<i>p</i>-adic path integrals

Zelenov, E. I.

doi:10.1063/1.529137

Cited by 26 publications

(22 citation statements)

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“…Обсуждение возможной роли теории чисел в физике можно найти в работах Манина [207], [208] и Варадараджана [255]. p-Адическая квантовая механика и теория поля обсужда-лись в [14], [18], [41], [44], [45], [46], [64], [79], [80], [81], [96], [138], [139], [140], [141], [170], [182], [189], [204], [227], [254], [255], [257], [258], [271], [272] и многих других работах.…”

Section: библиографический обзорunclassified

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Козырев¹

2008

Sovrem. Probl. Mat.

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Section: библиографический обзорunclassified

Untitled

Козырев¹

2008

Sovrem. Probl. Mat.

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“…Thus, p-adic path integral is the limit of the multiple Haar integral when N → ∞. To calculate (9) in this way one has to introduce some ordering in the time t ∈ Q p , and it is successfully done in [6]. On previous investigations of p-adic path integrals one can see [7,8,9,10] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Path integrals for quadratic lagrangians on p-adic and adelic spaces

Драгович

Rakic

2010

P-Adic Num Ultrametr Anal Appl

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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 p ↔ Q p ′ , p = p ′ . According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena. * Email address: dragovich@ipb.ac.rs † Email address: zrakic@matf.bg.ac.rs the classical Lagrangian. In the 1940's, Feynman developed Dirac's approach and shown that dynamical evolution of the wave function Ψ(x, t) iswhereandis the action for a path q(t) connecting points x ′ and x ′′ . The integral in (4) is known as the Feynman path integral. In the Feynman definition [1], discretizing the time t into equidistant subintervals, the path integral (4) is the limit of the corresponding multiple integral of N variables q i = q(t i ), (i = 1, 2, ..., N ), when N → ∞. It is the primary object of the Feynman's path integral approach to quantum mechanics which is related to the classical Lagrangian formalism. Feynman's, Schrödinger's and Heisenberg's approaches to ordinary quantum mechanics are equivalent, but their formalisms are not equally suitable in some generalizations.K(x ′′ , t ′′ ; x ′ , t ′ ) is the kernel of the corresponding unitary integral operator U (t ′′ , t ′ ) acting as follows:K(x ′′ , t ′′ ; x ′ , t ′ ) is also called the probability amplitude for a quantum particle to pass from a point x ′ at the time t ′ to the other point x ′′ at t ′′ . It is closely related to Green's function and the quantum-mechanical propagator. Starting from (3) one can easily derive the following three general properties:where integration is over all the configuration space. For all its history, the path integral has been a subject of great interest in theoretical and mathematical physics. It has became, not only in quantum mechanics (see, e.g. [2]) but also in the entire quantum theory, one of its the most profound and suitable approaches to foundations and elaborations. Feynman's path integral construction is also a natural and very successful instrument in formulation and investigation of p-adic [3] and adelic [4,5] quantum mechanics. Moreover there are no p-adic analogs of the differential equations (1) and (2).Adelic quantum mechanics contains complex-valued functions of real and all p-adic arguments in the adelic form. There is not the corresponding Schrödinger equation for p-adic dynamics, but Feynman's path integral method is quite appropriate. Feynman's path integral for probability amplitude in p-adic quantum mechanics K p (x ′′ , t ′′ ; x ′ , t ′ ) [3], where K p is complexvalued and x ′′ , x ′ , t ′′ , t ′ are p-adic variables, is a direct p-adic generalization of (4), i.e.

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“…(23) In (23) we take h ∈ Q and q, t ∈ Q p . This path integral is elaborated, for the first time, for the harmonic oscillator [11]. It was shown that there exists the limit…”

Section: P-adic Path Integralsmentioning

confidence: 99%

Adelic Quantum Mechanics: Nonarchimedean and Noncommutative Aspects

Djordjević

Драгович

Nešić

2001

Noncommutative Structures in Mathematics and Physics

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Abstract. This is an addendum to the user manual for authors using Kluwer Academic Publishers's L A T E X class file kluwer.cls.This document gives information about the various commands that are either needed or helpful in the creation of full Camera-Ready books using the book-specific class file crckbked.cls. IntroductionThe crckbked.cls class file is meant for the submission of camera-ready copies of books which are edited volumes rather than monographs. The main input fileThe overall structure of the main input file is the same as for the kluwer.cls file, an is demonstrated in Figure 1 of the User Manual. There is a "frontmatter" environment available, and this can be used to typeset the frontmatter of the issue. EXTRA COMMANDS DatesOther than some class files, the commands \date, \received, \accepted and \revised do not generate output.Running headlines crckapb.cls with the nato option does not generate running heads either. For the other document style options (\editedvolume and \proceedings) please make sure that they are available, correct, and fit into the horizontal space. * This e-mail address is available for all problems and questions. Commands for adjustments PAGE NUMBERSThe two 'opening' commands \firstpage and \lastpage can be used to adjust the page numbering if needed. BIBLIOGRAPHIC LISTSIf two articles use conflicting options for the references (e.g. All articles use 'namedreferences' except one), the commands \numreferences and \namedreferences can be used to switch styles 'on the fly'. Fonts for camera ready productionCheck with your editor which font should be used: for using Times rather than the default (Computer Modern), add a line \usepackage{klups} to the preamble. This also adds support for the appropriate math fonts. The klups package tries to guess what font support is available on your system and make the best possible use of it. However, it does not always get it right. In that case you can try, going from best to worst: Abstract.We present a short review of adelic quantum mechanics pointing out its non-Archimedean and noncommutative aspects. In particular, p-adic path integral and adelic quantum cosmology are considered. Some similarities between p-adic analysis and q-analysis are noted. The p-adic Moyal product is introduced.

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p-adic path integrals

Cited by 26 publications

References 18 publications

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Path integrals for quadratic lagrangians on p-adic and adelic spaces

Adelic Quantum Mechanics: Nonarchimedean and Noncommutative Aspects

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