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Representações e aspectos quânticos de sistemas não-comutativos

Júlio Glauber Ferreira dos Santos

Published Jun 28, 2013• 0 citations
OpenAlex

Abstract

This paper explores initially, in a intuitive form, some questions of non-commutativity in space well as the motivations for the study of non-commutative theories. Then be given an overight in some aspects of quantum theory non-commutative addressed in the plan. Following the basic rules of the non-commutative, see how to build states that saturate the uncertainty relations, i.e., get the least possible uncertainty in the observable in question, for a construction of non-commutative algebra of coordinates. In particular, following the search of states that saturate both relations Heisenberg uncertainty modied and show saturation of the Schrodinger's determinant for a given Gaussian state. Then, will be built the propagator for the free particle from the construction of noncommutative version of the plane wave, exploring the idea of average in coherent states. And also exploring the idea of the construction of wave packet in the non-commutative plane, get its evolved form using the path integral formalism, ending with the analysis of the Fourier's transform in non-commutative space. This paper analyzes the results and what can be done in a next project, with the tools presented.

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This paper explores initially, in a intuitive form, some questions of non-commutativity in space well as the motivations for the study of non-commutative theories. Then be given an overight in some aspects of quantum theory non-commutative addressed in the plan. Following the basic rules of the non-commutative, see how to build states that saturate the uncertainty relations, i.e., get the least possible uncertainty in the observable in question, for a construction of non-commutative algebra of coordinates. In particular, following the search of states that saturate both relations Heisenberg uncertainty modied and show saturation of the Schrodinger's determinant for a given Gaussian state. Then, will be built the propagator for the free particle from the construction of noncommutative version of the plane wave, exploring the idea of average in coherent states. And also exploring the idea of the construction of wave packet in the non-commutative plane, get its evolved form using the path integral formalism, ending with the analysis of the Fourier's transform in non-commutative space. This paper analyzes the results and what can be done in a next project, with the tools presented.
Our main focus is to explore different models in noncommutative spaces in higher dimensions. We provide a procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for the corresponding operators obey algebras whose uncertainty relations lead to minimal length, areas and volumes in phase space, which are in principle natural candidates of many different approaches of quantum gravity. We study some explicit models on these types of noncommutative spaces, first by utilising the perturbation theory, later in an exact manner. In many cases the operators are not Hermitian, therefore we use PT -symmetry and pseudo-Hermiticity property, wherever applicable, to make them self-consistent. Apart from building mathematical models, we focus on the physical implications of noncommutative theories too. We construct Klauder coherent states for the perturbative and nonperturbative noncommutative harmonic oscillator associated with uncertainty relations implying minimal lengths. In both cases, the uncertainty relations for the constructed states are shown to be saturated and thus imply to the squeezed coherent states. They are also shown to satisfy the Ehrenfest theorem dictating the classical like nature of the coherent wavepacket. The quality of those states are further underpinned by the fractional revival structure. More investigations into the comparison are carried out by a qualitative comparison between the dynamics of the classical particle and that of the coherent states based on numerical techniques. The qualitative behaviour is found to be governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures.
Inspired by a recent work that proposes using coherent states to evaluate the Feynman kernel in noncommutative space, we provide an independent formulation of the path-integral approach for quantum mechanics on the Moyal plane, with the transition amplitude defined between two coherent states of mean position coordinates. In our approach, we invoke solely a representation of the noncommutative algebra in terms of commutative variables. The kernel expression for a general Hamiltonian was found to contain Gaussian-like damping terms, and it is non-perturbative in the sense that it does not reduce to the commutative theory in the limit of vanishing θ—the noncommutative parameter. As an example, we studied the free particle's propagator which turned out to be oscillating with period being the product of its mass and θ. Further, it satisfies the Pauli equation for a charged particle with its spin aligned to a constant, orthogonal B field in the ordinary Landau problem, thus providing an interesting evidence of how noncommutativity can induce spin-like effects at the quantum mechanical level.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.
The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.