Abstract
The paper is a survey with modifications on the research of the so-called Newton–Nelson equation (the equation of motion in Nelson’s stochastic mechanics) on the total space of a bundle in two cases: where the base of the bundle is a Riemannian manifold and the bundle is real and where the base of the bundle is a Lorentz manifold and the bundle is complex. In the latter case, we describe the relations with the equation of motion of the quantum particle in the classical gauge field (the above-mentioned connection). Moreover, a certain second-order ordinary differential equation on the bundle with connection that is interpreted as the equation of motion of the classical particle in the classical gauge field is described.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ya. I. Belopolskaya and Yu. L. Dalecky, Stochastic Processes and Differential Geometry, Kluwer Academic, Dordrecht (1989).
R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, London (1964).
D. Dohrn, F. Guerra, and P. Ruggiero, “Spinning particles and relativistic particles in framework of Nelson’s stochastic mechanics,” in: Feynman Path Integrals. Proc. Int. Colloq. Held in Marseille, May 1978, Springer, Berlin (1979), Lect. Notes Phys., Vol. 106, pp. 165–181.
K. D. Elworthy, Stochastic Differential Equations on Manifolds, Cambridge Univ. Press, Cambridge (1982).
Yu. E. Gliklikh, Global and Stochastic Analysis in the Problems of Mathematical Physics [in Russian], Komkniga, Moscow (2005).
Yu. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics, Springer, London (2011).
Yu. E. Gliklikh and P. S. Ratiner, “On a certain type of second order differential equations on total spaces of fiber bundles with connections,” Nonlinear Analysis in Geometry and Topology, Hadronic Press, Palm Harbor (2000), pp. 99–106.
Yu. E. Gliklikh and N. V. Vinokurova, “On the motion of a quantum particle in the classical gauge field in the language of stochastic mechanics,” Commun. Stat. Theory Methods, 40, No. 19-20, 3630–3640 (2011).
Yu. E. Gliklikh and N. V. Vinokurova, “On the Newton–Nelson type equations on vector bundles with connections,” Rend. Sem. Mat. Univ. Politec. Torino, in print.
F. Guerra and P. Ruggiero, “A note on relativistic Markov processes,” Lett. Nuovo Cimento, 23, 529–534 (1978).
E. Nelson, “Derivation of the Schrödinger equation from Newtonian mechanics,” Phys. Rev., 150, No. 4, 1079–1085 (1966).
E. Nelson, Quantum Fluctuations, Princeton Univ. Press, Princeton (1985).
K. R. Parthasarathy, Introduction to Probability and Measure, Springer, New York (1978).
T. Zastawniak, “A relativistic version of Nelson’s stochastic mechanics,” Europhys. Lett., 13, 13–17 (1990).
T. Zastawniak, “Markov diffusion in relativistic stochastic mechanics,” in: A. Truman et al., ed., Proc. Swansea Conf. on Stochastic Mechanics, World Scientific, Singapore (1992), pp. 280–297.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 3, pp. 61–81, 2015.
Rights and permissions
About this article
Cite this article
Gliklikh, Y.E., Vinokurova, N.V. The Newton–Nelson Equation on Fiber Bundles with Connections. J Math Sci 225, 575–589 (2017). https://doi.org/10.1007/s10958-017-3479-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3479-0