p-ADIC AND ADELIC FREE RELATIVISTIC PARTICLE

Djordjević, Goran S.; Nešić, Ljubiša; Драгович, Бранко

doi:10.1142/s0217732399000365

Cited by 21 publications

(24 citation statements)

References 9 publications

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“…Remark 2.5. Since the determinant of a matrix is the product of all its eigenvalues, it is worth noting that according to (24) one can express Λ v (α 1 α 2 · · · α n ) in (26) in the following form:…”

Section: Let Us Change Variables Of Integration Bymentioning

confidence: 99%

“…In order to connect p-adic with standard quantum mechanics, adelic quantum mechanics was formulated [4]. Within adelic quantum mechanics a few basic physical systems [23,24], including some minisuperspace cosmological models [25], have been successfully considered. As a result of p-adic effects in the adelic approach, a space-time discreteness at the Planck scale is obtained.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Let Us Change Variables Of Integration Bymentioning

confidence: 99%

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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“…As an illustration of p-adic and adelic quantum-mechanical models, the following one-dimensional systems with the quadratic Lagrangians were considered: a free particle and a harmonic oscillator [5,14], a particle in a constant field [18], a free relativistic particle [19] and a harmonic oscillator with time-dependent frequency [20].…”

Section: Quantum Mechanics On Adelic Spacesmentioning

confidence: 99%

“…Adelic quantum mechanics takes into account ordinary as well as p-adic quantum effects and may be regarded as a starting point for construction of a more complete quantum cosmology and string/M-theory. In the low-energy limit adelic quantum mechanics effectively becomes the ordinary one [19].…”

Section: Quantum Mechanics On Adelic Spacesmentioning

confidence: 99%

Non-Archimedean Geometry and Physics on Adelic Spaces

Драгович¹

2004

Contemporary Geometry and Related Topics

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“…So far a rather small number of physical systems has been treated in p-adic and adelic quantum mechanics: a non-relativistic free particle [7], a harmonic oscillator [7,9], a particle in a constant field [8], the de Sitter minisuperspace model of the universe [10] and a relativistic free particle [11]. It is doubtless that evaluation of some other physical systems, which exhibit p-adic and adelic properties, will give new insights into this subject and new directions for future investigations at the Planck scale.…”

Section: Introductionmentioning

confidence: 99%