arXiv subjects

A. A. Deriglazov

Publications and source records attributed to A. A. Deriglazov.

At least 19 recordsLinked to original sources

arXiv2014

Variational problem for Hamiltonian system on so(k, m) Lie-Poisson manifold and dynamics of semiclassical spin

We describe the procedure for obtaining Hamiltonian equations on a manifold with $so(k, m)$ Lie-Poisson bracket from a variational problem. This implies identification of the manifold with base of a properly constructed fiber bundle embedded as a surface into the phase space with canonical Poisson bracket. Our geometric construction underlies the formalism used for construction of spinning particles in [24-27], and gives precise mathematical formulation of the oldest idea about spin as the "inner angular momentum".

math-ph↗

arXiv2014

Variational problem for the Frenkel and the Bargmann-Michel-Telegdi (BMT) equations

We propose Lagrangian formulation for the particle with value of spin fixed within the classical theory. The Lagrangian turns out to be invariant under non-abelian group of local symmetries. As the gauge-invariant variables for description of spin we can take either the Frenkel tensor or the BMT vector. Fixation of spin within the classical theory implies $O(\hbar)$-corrections to the corresponding equations of motion.

hep-th↗

arXiv2012

Classical-mechanical models without observable trajectories and the Dirac electron

We construct a non-Grassmann spinning-particle model which, by analogy with quantum mechanics, does not admit the notion of a trajectory within the position space. The pseudo-classical character of the model allows us to avoid the inconsistencies arising in the quantum-mechanical interpretation of a one-particle sector of the Dirac equation.

hep-th↗

arXiv2012

Non-Grassmann mechanical model of the Dirac equation

We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both $Γ^μ$ and $Γ^{μν}$\,-matrices in the course of quantization. Canonical quantization of the model implies the Dirac equation. We present the detailed analysis of both the Lagrangian and the Hamiltonian formulations of the model and obtain the general solution to the classical equations of motion. Comparing {\it Zitterbewegung} of the spatial coordinate with the evolution of spin, we ask on the possibility of space-time interpretation for the inner space of spin. We enumerate similarities between our analogous model of the Dirac equation and the two-body system subject to confining potential which admits only the elliptic orbits of the order of de Broglie wave-length. The Dirac equation dictates the perpendicularity of the elliptic orbits to the direction of center-of-mass motion.

hep-th↗

arXiv2012

Semiclassical Description of Relativistic Spin without use of Grassmann variables and the Dirac equation

We propose a relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Both $Γ$\,-matrices and the relativistic spin tensor are produced through the canonical quantization of the classical variables which parametrize the properly constructed relativistic spin surface. Although there is no mass-shell constraint in our model, our particle's speed cannot exceed the speed of light. The classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. In particular, the position variable experiences {\it Zitterbewegung} in noninteracting theory. The classical equations for the spin tensor are the same as those of the Barut-Zanghi model of a spinning particle.

hep-th↗

arXiv2012

Spinning-particle model for the Dirac equation and the relativistic Zitterbewegung

We construct the relativistic particle model without Grassmann variables which meets the following requirements. A) Canonical quantization of the model implies the Dirac equation. B) The variable which experiences {\it Zitterbewegung}, represents a gauge non-invariant variable in our model. Hence our particle does not experiences the undesirable {\it Zitterbewegung}. C) In the nonrelativistic limit spin is described by three-vector, as it could be expected.

hep-th↗

arXiv2011

Classical mechanics in reparametrization-invariant formulation and the Schrödinger equation

The dynamics of any classical-mechanics system can be formulated in the reparametrization-invariant (RI) form (that is we use the parametric representation for trajectories, ${\bf x}={\bf x}(τ)$, $t=t(τ)$ instead of ${\bf x}={\bf x}(t)$). In this pedagogical note we discuss what the quantization rules look like for the RI formulation of mechanics. We point out that in this case some of the rules acquire an intuitively clearer form. Hence the formulation could be an alternative starting point for teaching the basic principles of quantum mechanics. The advantages can be resumed as follows. a) In RI formulation both the temporal and the spatial coordinates are subject to quantization. b) The canonical Hamiltonian of RI formulation is proportional to the quantity $\tilde H=p_t+H$, where $H$ is the Hamiltonian of the initial formulation. Due to the reparametrization invariance, the quantity $\tilde H$ vanishes for any solution, $\tilde H=0$. So the corresponding quantum-mechanical operator annihilates the wave function, $\hat{\tilde H}Ψ=0$, which is precisely the Schrödinger equation $i\hbar\partial_tΨ=\hat HΨ$. As an illustration, we discuss quantum mechanics of the relativistic particle.

