Hamiltonization of theories with degenerate coordinates

We consider a class of Lagrangian theories where part of the coordinates does not have any time derivatives in the Lagrange function (we call such coordinates degenerate). We advocate that it is reasonable to reconsider the conventional definition of singularity based on the usual Hessian and, moreover, to simplify the conventional Hamiltonization procedure. In particular, in such a procedure, it is not necessary to complete the degenerate coordinates with the corresponding conjugate momenta.Comment: 14 pages, LaTex fil

Dirac Equation in Noncommutative Space for Hydrogen Atom

We consider the energy levels of a hydrogen-like atom in the framework of $\theta$-modified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels $2S_{1/2}, 2P_{1/2}$ and $2P_{3/2}$ is lifted completely, such that new transition channels are allowed.Comment: 9 pages, 1 figure; typos correcte

Particles with anomalous magnetic moment in external e.m. fields: the proper time formulation

In this paper we evaluate the expression for the Green function of a pseudo-classical spinning particle interacting with constant electromagnetic external fields by taking into account the anomalous magnetic and electric moments of the particle. The spin degrees of freedom are described in terms of Grassmann variables and the evolution operator is obtained through the Fock-Schwinger proper time method.Comment: 10 page

Gauge Symmetries on θ\theta-Deformed Spaces

A Hamiltonian formulation of gauge symmetries on noncommutative ($\theta$ deformed) spaces is discussed. Both cases- star deformed gauge transformation with normal coproduct and undeformed gauge transformation with twisted coproduct- are considered. While the structure of the gauge generator is identical in either case, there is a difference in the computation of the graded Poisson brackets that yield the gauge transformations. Our analysis provides a novel interpretation of the twisted coproduct for gauge transformations.Comment: LaTex, 20 pages, no figure

Canonical quantization of so-called non-Lagrangian systems

We present an approach to the canonical quantization of systems with equations of motion that are historically called non-Lagrangian equations. Our viewpoint of this problem is the following: despite the fact that a set of differential equations cannot be directly identified with a set of Euler-Lagrange equations, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. It turns out that in the general case the hamiltonization and canonical quantization of such an action are non-trivial problems, since the theory involves time-dependent constraints. We adopt the general approach of hamiltonization and canonical quantization for such theories (Gitman, Tyutin, 1990) to the case under consideration. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The proposed scheme is applied to the quantization of a general quadratic theory. In addition, we consider the quantization of a damped oscillator and of a radiating point-like charge.Comment: 13 page

Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action

It is known that actions of field theories on a noncommutative space-time can be written as some modified (we call them $\theta$-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and usual quantum mechanical features of the corresponding field theory. The $\theta$-modification for arbitrary finite-dimensional nonrelativistic system was proposed by Deriglazov (2003). In the present article, we discuss the problem of constructing $\theta$-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract $\theta$-modified actions of the relativistic particles from path integral representations of the corresponding noncommtative field theory propagators. We consider Klein-Gordon and Dirac equations for the causal propagators in such theories. Then we construct for the propagators path-integral representations. Effective actions in such representations we treat as $\theta$-modified actions of the relativistic particles. To confirm the interpretation, we quantize canonically these actions. Thus, we obtain the Klein-Gordon and Dirac equations in the noncommutative field theories. The $\theta$-modified action of the relativistic spinning particle is just a generalization of the Berezin-Marinov pseudoclassical action for the noncommutative case

On Modifications of the Sp(2) Covariant Superfield Quantization

We propose a modification of the Sp(2) covariant superfield quantization to realize a superalgebra of generating operators isomorphic to the massless limit of the corresponding superalgebra of the osp(1,2) covariant formalism. The modified scheme ensures the compatibility of the superalgebra of generating operators with extended BRST symmetry without imposing restrictions eliminating superfield components from the quantum action. The formalism coincides with the Sp(2) covariant superfield scheme and with the massless limit of the osp(1,2) covariant quantization in particular cases of gauge-fixing and solutions of the quantum master equations.Comment: 11 pages, to be published in Phys. Lett.

Field on Poincare group and quantum description of orientable objects

We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner's ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3+1 dimensions) amounts to introducing wave functions that depend on elements of the Poincare group $G$. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group $\Pi =G\times G$. All such transformations can be studied by considering a generalized regular representation of $G$ in the space of scalar functions on the group, $f(x,z)$, that depend on the Minkowski space points $x\in G/Spin(3,1)$ as well as on the orientation variables given by the elements $z$ of a matrix $Z\in Spin(3,1)$. In particular, the field $f(x,z)$ is a generating function of usual spin-tensor multicomponent fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.Comment: 46 page

Constraint algebra for Regge-Teitelboim formulation of gravity

We consider the formulation of the gravity theory first suggested by Regge and Teitelboim where the space-time is a four-dimensional surface in a flat ten-dimensional space. We investigate a canonical formalism for this theory following the approach suggested by Regge and Teitelboim. Under constructing the canonical formalism we impose additional constraints agreed with the equations of motion. We obtain the exact form of the first-class constraint algebra. We show that this algebra contains four constraints which form a subalgebra (the ideal), and if these constraints are fulfilled, the algebra becomes the constraint algebra of the Arnowitt-Deser-Misner formalism of Einstein's gravity. The reasons for the existence of additional first-class constraints in the canonical formalism are discussed.Comment: LaTeX, 12 pages; in this version the misprints in eq. (37) and (41) was correcte