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p-ADIC AND ADELIC FREE RELATIVISTIC PARTICLE

Goran S. Djordjević, Branko G Dragovich, Lj. Ne, Si Ć

Published Jan 1, 1999• 14 citations
OpenAlex

Abstract

We consider spectral problem for a free relativistic particle in p-adic and adelic quantum mechanics. In particular, we found p-adic and adelic eigenfunctions. Within adelic approach there exist quantum states that exhibit discrete structure of spacetime at the Planck scale. 1.

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p-ADIC AND ADELIC FREE RELATIVISTIC PARTICLE

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arXiv:hep-th/0404160•International Journal of Modern Physics A•Jun 30, 1995•59 citations
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.
Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and p-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of a simplest vacuum state leads to the well known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested. 1.
Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and p-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of the simplest vacuum state leads to the well-known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested.