Quantization of (2+1)-spinning particles and bifermionic constraint problem

This work is a natural continuation of our recent study in quantizing relativistic particles. There it was demonstrated that, by applying a consistent quantization scheme to a classical model of a spinless relativistic particle as well as to the Berezin-Marinov model of 3+1 Dirac particle, it is possible to obtain a consistent relativistic quantum mechanics of such particles. In the present article we apply a similar approach to the problem of quantizing the massive 2+1 Dirac particle. However, we stress that such a problem differs in a nontrivial way from the one in 3+1 dimensions. The point is that in 2+1 dimensions each spin polarization describes different fermion species. Technically this fact manifests itself through the presence of a bifermionic constant and of a bifermionic first-class constraint. In particular, this constraint does not admit a conjugate gauge condition at the classical level. The quantization problem in 2+1 dimensions is also interesting from the physical viewpoint (e.g. anyons). In order to quantize the model, we first derive a classical formulation in an effective phase space, restricted by constraints and gauges. Then the condition of preservation of the classical symmetries allows us to realize the operator algebra in an unambiguous way and construct an appropriate Hilbert space. The physical sector of the constructed quantum mechanics contains spin-1/2 particles and antiparticles without an infinite number of negative-energy levels, and exactly reproduces the one-particle sector of the 2+1 quantum theory of a spinor field.Comment: LaTex, 24 pages, no figure

Canonical form of Euler-Lagrange equations and gauge symmetries

The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler-Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proven that for local theories all the gauge generators are local in time operators.Comment: 27 pages, LaTex fil

Two-dimensional metric and tetrad gravities as constrained second order systems

Using the Gitman-Lyakhovich-Tyutin generalization of the Ostrogradsky method for analyzing singular systems, we consider the Hamiltonian formulation of metric and tetrad gravities in two-dimensional Riemannian spacetime treating them as constrained higher-derivative theories. The algebraic structure of the Poisson brackets of the constraints and the corresponding gauge transformations are investigated in both cases.Comment: replaced with revised version published in Mod.Phys.Lett.A22:17-28,200

Photon propagation in noncommutative QED with constant external field

We find dispersion laws for the photon propagating in the presence of mutually orthogonal constant external electric and magnetic fields in the context of the $\theta$-expanded noncommutative QED. We show that there is no birefringence to the first order in the noncommutativity parameter $$ By analyzing the group velocities of the photon eigenmodes we show that there occurs superluminal propagation for any direction. This phenomenon depends on the mutual orientation of the external electromagnetic fields and the noncommutativity vector. We argue that the propagation of signals with superluminal group velocity violates causality in spite of the fact that the noncommutative theory is not Lorentz-invariant and speculate about possible workarounds

Comments on ``A note on first-order formalism and odd-derivative actions'' by S. Deser

We argue that the obstacles to having a first-order formalism for odd-derivative actions presented in a pedagogical note by Deser are based on examples which are not first-order forms of the original actions. The general derivation of an equivalent first-order form of the original second-order action is illustrated using the example of topologically massive electrodynamics (TME). The correct first-order formulations of the TME model keep intact the gauge invariance presented in its second-order form demonstrating that the gauge invariance is not lost in the Ostrogradsky process.Comment: 6 pages, references are adde

Nilpotent noncommutativity and renormalization

We analyze renormalizability properties of noncommutative (NC) theories with a bifermionic NC parameter. We introduce a new 4-dimensional scalar field model which is renormalizable at all orders of the loop expansion. We show that this model has an infrared stable fixed point (at the one-loop level). We check that the NC QED (which is one-loop renormalizable with usual NC parameter) remains renormalizable when the NC parameter is bifermionic, at least to the extent of one-loop diagrams with external photon legs. Our general conclusion is that bifermionic noncommutativity improves renormalizablility properties of NC theories.Comment: 5 figures, a reference adde

Entanglement of two-qubit photon beam by magnetic field

We have studied the possibility of affecting the entanglement measure of 2-qubit system consisting of two photons with different fixed frequencies but with two arbitrary linear polarizations, moving in the same direction, by the help of an applied external magnetic field. The interaction between the magnetic field and the photons in our model is achieved through intermediate electrons that interact with both the photons and the magnetic field. The possibility of exact theoretical analysis of this scheme is based on known exact solutions that describe the interaction of an electron subjected to an external magnetic field (or a medium of electrons not interacting with each other) with a quantized field of two photons. We adapt these exact solutions to the case under consideration. Using explicit wave functions for the resulting electromagnetic field, we calculate the entanglement measure of the photon beam as a function of the applied magnetic field and parameters of the electron medium

On Quantization of Time-Dependent Systems with Constraints

The Dirac method of canonical quantization of theories with second class constraints has to be modified if the constraints depend on time explicitly. A solution of the problem was given by Gitman and Tyutin. In the present work we propose an independent way to derive the rules of quantization for these systems, starting from physical equivalent theory with trivial non-stationarity.Comment: 4 page

Pseudoclassical description of the massive Dirac particles in odd dimensions

A pseudoclassical model is proposed to describe massive Dirac (spin one-half) particles in arbitrary odd dimensions. The quantization of the model reproduces the minimal quantum theory of spinning particles in such dimensions. A dimensional duality between the model proposed and the pseudoclassical description of Weyl particles in even dimensions is discussed.Comment: 12 pages, LaTeX (RevTeX