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Branko Dragovich

Publications and source records attributed to Branko Dragovich.

At least 55 records · Page 3Linked to original sources
arXiv2006

Some Aspects of Noncommutativity on Real, p-Adic and Adelic Spaces

Classical and quantum mechanics for an extended Heisenberg algebra with canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by linear transformation of phase space coordinates and transmitted to the Hamiltonian (Lagrangian). This transformation does not change the quadratic form of Hamiltonian (Lagrangian) and Feynman's path integral maintains its well-known exact expression for quadratic systems. The compact matrix formalism is presented and can be easily employed in particular cases. Some p-adic and adelic aspects of noncommutativity are also considered.

hep-th↗
arXiv2006

Noncommutative Classical and Quantum Mechanics for Quadratic Lagrangians (Hamiltonians)

We consider classical and quantum mechanics for an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates. In our approach this additional noncommutativity is removed from the algebra by linear transformation of coordinates and transmitted to the Hamiltonian (Lagrangian). Since linear transformations do not change the quadratic form of Hamiltonian (Lagrangian), and Feynman's path integral has well-known exact expression for quadratic models, we restricted our analysis to this class of physical systems. The compact general formalism presented here can be easily realized in any particular quadratic case. As an important example of phenomenological interest, we explored model of a charged particle in the noncommutative plane with perpendicular magnetic field. We also introduced an effective Planck constant $\hbar_{eff}$ which depends on noncommutativity.

hep-th↗
arXiv2006

p-Adic and Adelic Cosmology: p-Adic Origin of Dark Energy and Dark Matter

A brief review of p-adic and adelic cosmology is presented. In particular, p-adic and adelic aspects of gravity, classical cosmology, quantum mechanics, quantum cosmology and the wave function of the universe are considered. p-Adic worlds made of p-adic matters, which are different from real world of ordinary matter, are introduced. Real world and p-adic worlds make the universe as a whole. p-Adic origin of the dark energy and dark matter are proposed and discussed.

hep-th↗
arXiv2005

p-Adic and Adelic Superanalysis

After a brief review of p-adic numbers, adeles and their functions, we consider real, p-adic and adelic superalgebras, superspaces and superanalyses. A concrete illustration is given by means of the Grassmann algebra generated by two anticommuting elements.

hep-th↗
arXiv2005

Noncommutative Quantum Mechanics with Path Integral

We consider classical and quantum mechanics related to an additional noncommutativity, symmetric in position and momentum coordinates. We show that such mechanical system can be transformed to the corresponding one which allows employment of the usual formalism. In particular, we found explicit connections between quadratic Hamiltonians and Lagrangians, in their commutative and noncommutative regimes. In the quantum case we give general procedure how to compute Feynman's path integral in this noncommutative phase space with quadratic Lagrangians (Hamiltonians). This approach is applied to a charged particle in the noncommutative plane exposed to constant homogeneous electric and magnetic fields.

hep-th↗
arXiv2004

On Uncertainty Relations at Planck's Scale

A brief review of the previous research on the Heisenberg uncertainty relations at the Planck scale is given. In this work, investigation of the uncertainty principle extends to p-adic and adelic quantum mechanics. In particular, p-adic analogs of the Heisenberg algebra and uncertainty relation are introduced. Unlike ordinary quantum theory, adelic quantum approach provides a promising framework to probe space below the Planck length.

hep-th↗
arXiv2004

On Generalized Functions in Adelic Quantum Mechanics

Some aspects of adelic generalized functions, as linear continuous functionals on the space of Schwartz-Bruhat functions, are considered. The importance of adelic generalized functions in adelic quantum mechanics is demonstrated. In particular, adelic product formula for Gauss integrals is derived, and the connection between the functional relation for the Riemann zeta function and quantum states of the harmonic oscillator is stated.

math-ph↗
arXiv2004

On Some Finite Sums with Factorials

The summation formula $$ \sum^{n-1}_{i=0}ε^i i! (i^k+u_k) = v_k+ε^{n-1} n! A_{k-1}(n) $$ $(ε=\pm 1; k=1,2,...; u_k, v_k\in \msbm\hbox{Z}; A_{k-1}$ is a polynomial) is derived and its various aspects are considered. In particular, divisibility with respect to $n$ is investigated. Infinitely many equivalents to Kurepa's hypothesis on the left factorial are found.

math.NT↗
arXiv2004

Adelic Harmonic Oscillator

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.

hep-th↗
arXiv2004

Adelic Model of Harmonic Oscillator

Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.

hep-th↗
arXiv2004

On p-Adic Power Series

We obtained the region of convergence and the summation formula for some modified generalized hypergeometric series (1.2). We also investigated rationality of the sums of the power series (1.3). As a result the series (1.4) cannot be the same rational number in all Z_p.

math-ph↗
arXiv2004

Path Integral Approach to Noncommutative Quantum Mechanics

We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Hamiltonians, as well as Lagrangians, in their commutative and noncommutative regimes. The general procedure to compute Feynman's path integral on this noncommutative phase space with quadratic Lagrangians (Hamiltonians) is presented. Using this approach, a particle in a constant field, ordinary and inverted harmonic oscillators are elaborated in detail.

hep-th↗
arXiv2004

Some p-Adic Aspects of Superanalysis

A brief review of a superanalysis over real and $p$-adic superspaces is presented. Adelic superspace is introduced and an adelic superanalysis, which contains real and $p$-adic superanalysis, is initiated.

hep-th↗
arXiv2003

p-Adic and Adelic Quantum Mechanics

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.

hep-th↗
arXiv2003

Path Integrals in Noncommutative Quantum Mechanics

Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.

hep-th↗
arXiv2003

Lagrangian Aspects of Quantum Dynamics on a Noncommutative Space

In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quadratic classical Lagrangians of these two systems. We also shown that there is a subclass of quadratic Lagrangians, which includes harmonic oscillator and particle in a constant field, whose connection between ordinary and noncommutative regimes can be expressed as a linear change of position in terms of a new position and velocity.

hep-th↗