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arXiv:quant-ph/0507136 · Physical Review A

Commuting position and momentum operators, exact decoherence, and emergent classicality

J. J. Halliwell

Published Oct 20, 2005• 6 citations
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Abstract

Inspired by an old idea of von Neumann, we seek a pair of commuting operators $X,\phantom{\rule{0.2em}{0ex}}P$ which are, in a specific sense, ``close'' to the canonical noncommuting position and momentum operators $x,\phantom{\rule{0.2em}{0ex}}p$. The construction of such operators is related to the problem of finding complete sets of orthonormal phase-space-localized states, a problem severely constrained by the Balian-Low theorem. Here these constraints are avoided by restricting attention to situations in which the density matrix is reasonably decohered (i.e., spread out in phase space). Commuting position and momentum operators are argued to be of use in discussions of emergent classicality from quantum mechanics. In particular, they may be used to give a discussion of the relationship between exact and approximate decoherence in the decoherent-histories approach to quantum theory.

DOI
10.1103/physreva.72.042109

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Inspired by an old idea of von Neumann, we seek a pair of commuting operators X,PX,\phantom{\rule{0.2em}{0ex}}PX,P which are, in a specific sense, ``close'' to the canonical noncommuting position and momentum operators x,px,\phantom{\rule{0.2em}{0ex}}px,p. The construction of such operators is related to the problem of finding complete sets of orthonormal phase-space-localized states, a problem severely constrained by the Balian-Low theorem. Here these constraints are avoided by restricting attention to situations in which the density matrix is reasonably decohered (i.e., spread out in phase space). Commuting position and momentum operators are argued to be of use in discussions of emergent classicality from quantum mechanics. In particular, they may be used to give a discussion of the relationship between exact and approximate decoherence in the decoherent-histories approach to quantum theory.
Any account of the emergence of classicality from quantum theory must address the fact that the quantum operators representing positions and momenta do not commute, whereas their classical counterparts suffer no such restrictions. To address this, we revive an old idea of von Neumann, and seek a pair of commuting operators X, P which are, in a specific sense, "close" to the canonical non-commuting position and momentum operators, x,p. The construction of such operators is related to the problem of finding complete sets of orthonormal phase space localized states, a problem severely limited by the Balian-Low theorem. Here these limitations are avoided by restricting attention to situations in which the density matrix is reasonably decohered (i.e., spread out in phase space).
In the decoherent histories approach to quantum theory, sets of histories are said to be decoherent when the decoherence functional, measuring interference between pairs of histories, is exactly diagonal. In realistic situations, however, only approximate diagonality is ever achieved, raising the question of what approximate decoherence actually means and how it is related to exact decoherence. This paper explores the possibility that an exactly decoherent set of histories may be constructed from an approximate set by small distortions of the operators characterizing the histories. In particular, for the case of histories of positions and momenta, this is achieved by doubling the set of operators and then finding, among this enlarged set, new position and momentum operators that commute, and so decohere exactly, and which are ``close'' to the original operators. Two derivations are given: one in terms of the decoherence functional, the second in terms of Wigner functions. The enlarged, exactly decoherent theory has the same classical dynamics as the original one, and coincides with the so-called deterministic quantum theories of the type recently studied by 't Hooft. These results suggest that the comparison of standard and deterministic quantum theories may provide an alternative method of characterizing emergent classicality. A side product is the surprising result that histories of momenta in the quantum Brownian motion model (for the free particle in the high-temperature limit) are exactly decoherent.
refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be
An hidden variable (hv) theory is a theory that allows globally dispersion free ensembles. We demonstrate that the Phase Space formulation of Quantum Mechanics (QM) is an hv theory with the position q, and momentum p as the hv. Comparing the Phase space and Hilbert space formulations of QM we identify the assumption that led von Neumann to the Hilbert space formulation of QM which, in turn, precludes global dispersion free ensembles within the theory. The assumption, dubbed I, is: "If a physical quantity A\mathbf{A}A has an operator A^\hat{A}A^ then f(A)f(\mathbf{A})f(A) has the operator f(A^)f(\hat{A})f(A^)". This assumption does not hold within the Phase Space formulation of QM. The hv interpretation of the Phase space formulation provides novel insight into the interrelation between dispersion and non commutativity of position and momentum (operators) within the Hilbert space formulation of QM and mitigates the criticism against von Neumann's no hidden variable theorem by, virtually, the consensus.
The emergence of classical behavior in quantum theory is often ascribed to the interaction of a quantum system with its environment, which can be interpreted as environmental monitoring of the system. As a result, off-diagonal elements of the density matrix of the system are damped in the basis of a preferred observable, often taken to be the position, leading to the phenomenon of decoherence. This effect can be modeled dynamically in terms of a Lindblad equation driven by the position operator. Here the question of decoherence resulting from a monitoring of position and momentum, i.e., a phase-space measurement, by the environment is addressed. There is no standard quantum observable corresponding to the detection of phase-space points, which is forbidden by Heisenberg's uncertainty principle. This issue is addressed by use of a coherent-state-based positive operator-valued measure for modeling phase-space monitoring by the environment. In this scheme, decoherence in phase space implies the diagonalization of the density matrix in both position and momentum representations. This is shown to be linked to a Lindblad equation where position and momentum appear as two independent Lindblad operators.