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Entropy

Quantum Mechanical Approach to the Khintchine and Bochner Criteria for Characteristic Functions

Leon W. Cohen

Published Jul 11, 2023• 0 citations
Publisher

Abstract

While it is generally accepted that quantum mechanics is a probability theory, its methods differ radically from standard probability theory. We use the methods of quantum mechanics to understand some fundamental aspects of standard probability theory. We show that wave functions and operators do appear in standard probability theory. We do so by generalizing the Khintchine and Bochner criteria for a complex function to be a characteristic function. We show that quantum mechanics clarifies these criteria and suggests generalizations of them.

DOI
10.3390/e25071042

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While it is generally accepted that quantum mechanics is a probability theory, its methods differ radically from standard probability theory. We use the methods of quantum mechanics to understand some fundamental aspects of standard probability theory. We show that wave functions and operators do appear in standard probability theory. We do so by generalizing the Khintchine and Bochner criteria for a complex function to be a characteristic function. We show that quantum mechanics clarifies these criteria and suggests generalizations of them.
Abstract We show that within classical probability theory there are mathematical quantities which are similar to quantum mechanical wave functions and operators. This is shown by generalizing a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function. We show that for the one dimensional case the methods of quantum mechanics for obtaining expectation values and distributions of observables follow simply. Particular difficulties arise for two non-commuting operators but nonetheless improper “quasi-distributions” can be defined and used with profit. Quasi-distributions can be thought of as two dimensional mappings of a one dimensional function. In the mathematical sense the distribution contains the same information as the wave function, since it is constructed from it and the wave function can be obtained from it uniquely. Nonetheless, an immense simplification occurs when one studies the wave function in a quasirepresentation: the physical nature of the wave function becomes much clearer. A number of explicit examples are given.
Statistical functions such as the moment-generating, characteristic, cumulant-generating, and second characteristic functions are standard tools in classical statistics and probability theory. They provide a systematic means to analyze the statistical properties of a system and find applications in diverse fields. While these functions are ubiquitous in classical theory, a quantum counterpart has remained underdeveloped because of the noncommutativity of operators. The absence of such a framework has obscured the connections between statistical quantities and the nonclassical features of quantum mechanics. Here, we construct a framework for quantum statistical functions that addresses these limitations and unifies the languages of quantum statistics. We show that the functions reproduce standard statistical quantities such as expectation values, variance, and covariance upon differentiation. By extending the framework to include pre- and post-selection, we define conditional functions that generate conditional statistical quantities, including the weak value and the weak variance. We further show that multivariable functions, defined with specific operator orderings, correspond to the Kirkwood--Dirac, Margenau--Hill, and Wigner distributions. By generalizing Bochner's theorem within the theory of compactly supported distributions, we obtain a criterion that separates classical statistics from quantum statistics, linking the failure of positive definiteness of the multivariable function to the emergence of quasiprobability. As an application, we import the classical method of moments and generalized method of moments into quantum estimation, introducing quantum estimators that exploit the proposed functions. Our framework reproduces quantum statistical quantities and incorporates the nonclassical features of quasiprobability, providing a basis for further study of quantum statistics.
Quantum mechanics is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Quantum mechanics is inescapable. Nearly all fields of modern physics, from elementary particles to the big bang, from semiconductors to solar energy cells, are related to quantum physics. Particles have characteristic trait of both particles and waves. The Schrodinger equation is the basic equation of quantum mechanics. The Schrodinger equation was established based on the quantization of energy conservation using quantum operators and the de Broglie relation. Wave function plays an important role in quantum mechanics. The probability interpretation assumes that the norm of the wave function measures the probability of the system in a particular coordinate or representation, in which the total probability is normalized.
The nature of quantum description is clarified. It is shown that complex-valued probability amplitudes are admissible within classical Hamiltonian mechanics. According to standard probability theory, such a description is always possible. The case of a spherical phase space is considered. It is shown that, in such a classical theory, there appears a universal constant that has dimensions of action ( h ), as well as Fock space and all attributes of quantum mechanics. Excitations of a chain of such systems are described by the equations of quantum mechanics with a correct normalization condition. It is shown that an answer to the question of what a particle and its wave function are is provided by quantum field theory (these are a single-particle field excitation and a function that describes it). Experiments are proposed that would make it possible to solve the problem of the physical nature of the wave function.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
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.