math-ph↗

arXiv2011

Five-Dimensional Mechanics as the Starting Point for the Magueijo-Smolin Doubly Special Relativity

We discuss a way to obtain the doubly special relativity kinematical rules (the deformed energy-momentum relation and the nonlinear Lorentz transformations of momenta) starting from a singular Lagrangian action of a particle with linearly realized SO(1,4) symmetry group. The deformed energy-momentum relation appears in a special gauge of the model. The nonlinear transformations of momenta arise from the requirement of covariance of the chosen gauge.

hep-th↗

arXiv2011

Generalization of the Extended Lagrangian Formalism on a Field Theory and Applications

Formalism of extended Lagrangian represent a systematic procedure to look for the local symmetries of a given Lagrangian action. In this work, the formalism is discussed and applied to a field theory. We describe it in detail for a field theory with first-class constraints present in the Hamiltonian formulation. The method is illustrated on examples of electrodynamics, Yang-Mills field and non-linear sigma model.

hep-th↗

arXiv2011

Formal similarity between mathematical structures of electrodynamics and quantum mechanics

Electromagnetic phenomena can be described by Maxwell equations written for the vectors of electric and magnetic field. Equivalently, electrodynamics can be reformulated in terms of an electromagnetic vector potential. We demonstrate that the Schrödinger equation admits an analogous treatment. We present a Lagrangian theory of a real scalar field $ϕ$ whose equation of motion turns out to be equivalent to the Schrödinger equation with time independent potential. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics. The field $ϕ$ is in the same relation to the real and imaginary part of a wave function as the vector potential is in respect to electric and magnetic fields. Preservation of quantum-mechanics probability is just an energy conservation law of the field $ϕ$.

quant-ph↗

arXiv2009

On singular Lagrangian underlying the Schrödinger equation

We analyze the properties that manifest Hamiltonian nature of the Schrödinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation). It is used to show that any solution to the Schrödinger equation with time independent potential can be presented in the form $Ψ=(-\frac{\hbar^2}{2m}\triangle+V)ϕ+i\hbar\partial_tϕ$, where the real field $ϕ(t, x^i)$ is some solution to nonsingular Lagrangian theory being specified below. Preservation of probability turns out to be the energy conservation law for the field $ϕ$. After introducing the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics: the real field $ϕ$ turns out to be a kind of potential for a wave function.

math-ph↗

arXiv2009

Improved extended Hamiltonian and search for local symmetries

We analyze a structure of the singular Lagrangian $L$ with first and second class constraints of an arbitrary stage. We show that there exist an equivalent Lagrangian (called the extended Lagrangian $\tilde L$) that generates all the original constraints on second stage of the Dirac-Bergmann procedure. The extended Lagrangian is obtained in closed form through the initial one. The formalism implies an extension of the original configuration space by auxiliary variables. Some of them are identified with gauge fields supplying local symmetries of $\tilde L$. As an application of the formalism, we found closed expression for the gauge generators of $\tilde L$ through the first class constraints. It turns out to be much more easy task as those for $L$. All the first class constraints of $L$ turn out to be the gauge symmetry generators of $\tilde L$. By this way, local symmetries of $L$ with higher order derivatives of the local parameters decompose into a sum of the gauge symmetries of $\tilde L$. It proves the Dirac conjecture in the Lagrangian framework.

hep-th↗

arXiv2007

Formulation of a constrained system in terms of extended Lagrangian and its local symmetries

