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arXiv:1302.4284 · Journal of Mathematical Physics

Classical limits of quantum mechanics on a non-commutative configuration space

Fabio Benatti, Laure Gouba

Published Jun 1, 2013• 6 citations
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Abstract

We consider a model of non-commutative quantum mechanics given by two harmonic oscillators over a non-commutative two dimensional configuration space. We study possible ways of removing the non-commutativity based on the classical limit context known as anti-Wick quantization. We show that removal of non-commutativity from the configuration space and from the canonical operators is not commuting operation.

DOI
10.1063/1.4808181

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We consider a model of non-commutative quantum mechanics given by two harmonic oscillators over a non-commutative two dimensional configuration space. We study possible ways of removing the non-commutativity based on the classical limit context known as anti-Wick quantization. We show that removal of non-commutativity from the configuration space and from the canonical operators is not commuting operation.
The classical limit of two quantum mechanical oscillators on a non-commutative configuration space has been recently studied based on the anti-Wick quantization scheme which makes use of completely positive maps constructed by means of coherent states. The limits corresponding to the removal of configuration-space non-commutativity and position-momentum non-commutativity do not commute. While reducing to a commutative configuration-space and then to classical position and momentum yields a classical theory on a 444-dimensional phase-space, the inverted sequence of limits does instead provide a classical theory on a 222-dimensional phase-space. Moreover, if the free dynamics of the two non-commutative quantum oscillators is also taken into account, in the first case one again obtains the dynamics of two classical free oscillators dynamics; instead, in the second case, one loses completely track of the dynamics and reduces to a purely kinematical 111-dimensional description. We herewith explain these results by examining the fate of the phase-space localisation properties of the Wigner functions of coherent states under the various limits.
In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.
In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation.In coordinates {q i , p k } the canonical symplectic two-form is ω 0 = dq i ∧ d p i .It is well known in symplectic mechanics [5, 6,9] that the interaction of a charged particle with a magnetic field can be described in a Hamiltonian formalism without a choice of a potential.This is done by means of a modified symplectic twoform ω = ω 0 -eF, where e is the charge and the (time-independent) magnetic field F is closed: dF = 0.With this symplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: {p k , p l } = e F kl (q).Similarly a closed two-form in p-space G may be introduced.Such a dual magnetic field G interacts with the particle's dual charge r.A new modified symplectic two-form ω = ω 0 -eF + rG is then defined.Now, both p-and q-variables will cease to Poisson commute and upon quantisation they become noncommuting operators.In the particular case of a linear phase space R 2N , it makes sense to consider constant F and G fields.It is then possible to define, by a linear transformation, global Darboux coordinates: {ξ i , π k } = δ i k .These can then be quantised in the usual way [ ξ i , π k ] = i δ i k .The case of a quadratic potential is examined with some detail when N equals 2 and 3.
It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant ℏ\hbar ℏ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn\mathbb R^nRn. So that in the Hamiltonian formalism, the phase space is Rn×Rn\mathbb R^n\times \mathbb R^nRn×Rn with its canonical symplectic form σ\sigma σ. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.
It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock–Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock–Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.
In this article, we continue our investigation on the role of non-commutativity in quantum theory. Using the method explained in "On non-commutativity in quantum theory (I): from classical to quantum probability", we analyze two toy models which exhibit non-commutativity between the corresponding position and velocity random variables. In particular, using ordinary probability theory, we study the kinematics of a point-like particle jumping at random over a discrete random space. We show that, after the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space.