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p-adic Feynman's path integrals

Branko G Dragovich, Lj. Nesic

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The Feynman path integral method plays even more important role in p-adic and adelic quantum mechanics than in ordinary quantum theory. p-Adic path integral is defined as a natural generalization of the standard one. Here 1 , we give a brief review of our recent investigations on this subject. In particular, the path integrals for two-dimensional models with quadratic Lagrangians are evaluated.

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The Feynman path integral method plays even more important role in p-adic and adelic quantum mechanics than in ordinary quantum theory. p-Adic path integral is defined as a natural generalization of the standard one. Here 1 , we give a brief review of our recent investigations on this subject. In particular, the path integrals for two-dimensional models with quadratic Lagrangians are evaluated.
Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of the number fields ℝ ↔ ℚ p and ℚ ↔ ℚ p , p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.
p-Adic mathematical physics emerged as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale. One of its main achievements is a successful formulation and development of p-adic and adelic quantum mechanics, which have complex-valued wave functions of p-adic and adelic arguments, respectively. Various aspects of these quantum mechanics are reviewed here. In particular, the corresponding Feynman's path integrals, some minisuperspace cosmological models, and relevant approache to string theory, are presented. As a result of adelic approach, p-adic effects exhibit a spacetime and some other discreteness, which depend on the adelic quantum state of the physical system under consideration. Besides review, this article contains also some new results.