Noncommutative version of D-dimensional relativistic particle is proposed. We consider the particle interacting with the configuration space variable $\theta^{\mu\nu}(\tau)$ instead of the numerical matrix. The corresponding Poincare invariant action has a local symmetry, which allows one to impose the gauge $\theta^{0i}=0, ~ \theta^{ij}=const$. The matrix $\theta^{ij}$ turns out to be the noncommutativity parameter of the gauge fixed formulation. Poincare transformations of the gauge fixed formulation are presented in the manifest form. Consistent coupling of NC relativistic particle to the electromagnetic field is constructed and discussed.Comment: 12 pages, LaTex file, misprints correcte
We construct a relativistic spinning-particle Lagrangian where spin is considered as a composite quantity constructed on the base of a non-Grassmann vector-like variable. The variational problem guarantees both a fixed value of the spin and the Frenkel condition on the spin-tensor. The Frenkel condition inevitably leads to relativistic corrections of the Poisson algebra of the position variables: their classical brackets became noncommutative. We construct the relativistic quantum mechanics in the canonical formalism (in the physical-time parametrization) and in the covariant formalism (in an arbitrary parametrization). We show how state vectors and operators of the covariant formulation can be used to compute the mean values of physical operators in the canonical formalism, thus proving its relativistic covariance. We establish relations between the Frenkel electron and positive-energy sector of the Dirac equation. Various candidates for the position and spin operators of an electron acquire clear meaning and interpretation in the Lagrangian model of the Frenkel electron. Our results argue in favor of Pryce's (d)-type operators as the spin and position operators of Dirac theory. This implies that the effects of non-commutativity could be expected already at the Compton wavelength. We also present the manifestly covariant form of the spin and position operators of the Dirac equation.
We present Lagrangian which implies both necessary constraints and dynamical equations for position and spin of relativistic spin one-half particle. The model is consistent for any value of magnetic moment µ and for arbitrary electromagnetic background. Our equations coincide with those of Frenkel in the approximation in which the latter have been obtained by Frenkel. Transition from approximate to exact equations yields two structural modifications of the theory. First, Frenkel condition on spin-tensor turns into the Pirani condition. Second, canonical momentum is no more proportional to velocity. Due to this, even when µ = 1 (Frenkel case), the complete and approximate equations predict different behavior of particle. The difference between momentum and velocity means extra contribution into spin-orbit interaction. To estimate the contribution, we found exact solution to complete equations for the case of uniform magnetic field. While Frenkel electron moves around the circle, our particle experiences magnetic Zitterbewegung, that is oscillates in the direction of magnetic field with amplitude of order of Compton wavelength for the fast particle. Besides, the particle has dipole electric moment. * Electronic address: alexei.deriglazov@ufjf.edu.br † Electronic address: pupasov@phys.tsu.ru arXiv:1401.7641v3 [hep-th] 26 Jun 2014 to commutatoras well as with respect to anticommutatorThese equations prompt that spin-space in classical model can be described by either even or odd (Grassmann) variables. The pioneer model based on odd variables have been constructed by Berezin and Marinov [26]. This gives very economic and elegant scheme for semiclassical description of spin. For non relativistic spin, the Lagrangian readsSince the Lagrangian is linear onξ i , their conjugate momenta coincide with ξ, π i = ∂L ∂ξi = iξ i . The relations represent the Dirac second-class constraints and are taken into account by transition from the grassmannian Poisson bracket to the Dirac one. After that, the constraints can be used to exclude π i . Dirac bracket of the remaining variables reads {ξ i , ξ j } DB = iδ ij . Comparing this with Eq. (2), we quantize the model replacing ξ i → 2 σ i . Relativistic spin is described in a similar way [26][27][28][29]. The problem here is that Grassmann classical mechanics represents a rather formal mathematical construction. It leads to certain difficulties [26,29] in attempts to use it for description the spin effects on the semiclassical level, before the quantization. Besides, generalization of Grassmann mechanics to higher spins is not known [30]. Hence it would be interesting to describe spin on a base of usual variables, that is we intend to arrive at the commutator algebra (1) instead of (2).Contrary to the models based on commuting spinors [32,33], in the Berezin-Marinov approach the σ i (or γ µ ) matrices do not appear in classical theory but produced through the quantization process. The same turns out to be true in our model based on non-Grassmann vector for description of spin.Very ge...
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