Abstract
After recalling the crucial role of first integrals (or constants of the motion) in classical and quantum dynamics, we show why the concept of martingale is their natural probabilistic counterpart. This is used in the framework of Euclidean Quantum Mechanics, a probabilistic approach to quantization. The relevant Theorem of Noether is shown; it is founded on a new probabilistic interpretation of the symmetry group of some heat equations.
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References
V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, “Mathematical Aspects of Classical and Celestial Mechanics”, Encycl. of Math. Sciences. Vol. 3, Springer-Verlag, Berlin (1993)
T.F. Jordan, “Linear Operators for Quantum Mechanics”, John Wiley, N.Y.
R. P. Feynman, A. R. Hibbs, “Quantum Mechanics and Path Integral”, Mc Graw-Hill, N.Y. (1965)
R.H. Cameron, J. Math. Physics 39, 126 (1960)
E. Nelson, “Dynamical Theories of Brownian Motion”, Princeton Univ. Press, N.Y. (1967)
J.C. Zambrini, J. Math. Physics, 27, 2307 (1986)
S. Albeverio, K. Yasue, J. C. Zambrini, Ann. Inst. Henri Poincaré. (Phys. Théor.) 49, 259 (1989)
W. H. Fleming, H. M. Soner, “Controlled Markov processes and viscosity solutions”, Springer-Verlag (1993)
P.J. Olver, “Applications of Lie groups to differential equations”, Springer-Verlag (1986)
A.B. Cruzeiro, J.C. Zambrini, “Malliavin Calculus and Euclidean Quantum Mechanics: I”, J. Funct. Anal. 96, 62 (1991)
T. Kolsrud, J. C. Zambrini, in “Stochastic Analysis and Applications”, Progress in Probability Vol. 26. A. B. Cruzeiro, J. C. Zambrini Edit., Birkhäuser Verlag, Boston (1991)
M. Thieullen, J. C. Zambrini: “Symmetries in stochastic calculus of variations”, to appear in Prob. Theory and Related Fields
M. Thieullen, J. C. Zambrini: “Probability and quantum symmetries. I The Theorem of Noether in Schrödinger’s Euclidean Quantum mechanics” to appear in Ann. Inst. Henri Poincaré (Phys. Théorique)
W. Miller Jr, “Symmetry and Separation of variables”, Encycl. of Mathematics and Applic. Vol. 4, Addison-Wesley, Reading, Mass (1977)
S. Albeverio, J. Rezende, J. C. Zambrini, “Probability and quantum symmetries II. The Theorem of Ncether in Quantum Mechanics.” In preparation.
J.C. Zambrini, “Probability and quantum symmetries in a Riemannian manifold”, to appear in Second seminar on stochastic analysis, random fields and applications, Ascona, Sept. 1996, Eds. M. Dozzi, R. Dalang, F. Russo.
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Zambrini, JC. (1998). Probabilistic Interpretation of the Symmetry Group of Heat Equations. In: Decreusefond, L., Øksendal, B., Gjerde, J., Üstünel, A.S. (eds) Stochastic Analysis and Related Topics VI. Progress in Probability, vol 42. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2022-0_20
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DOI: https://doi.org/10.1007/978-1-4612-2022-0_20
Publisher Name: Birkhäuser, Boston, MA
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