It is shown that an arbitrary singular Lagrangian theory (with first and second class constraints up to $N$-th stage in the Hamiltonian formulation) can be reformulated as a theory with at most third-stage constraints. The corresponding Lagrangian $\tilde L$ can be obtained by pure algebraic methods, its manifest form in terms of quantities of the initial formulation is found. Local symmetries of $\tilde L$ are obtained in closed form. All the first class constraints of the initial Lagrangian turn out to be gauge symmetry generators for $\tilde L$.

hep-th↗

arXiv2007

Reconstruction of higher stage first class constraints into the secondary ones

We analyze a singular theory with first class constraints of an arbitrary stage. Relation among the formulations of the constrained system in terms of complete and extended Hamiltonians is clarified. We replace the extended Hamiltonian $H_{ext}$ by an improved one. The improved Hamiltonian has the same structure as $H_{ext}$ (higher stage constraints enter into the Hamiltonian in the manifest form), but, in contrast to $H_{ext}$, it arises as the complete Hamiltonian for some Lagrangian $\tilde L$, called the extended Lagrangian. This implies, in particular, that all the quantities appearing in the improved Hamiltonian have a clear meaning in the Dirac framework. $\tilde L$ is obtained in a closed form in terms of quantities of the initial formulation $L$. The formulations with $L$ and $\tilde L$ turn out to be equivalent. As an application of the formalism, we found local symmetries of $\tilde L$ in a closed form. All the constraints of $L$ turn out to be gauge symmetry generators for $\tilde L$. The procedure is illustrated with an example of a model with fourth-stage constraints.

hep-th↗

arXiv2007

Open string with a background B-field as the first order mechanics, noncommutativity and soldering formalism

To study noncommutativity properties of the open string with constant B-field we construct a mechanical action which reproduces classical dynamics of the string sector under consideration. It allows one to apply the Dirac quantization procedure for constrained systems in a direct and unambiguous way. The mechanical action turns out to be the first order system without taking the strong field limit $B\longrightarrow\infty$. In particular, it is true for zero mode of the string coordinate which means that the noncommutativity is intrinsic property of this mechanical system. We describe the arbitrariness in the relation existent between the mechanical and the string variables and show that noncommutativity of the string variables on the boundary can be removed. It is in correspondence with the result of Seiberg and Witten on relation among noncommutative and ordinary Yang-Mills theories. The recently developed soldering formalism helps us to establish a connection between the original open string action and the Polyakov action.

hep-th↗

arXiv2007

Construction of Lagrangian local symmetries for general quadratic theory

We propose a procedure which allows one to construct local symmetry generators of general quadratic Lagrangian theory. Manifest recurrence relations for generators in terms of so-called structure matrices of the Dirac formalism are obtained. The procedure fulfilled in terms of initial variables of the theory, and do not implies either separation of constraints on first and second class subsets or any other choice of basis for constraints.

hep-th↗

arXiv2007

Conversion of second class constraints by deformation of Lagrangian local symmetries

For a theory with first and second class constraints, we propose a procedure for conversion of second class constraints based on deformation the structure of local symmetries of the Lagrangian formulation. It does not require extension or reduction of configuration space of the theory. We give examples in which the initial formulation implies a non linear realization of some global symmetries, therefore is not convenient. The conversion reveals hidden symmetry presented in the theory. The extra gauge freedom of conversed version is used to search for a parameterization which linearizes the equations of motion. We apply the above procedure to membrane theory (in the formulation with world-volume metric). In the resulting version, all the metric components are gauge degrees of freedom. The above procedure works also in a theory with only second class constraints presented. As an examples, we discuss arbitrary dynamical system of classical mechanics subject to kinematic constraints, O(N)-invariant nonlinear sigma-model, and the theory of massive vector field with Maxwell-Proca Lagrangian.

hep-th↗

arXiv2005

Analysis of constrained theories without use of primary constraints

It is shown that the Dirac approach to Hamiltonization of singular theories can be slightly modified in such a way that primary Dirac constraints do not appear in the process. According to the modified scheme, Hamiltonian formulation of singular theory is first order Lagrangian formulation, further rewritten in special coordinates.

hep-th↗