arXiv subjects

M. V. Gorbatenko

Publications and source records attributed to M. V. Gorbatenko.

28 records · Page 2Linked to original sources

arXiv2011

Uniqueness and Self-Conjugacy of Dirac Hamiltonians in arbitrary Gravitational Fields

Proofs of two statements are provided in this paper. First, the authors prove that the formalism of the pseudo-Hermitian quantum mechanics allows describing the Dirac particles motion in arbitrary stationary gravitational fields. Second, it is proved that using the Parker weight operator and the subsequent transition to the η-representation gives the transformation of the Schroedinger equation for nonstationary metric, when the evolution operator becomes self-conjugate. The scalar products in the η-representation are flat, which makes possible the use of a standard apparatus for the Hermitian quantum mechanics. Based on the results of this paper the authors draw a conclusion about solution of the problem of uniqueness and self-conjugacy of Dirac Hamiltonians in arbitrary gravitational fields including those dependent on time. The general approach is illustrated by the example of Dirac Hamiltonians for several stationary metrics, as well as for the cosmologically flat and the open Friedmann models.

gr-qc↗

arXiv2010

Solution of the problem of uniqueness and hermiticity of hamiltonians for Dirac particles in gravitational fields

The authors prove that the dynamics of spin 1/2 particles in stationary gravitational fields can be described using an approach, which builds upon the formalism of pseudo-Hermitian Hamiltonians. The proof consists in the analysis of three expressions for Hamiltonians, which are derived from the Dirac equation and describe the dynamics of spin 1/2 particles in the gravitational field of the Kerr solution. The Hamiltonians correspond to different choices of tetrad vectors and differ from each other. The differences between the Hamiltonians confirm the conclusion known from many studies that the Hamiltonians derived from the Dirac equation are non-unique. Application of standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians produces the same Hermitian Hamiltonian. The eigenvalue spectrum of the resulting Hamiltonian is the same as that of the Hamiltonians derived from the Dirac equation with any chosen system of tetrad vectors. For description of the dynamics of spin 1/2 particles in stationary gravitational fields can be used not only the formalism of pseudo-Hermitian Hamiltonians, but also an alternative approach, which employs the Parker scalar product. The authors show that the alternative approach is equivalent to the formalism of pseudo-Hermitian Hamiltonians.

gr-qc↗

arXiv2010

On physical meaning of Weyl vector

The present paper considers if the new proposed conformal geometrodynamics (CGD) can extend the Nature features compared with general theory of relativity (GTR). The answer for this question can be connected with unique phenomenon arising from Riemann space transition used in GTR, to Weyl space used in CGD. We have in mind the possibility to set up the perfect correspondence in certain spatial regions between the equations of different physical phenomena: (1) Phenomena associated with Weyl degrees of freedom in a plane space. (2) Phenomena described in terms of half-integer spin particles and observed quantities corresponding to the full set of bispinors. The said phenomenon is described in the present paper. Analyzed here are the new prospects in the problem of combination of quantum physics concepts and GTR, unification of physical interactions and understanding of many of the effects known from the experiments but not properly understood yet.

gr-qc↗

arXiv2008

Dirac particle in gravitation field

Being considered is the motion of Dirac particle in gravitational field, described by Kerr solution. It is proved, that evolution of the wave function is determined by Hermitian Hamiltonian, if the concomitant reference frame is involved.

gr-qc↗

arXiv2007

A Conformally Invariant Approach to Estimation of Relations Between Physical Quantities

A.V.Pushkin's approach based on conformal geometrodynamics (CGD) to calculation of quantitative relations between physical quantities is presented and analyzed. In the simplest cases of the stationary solutions to the CGD equations the approach implies separation of internal and external parts (relative to a certain boundary) from the solutions and using inverse transformations transforming the parts into each other. For the quasi-stationary (metastable) states, the possibility of the nonperturbative calculation of their lifetimes is shown. The approach is illustrated by several examples. In particular, it is shown that the Dirac ``large number hypothesis'' is a consequence of the approach. Also, the evaluated radiation lifetime of the first excited level of 2p hydrogen atom and neutron lifetime are presented.

