In this work we have shown precisely that the curvature of a 2-sphere introduces quantum features in the system through the introduction of the noncommutative (NC) parameter that appeared naturally via equations of motion. To obtain this result we us
ed the fact that quantum mechanics can be understood as a NC symplectic geometry, which generalized the standard description of classical mechanics as a symplectic geometry. In this work, we have also analyzed the dynamics of the model of a free particle over a 2-sphere in a NC phase-space. Besides, we have shown the solution of the equations of motion allows one to show the equivalence between the movement of the particle physical degrees of freedom upon a 2-sphere and the one described by a central field. We have considered the effective force felt by the particle as being caused by the curvature of the space. We have analyzed the NC Poisson algebra of classical observables in order to obtain the NC corrections to Newtons second law. We have demonstrated precisely that the curvature of the space acted as an effective potential for a free particle in a flat phase-space. Besides, through NC coherent states quantization we have obtained the Green function of the theory. The result have confirmed that we have an UV cutoff for large momenta in the NC kernel. We have also discussed the relation between affine connection and Dirac brackets, as they describe the proper evolution of the model over the surface of constraints in the Lagrangian and Hamiltonian formalisms, respectively. As an application, we have treated the so-called textit{Zitterbewegung} of the Dirac electron. Since it is assumed to be an observable effect, then we have traced its physical origin by assuming that the electron has an internal structure.
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 $Gamma^mu$ and $Gamma^{mu u}$,-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.
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.
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.
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, ${bf x}={bf x}(tau)$, $t=t(tau)$ instead of ${bf x}={bf x}(t)$). In this ped
agogical 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, $hat{tilde H}Psi=0$, which is precisely the Schrodinger equation $ihbarpartial_tPsi=hat HPsi$. As an illustration, we discuss quantum mechanics of the relativistic particle.
In this work, we comment on two special points in the paper by S. Ghosh [Phys. Rev. D 74, 084019 (2006)]. First of all, the Lagrangian presented by the author does not describe the Magueijo- Smolin model of Doubly Special Relativity since it is equiv
alent to the Lagrangian of the standard free relativistic particle. We also show that the introduction of noncommutative structures is not relevant to the problem of Lorentz covariance in Ghosh formalism.
