Phase-space formalism: Generalized Hermite polynomials, orthogonal functions and creation–annihilation operators

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

The theory of Hermite polynomials of two variables and two indices is discussed herein. Within the context of phase-space formulation of classical and quantum mechanics, they play the same role as conventional Hermite polynomials in ordinary quantum mechanics. Finally their extension to m variables and m indices is analyzed.

The paper gives a review of recent progress in the classification of monodromy-free Schr\"odinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials defined as Wronskians of classic Hermite polynomials appear in a number of mathematical physics problems as well as in the theory of random matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those potentials demonstrate localized oscillatory behavior near the origin. We derive an explicit condition of non-singularity of the corresponding potentials and estimate a localization range with respect to indices of polynomials and distribution of their zeros in the complex plane. It turns out that 1D SUSY quantum non-singular potentials come as a dressing of the harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this end, all generalized Hermite polynomials are produced by appropriate periodic closure of this algebra which leads to rational solutions of the Painlev\'e IV equation. We discuss the structure of the discrete spectrum of Schr\"odinger operators and its link to the monodromy-free condition.

The classical limits of phase-space formulation of quantum mechanics are studied. As a special example, some properties of both quantum mechanical and classical entropies are discussed in detail.

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.

The known methods, due for instance to G.W. Mackey and T.F. Jordan, which exploit the transformation properties with respect to the Euclidean and Galileian group to determine the formalism of the Quantum Theory of a localizable particle, fail in the case that the considered transformations are not symmetries of the physical system. In the present work we show that the formalism of standard Quantum Mechanics for a particle without spin can be completely recovered by exploiting the covariance properties with respect to the group of Euclidean transformations, without requiring that these transformations are symmetries of the physical system.

Generalized Hermite polynomials

2 - The Formalism of Quantum Mechanics

Quantum mechanical irrebersibility

When do we utilize the structure of a deformed Lie algebra in Physics? As a matter of fact and a posteriori one can say that almost always when we passed from one level of description of phenomena to another one the deformation structure was present. This is the case when one passes from non-relativistic Galilean mechanics to relativistic (Lorentz invariant) mechanics, as the Poincare Lie algebra is a deformation of the Galilean Lie algebra (when the light velocity c becomes finite). This is also the case, as will be shown in the next chapter, when one passes from classical mechanics, which is described by the infinite-dimensional Lie algebra of C ∞ functions on phase-space endowed with the Poisson bracket structure, to a phase-space formulation of quantum mechanics (in this case it is ℏ - the Planck constant - which becomes a finite positive number).

States in classical mechanics are probability measures, and their Fourier transforms are continuous functions of positive type. States in the phase-space formulation of quantum mechanics are Wigner distribution functions, and their (symplectic) Fourier transforms have been characterized by Kastler [Commun. Math. Phys. 1, 14 (1965)] and Loupias and Miracle-Sole [Commun. Math. Phys. 2, 31 (1966); Ann. Insto. H. Poincaré 6, 39 (1967)] as being continuous functions of ‘‘ℏ-positive type.’’ In this paper, (Schwartz) distributions of ℏ-positive type, are defined and studied. It is shown that if such distributions are bounded on a certain sequence of test functions, then they are symplectic Fourier transforms of Wigner distribution functions. These results, are applied to a variety of problems ranging from ones involving the quantum Liouville equation to a problem in signal analysis.

The real number line has served classical physics well as a model continuum for both space and time. Quantum mechanics preserves this continuum picture of space and time while extending the range of functions on spacetime to complex numbers. The resulting theory is precise, elegant, experimentally verified and universally accepted. However, since the existence of a ‘realistic’ basis for quantum mechanics remains an open question, there is as yet no consensus on a satisfactory interpretation of the theory. The step in the transition from classical to quantum physics that characteristically impedes attempts to construct realistic models is the special extension of functions from R to C. In quantum mechanics this step is formal and algebraic in nature. This talk discusses an alternative in which the geometry of a spacetime path automatically builds the extension from Z to C, with all spacetime functions simply counting contributions of a single path. The model produces the correct continuum propagator in the continuum limit, but the realistic origin of the propagator is only apparent when the smallest scale in the system (Planck scale) remains finite. This type of model is consequently missed by current formulations of quantum mechanics because the continuum limit is assumed at the outset. This suggests that it may be fruitful to look to discrete formulations of quantum mechanics in the quest for an ontological version of the theory.

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a ‘limiting case’ of the category Hilb packaging quantum mechanics. Following from this assumption, quantum–classical correspondence can be considered as a mapping of the category Hilb to the category Set, i.e., a functor from Hilb to Set, taking place in the macroscopic limit. As a procedure, which takes us from an object of the category Hilb (i.e., a Hilbert space) in the macroscopic limit to an object of the category Set (i.e., a set of values that describe the configuration of a system), this functor must take a finite number of steps in order to make the equivalence of Hilb and Set verifiable. However, as it is shown in the present paper, such a constructivist requirement cannot be met in at least one case of an Ising model of a spin glass. This could mean that it is impossible to demonstrate the emergence of classicality totally from the formalism of standard quantum mechanics.

This Element explores the formal and conceptual foundations of phase space formulations of classical and quantum mechanics. It provides an overview of the core mathematical and physical content of Hamiltonian mechanics, stochastic phase space mechanics, contact Hamiltonian mechanics, and open and closed quantum mechanics on phase space. The formal material is unified via three interpretative themes relating to structured possibility spaces, Liouville's theorem and its failure, and the classical and quantum notions of open and closed systems. This Element book is intended for researchers and graduate students in the philosophy and foundations of physics with an interest in the conceptual foundations of physical theory.