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AIP conference proceedings

Fundamental phase space formula for the similitude group

Laarni B. Natividad, Job A. Nable

Published Jan 1, 2024• 0 citations
Publisher

Abstract

In this work, the statement and proof of a fundamental formula in the phase space representation of quantum systems will be carried out for the similitude group, 𝑆𝑖𝑚(2).This formula takes the form∫ 𝒂(𝒀)𝑷(𝒀)𝒅(𝒀) = 〈𝑨 ̂〉, where 𝑌 is the phase space variable and 〈𝐴 ̂〉 is a linear operator on Hilbert space representing a quantum dynamical observable.〈𝐴 ̂〉 is the quantum expected value of the observable in a state of the system.The focus on the similitude group is due to current interest in signal analysis, localization operators and pseudo-differential operators.The fundamental formula states that this may be computed in a classical manner, as an integral against a probability distribution.The formula is intimately related to the quantization-dequantization problem 𝑎(𝑌) ⟷ 𝐴 ̂ which assigns a quantum operator to the classical phase space function 𝑎(𝑌).

DOI
10.1063/5.0192115

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In this work, the statement and proof of a fundamental formula in the phase space representation of quantum systems will be carried out for the similitude group, 𝑆𝑖𝑚(2).This formula takes the form∫ 𝒂(𝒀)𝑷(𝒀)𝒅(𝒀) = 〈𝑨 ̂〉, where 𝑌 is the phase space variable and 〈𝐴 ̂〉 is a linear operator on Hilbert space representing a quantum dynamical observable.〈𝐴 ̂〉 is the quantum expected value of the observable in a state of the system.The focus on the similitude group is due to current interest in signal analysis, localization operators and pseudo-differential operators.The fundamental formula states that this may be computed in a classical manner, as an integral against a probability distribution.The formula is intimately related to the quantization-dequantization problem 𝑎(𝑌) ⟷ 𝐴 ̂ which assigns a quantum operator to the classical phase space function 𝑎(𝑌).
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
Phase-space representation of quantum state vectors is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relative-position state and relative-momentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained. Phase-space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative-position states and the relative-momentum states. The absolute-square of the phase-space function represents the probability distribution of the phase-space variables. Time-evolution of a quantum system is investigated in terms of the phase-space functions. The relations to the phase-space representations formulated by the other methods are obtained.
The objective of this work is to present some aspects of the phase space representation of quantum mechanics on the Euclidean Motion Group of rank three. In particular, the main contribution of this paper is the statement and proof of fundamental formula relating the Weyl transform for the group and the Wigner functions on the group, given by 〈F1|WkF2〉=∫R3∫so(3)k(c,C)WE(3)(F1,F2)dc dC. Here, Wk is the Weyl transform of the function k on E(3) and WE(3) is the Wigner transform. Thus, the various phase space objects such as the Wigner transform and the Weyl transform are also explicitly formulated on the Euclidean Motion Group.
The problem of quantization in the phase-space formulation of quantum mechanics is considered. An integral equation for the phase-space eigenfunctions is derived which is equivalent to the standard eigenvalue equation for a quantum mechanical operator. A differential form is also given. A variational principle is derived for quasiprobability distributions. It is shown that the expected value of the classical Hamiltonian calculated with a trial quasiprobability distribution will be greater than the ground state energy only if the distribution is chosen from a certain class of functions. The notion of ψ-representability is introduced to classify these functions. They represent distributions which correspond to possible quantum mechanical states. Also, a general relation is given between different distribution functions.
The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.
An operator-valued quantum phase space formula is constructed. The phase space formula of Quantum Mechanics provides a natural link between first and second quantization, thus contributing to the understanding of quantization problem. By the combination of quantization and hamiltonization of dynamics, a quantization hierarchy is introduced, beyond the framework of first and second quantization and generalizing the standard quantum theory. We apply our quantization method to quantum many-body system and propose an eclectic model, in which the dimension of Hilbert space does not scale exponentially with the number of particles due to the locality of interaction, and the evolution is a constrained Hamiltonian dynamics.