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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Smeared Wigner functions and quantum-mechanical histories

J. J. Halliwell

Published Aug 15, 1992• 37 citations
Publisher

Abstract

We calculate the probability for a quantum-mechanical history consisting of imprecise samplings of position at two moments of time. In the limit of small time separation, this leads to an imprecise sampling of position together with a time-of-flight sampling of momentum. We also calculate the probability for the history consisting of direct momentum and position samplings a short time apart. In each case, we find that the resulting probability distribution on phase space is a smeared version of the Wigner function, and is positive. We show that these smearings belong to a class of smearings which make the Wigner function positive. In the case of the time-of-flight momentum sampling, it is more general than previously considered smearings, such as that of Husimi.

DOI
10.1103/physrevd.46.1610

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We calculate the probability for a quantum-mechanical history consisting of imprecise samplings of position at two moments of time. In the limit of small time separation, this leads to an imprecise sampling of position together with a time-of-flight sampling of momentum. We also calculate the probability for the history consisting of direct momentum and position samplings a short time apart. In each case, we find that the resulting probability distribution on phase space is a smeared version of the Wigner function, and is positive. We show that these smearings belong to a class of smearings which make the Wigner function positive. In the case of the time-of-flight momentum sampling, it is more general than previously considered smearings, such as that of Husimi.
We study a formulation of quantum mechanics in which the central notion is that of a quantum-mechanical history---a sequence of events at a succession of times. The primary aim is to identify sets of ``decoherent'' (or ``consistent'') histories for the system. These are quantum-mechanical histories suffering negligible interference with each other, and, therefore, to which probabilities may be assigned. These histories may be found for a given system using the so-called decoherence functional. When the decoherence functional is exactly diagonal, probabilities may be assigned to the histories, and all probability sum rules are satisfied exactly. We propose a condition for approximate decoherence, and argue that it implies that most probability sum rules will be satisfied to approximately the same degree. We also derive an inequality bounding the size of the off-diagonal terms of the decoherence functional. We calculate the decoherence functional for some simple one-dimensional systems, with a variety of initial states. For these systems, we explore the extent to which decoherence is produced using two different types of coarse graining. The first type of coarse graining involves imprecise specification of the particle's position. The second involves coupling the particle to a thermal bath of harmonic oscillators and ignoring the details of the bath (the Caldeira-Leggett model). We argue that both types of coarse graining are necessary in general. We explicitly exhibit the degree of decoherence as a function of the temperature of the bath, and of the width to within which the particle's position is specified. We study the diagonal elements of the decoherence functional, representing the probabilities for the possible histories of the system. To the extent that the histories decohere, we show that the probability distributions are peaked about the classical histories of the system, with the distribution of their initial positions and momenta given by a smeared version of the Wigner function. We discuss this result in connection with earlier uses of the Wigner function in this context. We find that there is a certain amount of tension between the demands of decoherence and peaking about classical paths.
Wigner function is a ``quasidistribution'' that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between the Wigner function and appropriate measurements involving the system's position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system's position and momentum. We show that one can relate the Wigner function to the Kirkwood joint quasidistribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space and then turn to the case of a discrete, finite-dimensional Hilbert space.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon's information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at nnn moments of time. We derive a lower bound on the information of the joint probability for nnn position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form ln(VHVS)\mathrm{ln}(\frac{{V}_{H}}{{V}_{S}})ln(VS​VH​​), where VH{V}_{H}VH​ is a volume element of history space, which we define, and VS{V}_{S}VS​ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.
A set of quasi-probability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The phase-space distribution does not have to be bilinear in the state function. The Wigner distribution is a special case. A general relationship between the phase-space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. This allows the writing of correspondence rules at will,of which the ones presently known are particular cases. The dynamics and other properties of the generalized phase-space distribution are considered.
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx\ifmmode\ddot\else\textasciidieresis\fi{}+V'(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak \ensuremath{\Delta}F is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\ensuremath{\sigma}}}^{2}$(\ensuremath{\Delta}F)2{)}^{2})2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/t2{\mathit{t}}^{2}t2, where \ensuremath{\sigma} is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.