.377
[5] B.S. De Witt , C.F. Hart and C.J. Isham , Topology and quantum field theory . To appear in the Schwinger Festschrift.
[6] C. De Witt-Morette , A. Maheshwari and B. Nelson , Path integration in non-relativistic mechanics , Physics Reports , t. 50 , 1979 , p. 257 - 372 .
[7] C. De Witt-Morette and K.D. Elworthy , Stochastic differential equations . Lecture notes and problems , Preprint of the Center for Relativity, University of Texas , Austin , TX 78712, 1979 .
[8] J. Dollard and C. Friedman , Product integration , Addison-Wesley , 1979 . MR 552941
[9] J.S. Dowker , Quantum mechanics and field theory on multiply connected and on homogeneous spaces , J. Phys. A. Gen. Phys. , t. 5 , 1972 , p. 936 - 943 . MR 418715
[10] J. Eells and K.D. Elworthy , Stochastic Dynamical Systems, in Control theory and topics in functional analysis , vol. III , International Atomic Energy Agency , Vienna , 1976 , p. 179 - 185 . MR 516923 | Zbl 0355.60053 · Zbl 0355.60053
[11] K.D. Elworthy , Stochastic differential equations on manifolds . Notes based on seminars given by J. Eells and K. D. Elworthy, available from the Department of Mathematics, University of Warwick , Coventry CV4 7AL, and references therein, 1978 .
[12] K.D. Elworthy and A. Truman , Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation on riemannian manifolds . In preparation, 1979 . Zbl 0485.70024 · Zbl 0485.70024 · doi:10.1063/1.524784
[13] W. Garczynski , On alternative ways of including an external electromagnetic field , Bull. Acad. Polonaise Sciences , t. XXI , 1973 , p. 355 - 358 .
[14] M.G.G. Laidlaw , Ph. D. Thesis , University of North Carolina , 1971 .
[15] M.G.G. Laidlaw and C.M. Dewitt , Feynman functional integrals for systems of indistinguishable particles , Phys. Rev. , t. D 3 , 1971 , p. 1375 - 1377 .
[16] D.W. Mclaughlin and L.S. Schulman , Path integrals in curved spaces , J. Math. Phys. , t. 12 , 1971 , p. 2520 - 2524 . MR 289052
[17] C. Morette , On the definition and approximation of Feynnman’s path integrals , Phys. Rev. , t. 81 , 1951 , p. 848 - 852 . MR 41009 | Zbl 0042.45506 · Zbl 0042.45506 · doi:10.1103/PhysRev.81.848
[18] B.L. Nelson and B. Sheeks , Path integration for velocity dependent potentials , Preprint of the Center for Relativity, University of Texas , Austin , TX 78712, 1979 . · Zbl 0509.70012
[19] W. Pauli , Asugewahlte Kapitel aus Der Feldquantisierung, Gehalten an Der E. T. H. in Zürich, 1950-1951 . Reprinted in Pauli Lectures in Physics , Ed. C. P. Enz, MIT Press , 1952 .
[20] H.R. Petry , Exotic spinors in superconductivity , J. Math. Phys. , t. 20 , 1979 , p. 231 - 240 . MR 519205
[21] M.A. Pinsky , Stochastic riemannian geometry , Probabilistic analysis and related topics , t. 1 , 1978 , p. 199 - 236 . MR 501385 | Zbl 0452.60083 · Zbl 0452.60083
[22] L.S. Schulman , A path integral for spin , Phys. Rev. , t. 176 , 1968 , p. 1558 - 1567 . MR 237149
[23] L.S. Schulman , Green’s function for an electron in a lattice , Phys. Rev. , t. 188 , 1969 , p. 1139 - 1142 .
[24] B. Simon , Functional integration and quantum physics , Academic Press , 1979 . MR 544188 | Zbl 0434.28013 · Zbl 0434.28013
[25] N. Steenrod , The topology of fibre bundles , Princeton University Press , 1951 . MR 39258 | Zbl 0054.07103 · Zbl 0054.07103
[26] A. Truman , Classical mechanics, the diffusion (heat) equation and the Schrödinger equation , J. Math. Phys. , t. 18 , 1977 , p. 2308 - 2315 . MR 468817 | Zbl 0366.35042 · Zbl 0366.35042 · doi:10.1063/1.523240
[27] S.K. Wong , Field and particle equations for the classical Yang-Mills field and particles with isotropic spin , Nuovo Cimento , t. 65 , 1970 , p. 689 - 694 .
[28] C.N. Yang and R.L. Mills , Conservation of isotopic spin and isotopic gauge invariance , Phys. Rev. , t. 96 , 1954 , p. 191 - 195 . MR 65437 · Zbl 1378.81075
[29] S.T. Yau , On the heat kernel of a complete riemannian manifold , J. Math. pures et appl. , t. 57 , 1978 , p. 191 - 207 . MR 505904 | Zbl 0405.35025 · Zbl 0405.35025
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