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arXiv:1004.1984 · Journal of Physics A Mathematical and Theoretical

Noncommutative quantum mechanics—a perspective on structure and spatial extent

Christian M. Rohwer, Konstantin G. Zloshchastiev, Laure Gouba, Frederik G. Scholtz

Published Jul 20, 2010• 29 citations
arXivHTMLPDF

Abstract

We explore the notion of spatial extent and structure, already alluded to in earlier literature, within the formulation of quantum mechanics on the noncommutative plane. Introducing the notion of position and its measurement in the sense of a weak measurement (positive operator-valued measure), we find two equivalent pictures in a position representation: a constrained local description in position containing additional degrees of freedom and an unconstrained nonlocal description in terms of the position without any other degrees of freedom. Both these descriptions have a corresponding classical theory which shows that the concept of extended, structured objects emerges quite naturally and unavoidably there. It is explicitly demonstrated that the conserved energy and angular momentum contain corrections to those of a point particle. We argue that these notions also extend naturally to the quantum level. The local description is found to be the most convenient as it manifestly displays additional information about the structure of quantum states that is more subtly encoded in the nonlocal, unconstrained description. Subsequently, we use this picture to discuss the free particle and harmonic oscillator as examples.

DOI
10.1088/1751-8113/43/34/345302

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We explore the notion of spatial extent and structure, already alluded to in earlier literature, within the formulation of quantum mechanics on the noncommutative plane. Introducing the notion of position and its measurement in the sense of a weak measurement (positive operator-valued measure), we find two equivalent pictures in a position representation: a constrained local description in position containing additional degrees of freedom and an unconstrained nonlocal description in terms of the position without any other degrees of freedom. Both these descriptions have a corresponding classical theory which shows that the concept of extended, structured objects emerges quite naturally and unavoidably there. It is explicitly demonstrated that the conserved energy and angular momentum contain corrections to those of a point particle. We argue that these notions also extend naturally to the quantum level. The local description is found to be the most convenient as it manifestly displays additional information about the structure of quantum states that is more subtly encoded in the nonlocal, unconstrained description. Subsequently, we use this picture to discuss the free particle and harmonic oscillator as examples.
In this thesis we shall demonstrate that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the formalism used entails a description that is non-local in that it requires all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary to specify quantum states completely. The remainder of the thesis, will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic field found by Susskind. It will be argued that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, with an interpretation in terms of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions make explicit the additional structure which is encoded more subtly in the non-local description. Additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory.
Recent works in foundations of quantum (field) theory and relativistic quantum information try to better grasp the interplay between the structure of quantum correlations and the constraints imposed by causality on physical operations. Chapter 1 is dedicated to the study of the conceptual implications of quantum nonlocality, a concept that subsumes that of entanglement in a certain way. We detail the recent information-theoretic approaches to understanding the structure of nonlocal correlations, and the issues the latter raise concerning the ability of local observers to isolate a system from its environment. Chapter 2 reviews in what sense imposing Poincare invariance affects entanglement detection and quantification procedures. This invariance ultimately forces a description of all quantum systems within the framework of quantum field theory, which leads to the impossibility of localized finite-energy states and to the divergence of all entanglement measures for local observers. We provide a solution to these two problems by showing that there exists a generic equivalence between a finite spatial resolution of the measurement apparatus and the exclusion of high-energy degrees of freedom from the definition of the observed system. This equivalence allows for an epistemic interpretation of the standard quantum formalism describing nonrelativistic localized systems and their correlations, hence a clarification of the origin of the finite measures of entanglement between such systems. Chapter 3 presents a recent theoretical framework that predicts the existence of correlations with indefinite causal order. In analogy to the information-theoretic approaches to nonlocal correlations, we introduce some principles that constrain the structure of such correlations, which is a first step toward a clear understanding of their physical origin.
With approaching quantum/noncommutative models for the deep microscopic spacetime in mind, and inspired by our recent picture of the (projective) Hilbert space as the model of physical space behind basic quantum mechanics, we reformulate here the WWGM formalism starting from the canonical coherent states and taking wavefunctions as expansion coefficients in terms of this basis. This provides us with a transparent and coherent story of simple quantum dynamics where both the wavefunctions for the pure states and operators acting on them arise from the single space/algebra, which exactly includes the WWGM observable algebra. Altogether, putting the emphasis on building our theory out of the underlying relativity symmetry -- the centrally extended Galilean symmetry in the case at hand -- allows one to naturally derive both a kinematical and a dynamical description of a quantum particle, which moreover recovers the corresponding classical picture (understood in terms of the Koopman-von Neumann formalism) in the appropriate (relativity symmetry contraction) limit. Our formulation here is the most natural framework directly connecting all of the relevant mathematical notions and we hope it may help a general physicist better visualize and appreciate the noncommutative-geometric perspective behind quantum physics. It also helps to inspire and illustrate our perspective on looking at quantum mechanics and quantum physics in general in direct connection to the notion of quantum (deformed) relativity symmetries and the corresponding quantum/noncommutative models of spacetime as various levels of approximations all the way down to the Newtonian.
Quantum mechanical descriptions in terms of momentum and position are identified as alternatives under the condition of equal complex-vaulted Lebesgue square integrability. While this does not change any of the formal results obtained in quantum mechanics, it does shed a different interpretive light on the steps that lead up to these results. Instead of being independent, even in concept, momentum and position are identified as being the same thing, merely seen from different views. Neither is required to complement the descriptive capability of the other, since each forms a complete alternative in its own right. Apparent complementarity, as well as mutual indeterminacy of codescription, comes about whenever an attempt is made to overlap these two descriptions to form a third codescription which tries to maintain both the terminology and the net integration values of the separate alternatives. Noncommutation of associated operators is a direct consequence of the alternative nature of momentum and position, and the local to global character of the particular map between these alternatives leads to an apparent point-wave duality in their descriptions. It is suggested that challenges to the contemporary interpretation of quantum theory should begin with the assumptions that lead to its formulation involving alternative representations.
Based on a recent relational formulation of quantum reference frame transformations, especially with a case of quantum spatial translations in particular, we analyzed how the ‘value’ of an observable for a fixed state change. That is the exact analog of the classical description, for example, of the value of the x-coordinate for a particle decrease by 2 units when we perform a translation of the reference frame putting the new origin at x=2. The essence of the quantum reference frame transformations is to have the quantum fluctuations, and even entanglement, of the physical object which serves as the (new) reference frame, taken into account. We illustrate how the recently introduced notion of the noncommutative values of quantum observables gives such a definite description successfully. Formulations, and an analysis of a case example in qubit systems, of analog transformations for observables with a discrete or finite spectrum is also presented. Issues about the evolving picture of the symmetry system of all quantum reference frame transformations discussed.
In the Hamiltonian formulation, classical mechanics employs the commutative algebra of functions that are defined on phase space a point of which could be represented using Dirac delta distributions. In the absence of such a concrete existence of the notion of point in the quantum domain, we show how the quantum mechanical formalism emerges by replacing the commutative algebra by a noncommutative algebra of functions and introducing the quantum condition. The noncommutativity is achieved by deforming the product of two functions and by introducing the Moyal brackets, the basic Moyal brackets between two spatial coordinates and between position and momentum coordinates being noncommutative. The only other input we make use of is the quantum condition which is motivated from its classical analogue—the square of Dirac delta distributions.