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arXiv:1003.5902 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum and pseudoclassical descriptions of nonrelativistic spinning particles in noncommutative space

Tiago C. Adorno, Mario Cesar Baldiotti, Dmitri M Gitman

Published Dec 15, 2010• 7 citations
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Abstract

We construct a nonrelativistic wave equation for spinning particles in the noncommutative space (in a sense, a $\ensuremath{\theta}$ modification of the Pauli equation). To this end, we consider the nonrelativistic limit of the $\ensuremath{\theta}$-modified Dirac equation. To complete the consideration, we present a pseudoclassical model (\`a la Berezin-Marinov) for the corresponding nonrelativistic particle in the noncommutative space. To justify the latter model, we demonstrate that its quantization leads to the $\ensuremath{\theta}$-modified Pauli equation. We extract $\ensuremath{\theta}$-modified interaction between a nonrelativistic spin and a magnetic field from such a Pauli equation and construct a $\ensuremath{\theta}$ modification of the Heisenberg model for two coupled spins placed in an external magnetic field. In the framework of such a model, we calculate the probability transition between two orthogonal Einstein-Podolsky-Rosen states for a pair of spins in an oscillatory magnetic field and show that some of such transitions, which are forbidden in the commutative space, are possible due to the space noncommutativity. This allows us to estimate an upper bound on the noncommutativity parameter.

DOI
10.1103/physrevd.82.123516

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We construct a nonrelativistic wave equation for spinning particles in the noncommutative space (in a sense, a \ensuremathθ\ensuremath{\theta}\ensuremathθ modification of the Pauli equation). To this end, we consider the nonrelativistic limit of the \ensuremathθ\ensuremath{\theta}\ensuremathθ-modified Dirac equation. To complete the consideration, we present a pseudoclassical model (`a la Berezin-Marinov) for the corresponding nonrelativistic particle in the noncommutative space. To justify the latter model, we demonstrate that its quantization leads to the \ensuremathθ\ensuremath{\theta}\ensuremathθ-modified Pauli equation. We extract \ensuremathθ\ensuremath{\theta}\ensuremathθ-modified interaction between a nonrelativistic spin and a magnetic field from such a Pauli equation and construct a \ensuremathθ\ensuremath{\theta}\ensuremathθ modification of the Heisenberg model for two coupled spins placed in an external magnetic field. In the framework of such a model, we calculate the probability transition between two orthogonal Einstein-Podolsky-Rosen states for a pair of spins in an oscillatory magnetic field and show that some of such transitions, which are forbidden in the commutative space, are possible due to the space noncommutativity. This allows us to estimate an upper bound on the noncommutativity parameter.
Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory.It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices.Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables.The first model implies Pauli equation upon the canonical quantization.The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.
Reparametrization invariant dynamics on a sphere, being parametrized by angular momentum coordinates, represents an appropriate framework for semiclassical description of nonrelativistic spin. The space can be quantized according to Berezin–Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for describing spin without the use of Grassmann variables. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.
In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in a noncommutative phase-space as well as the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic fields are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.
In this work, we study the noncommutative nonrelativistic quantum dynamics of a neutral particle, which possesses permanent magnetic and electric dipole moments, in the presence of external electric and magnetic fields. We use the Foldy-Wouthuysen transformation of the Dirac spinor with a nonminimal coupling to obtain the nonrelativistic limit. In this limit, we study the noncommutative quantum dynamics and obtain the noncommutative Anandan geometric phase. We analyze the situation where the magnetic dipole moment of the particle is zero, and we obtain the noncommutative version of the He-McKellar-Wilkens effect. We demonstrate that this phase in the noncommutative case is a geometric dispersive phase. We also investigate this geometric phase by considering the noncommutativity in the phase space, and the Anandan phase is obtained.
We study the noncommutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric qaudrupole moment, in the presence of an external magnetic field. First, by introducing a shift for the magnetic field, we give the Schrödinger equations in the presence of an external magnetic field both on a noncommutative space and a noncommutative phase space, respectively. Then by solving the Schrödinger equations both on a noncommutative space and a noncommutative phase space, we obtain quantum phases of the electric quadrupole moment, respectively. We demonstrate that these phases are geometric and dispersive.