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Joint Representations in Quantum Mechanics and Signal Processing Theory: Why a Probability Function of Time and Frequency is Disallowed,

David Lowe

Published Dec 1, 1986• 4 citations
OpenAlex

Abstract

Abstract : A discussion is presented which attempts to clarify several misconceptions regarding the nature of distribution functions constructed as simultaneous functions of time and frequency. The problems associated with trying to interpret such densities as probability distributions are pointed out and the close correspondence between signal processing theory and quantum mechanics is exploited in order to address the problems raised. On the basis of the arguments presented, a preferred joint distribution is constructed, its properties outlined and its relevance to the representation of complex time-varying signals is discussed.

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•Feb 1, 1976•0 citations
Abstract : A discussion is presented which attempts to clarify several misconceptions regarding the nature of distribution functions constructed as simultaneous functions of time and frequency. The problems associated with trying to interpret such densities as probability distributions are pointed out and the close correspondence between signal processing theory and quantum mechanics is exploited in order to address the problems raised. On the basis of the arguments presented, a preferred joint distribution is constructed, its properties outlined and its relevance to the representation of complex time-varying signals is discussed.
Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the "covariant" transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual.
The paper considers the problem of finding the joint probability distribution of position, P(q 0,q 1,…, qN ), at the different times t 0,t 1, …, tN . The problem is addressed both classically and quantum mechanically. The characteristic function approach is used and it is shown that for certain cases the quantum mechanical answer is the same as the classical answer with replacement of the initial classical probability distribution by the Wigner distribution of the initial wave function. This is the case for taking a certain form of the quantum characteristic function. Other choices are also discussed.
Using the recently developed phase-space techniques for the treatment of quantum-mechanical problems, we set up a procedure for calculating multitime-correlation functions in terms of the joint distribution functions. The correlation functions are then expressed simply as integrals over the associated phase space. Explicit expressions are given for these joint distribution functions, in terms of Green's functions of the ccc-number equations of motion for the phase-space equivalent of the density operator. Using these joint distribution functions, an exact regression theorem is rederived, and the connection with the multitime correspondence between classical and quantum stochastic processes is discussed.
A viewpoint is presented which, taken in probability theory and/or in quantum theory, will help place certain facts in a unified perspective and clarify their apparently enigmatic status. These facts are: (1) while the probabilities in probability theory are squared moduli of complex numbers; (2) the measure-theoretic apparatus of probability theory cannot accommodate the quantization rules for physical measurables and, above all; (3) the notion of limits on the accuracy attainable in simultaneous measurements on canonically conjugate variables does not appear in the theorems and algorithms of probability theory. The viewpoint rests on the intuitive but fundamental idea that measurable functions have non-unique inverses, and that an inner product representation of measures can be obtained whenever the function is differentiable. This idea is generalized to functions measurable on real rectangles and on complex plane. The formulas derived for measures are formally the same as those in quantum mechanics. The remainder of the development is based on the notion of a measurable-preserving point-to-point mapping from the range of these special measurable functions to an appropriate subset of a Hilbert space. Physical interpretations are attempted for each of the measure-theoretic constructs.
Review of the probability representation of quantum mechanics, in which a fair probability distribution is used instead of the density matrix or wave function to describe quantum states is presented. Bell inequalities are formulated as properties of joint probability distributions describing the states of multipartite spin systems in the probability representation. Probability distributions depending on extra parameters are discussed in the connection with problems of classical and quantum statistical mechanics, where these distributions can be considered in the context of non‐Kolmogorov probability theory.
In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.