gr-qc↗

arXiv2004

Physical Vacuum Properties and Internal Space Dimension

The paper addresses matrix spaces, whose properties and dynamics are determined by Dirac matrices in Riemannian spaces of different dimension and signature. Among all Dirac matrix systems there are such ones, which nontrivial scalar, vector or other tensors cannot be made up from. These Dirac matrix systems are associated with the vacuum state of the matrix space. The simplest vacuum system realization can be ensured using the orthonormal basis in the internal matrix space. This vacuum system realization is not however unique. The case of 7-dimensional Riemannian space of signature 7(-) is considered in detail. In this case two basically different vacuum system realizations are possible: (1) with using the orthonormal basis; (2) with using the oblique-angled basis, whose base vectors coincide with the simple roots of algebra E_{8}. Considerations are presented, from which it follows that the least-dimension space bearing on physics is the Riemannian 11-dimensional space of signature 1(-)& 10(+). The considerations consist in the condition of maximum vacuum energy density and vacuum fluctuation energy density.

gr-qc↗

arXiv2003

Discontinuity surface instead of singularity

Einstein equations are addressed with the energy-momentum tensor that appears if the equations under discussion are required to possess conformal invariance. It is proved that thus derived equations (equations of conformally invariant geometrodynamics) can have not only smooth solutions, but also solutions with discontinuities on space-like hypersurfaces. The solutions obtained are similar to the well-known discontinuous Einstein equation solutions like shock-wave solutions, extended-source solutions, etc. For the centrally symmetric stationary solution discussed in the paper, the discontinuity surface removes the singularity. The degree of generality of this solution regularization mechanism is discussed. The issue of the mechanism that forces any smooth solution in the conformally invariant geometrodynamics to be rearranged into the discontinuous one when certain conditions are met is also discussed. The conditions can be: (1) sound speed becoming to be higher than light speed; (2) the solution becoming intolerant to smaller and smaller-scale perturbation modes.

gr-qc↗

arXiv2002

Fermions in conformally invariant geometrodynamics

Dynamic equations that are the simplest conformally invariant generalization of Einstein equations with cosmological term are considered. Dimensions and Weyl weights of the additional geometrical fields (the vector and the antisymmetric tensor) appearing in the scheme are such, that they admit an unexpected interpretation. It is proved that the fields can be interpreted as observed, generated by bispinor degrees of freedom. The vacuum polarization density matrix leads to different probabilities of different helicity particle generation.

gr-qc↗

arXiv2001

On correspondence between tensors and bispinors

It is known that in the four-dimensional Riemannian space the complex bispinor generates a number of tensors: scalar, pseudo-scalar, vector, pseudo-vector, antisymmetric tensor. This paper solves the inverse problem: the above tensors are arbitrarily given, it is necessary to find a bispinor (bispinors) reproducing the tensors. The algorithm for this mapping constitutes construction of Hermitean matrix $M$ from the tensors and finding its eigenvalue spectrum. A solution to the inverse problem exists only when $M$ is nonnegatively definite. Under this condition a matrix $Z$ satisfying equation $M=ZZ^{+}$ can be found. One and the same system of tensor values can be used to construct the matrix $Z$ accurate to an arbitrary factor on the left-hand side, viz. unitary matrix $U$ in polar expansion $Z=HU$. The matrix $Z$ is shown to be expandable to a set of bispinors, for which the unitary matrix $U$ is responsible for the internal (gauge) degrees of freedom. Thus, a group of gauge transformations depends only on the Riemannian space dimension, signature, and the number field used. The constructed algorithm for mapping tensors to bispinors admits extension to Riemannian spaces of a higher dimension.

math-ph↗

arXiv2001

On a relation between the Bach equation and the equation of geometrodynamics

The Bach equation and the equation of geometrodynamics are based on two quite different physical motivations, but in both approaches, the conformal properties of gravitation plays the key role. In this paper we present an analysis of the relation between these two equations and show that the solutions of the equation of geometrodynamics are of a more general nature. We show the following non-trivial result: there exists a conformally invariant Lagrangian, whose field equation generalizes the Bach equation and has as solutions those Ricci tensors which are solutions to the equation of geometrodynamics.

gr-qc↗