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 (2016) and Phys. Rev. A 96, 022117 (2017)] 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. First 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.
This thesis addresses aspects of quantum phase and sets out to draw attention to a collection of new and extend some not so new techniques utilizing Weyl's correspondence rule and its inverse. The phase formulation of quantum mechanics is as old as quantum mechanics itself and evolved over a period of two decades due to the contributions of H. Weyl \cite{Weyl1950}, E. P. Wigner \cite{Wigner1932}, H. Groenewold \cite{Groenewold1946} and J. E. Moyal \cite{Moyal1949}. Over the ensuing years the phase picture has matured into a practical analytical tool, and has generated a multitude of technical papers, reports, reviews, and expositions from many authors and commentators. One reason for its success is that it offers an intuitive picture of quantum mechanics owing to its \it{classical look and feel}, and it has made an impression, in one way or another, in many fields such as mathematical physics, quantum optics, and physical chemistry.The thesis opens with a brief history of classical phase space, tracing the possible origin of the term ``phase space itself, and chronicles the events leading to the conceptualization (and generalization) of quantum phase space. In Chapter 2 the Weyl correspondence is recast in terms of von Neumann's integral kernel transform and a heuristic analysis is given from this perspective. A characteristic of Weyl's correspondence rule and its inverse is that there are two inherent representations of the same operator. These have been referred to as the Weyl representation and Wigner representation and are related by a symplectic Fourier transform. For an arbitrary outer dyadic operator it is shown that the Weyl representation leads to the \it{modified} Gabor transform (known to Signal Processing), while the Wigner representation specializes to the Bargmann transform and a representation in terms of analytic functions. Calculating Weyl-Wigner (WW) transforms of products of
n operators, from first principles, has proved to be a technically difficult problem. However, with the machinery developed in Chapter 2 the problem is reduced to a straightforward algebraic exercise. Finally, forming an
s-parametrized generalization of Weyl's correspondence rule and focusing on the Weyl representation it is shown to lead to an intimate relationship with the relative state formalism \cite{Ban1998}. Weyl's correspondence rule is only applicable to operators and consequently state vectors
∣ψ⟩ are not represented in the usual formulation of quantum phase space. This problem is circumvented in Chapter 3 by introducing the notion of a normalized fixed state (or window) vector
∣φ0⟩ to form an outer dyadic product $\wh{\Psi} = |\psi\rangle\langle\varphi_0|$. The WW transform of this product under the action of the Weyl's rule is a complex valued function of real variables called a phase amplitude and denoted by the symbol
Ψ. The phase amplitudes have many (and sometimes surprising) properties. For example they: \begin{itemize} \item form the building blocks of Wigner functions
W by admitting a generalized Born rule
W=Ψ⋆Ψˉ, and as such can be considered to be a more fundamental objects than Wigner functions; \item superpose and interfere like the state vectors they represent; \item specialize to the Bargmann and Gabor transforms when a Gaussian window is chosen; \end{itemize} \ni In Chapter 4 further aspects of phase amplitudes are explored and discussed. For example it is shown that they: \begin{itemize} \item define phase state vectors which form an over complete set and are elements of a reproducing kernel Hilbert space; \item allow unitary ray representations of the group symmetries of translation, rotation and time reversal; \item evolve in phase under the influence of a propagator
R which is intimately related to the Wigner propagator
P. \end{itemize} Chapter 5 considers an extension of phase amplitudes to include systems with finitely many degrees of freedom. These are called spin amplitudes and are obtained by first forming the outer dyadic product of a spin state vector with an arbitrary fixed spin state vector, and then applying a finite-dimensional analogue of the WW transform. The spin amplitude operator is represented as an
n×n matrix, whose elements are products of the spin components of the spin and fixed state vectors. Two kinds of spin amplitudes are considered, arising on the sphere
S2 or on a
(2j+1) dimensional lattice. In the special case of spin-$\half$ systems it is shown that the spin amplitudes transform as spinors under rotations on both the sphere and lattice. As a physical application the spin amplitudes for a spin-$\half$ magnetic dipole are found for a Nuclear Magnetic Resonance (NMR) environment. Chapter 6 considers Schwinger's abstract approach to quantum mechanics which takes into account the inability to indefinitely compensate for fluctuations in the measurement process. The resulting measurement algebra provides a symbolic representation of the physical laws as opposed to the mathematical abstraction (of von Neumann and Dirac) that was justified \it{\'a posteriori} by being in agreement with experiment. Despite Schwinger's prolific work on the subject of measurement algebra, its interpretation in quantum phase remained an open issue. It is shown that the key to formulating a quantum phase in terms of the measurement operators and transformation functions is the determination of a suitable set of basis operators for an arbitrary measurement operator. Many new results follow as a consequence of expanding an arbitrary measurement operator in this basis. In Chapter 7 the quantum-classical interface is explored by considering Hamilton's principal function or action
S(q,v,t). Starting from a purely heuristic classical argument, a phase function
U(p,q,t) is derived in terms of the classical action and is shown to be proportional to the WW transform of the quantum evolution operator. This association leads to the transform \be U(p,q,t) = A(t) \int e^{ipv/\hbar} e^{iS(q,v,t)/\hbar} \; \rm{d}v, \ee which enables one, in principle, to deduce the Wigner functions of a given system when only the classical action is known. Simple physical systems are used to provide examples illustrating this approach.
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.
We present a group-theoretic reconstruction of quantum mechanics in which the standard formalism emerges from the representation theory of symmetry groups rather than being introduced axiomatically. Starting from physically motivated symmetry principles, we show how the conventional postulates of quantum mechanics including the Hilbert space structure of states, observables as generators of symmetry transformations, the Born probability rule, unitary time evolution, the composition of multipartite systems, and the measurement framework can be recovered within a unified representation-theoretic framework. We then examine the measurement problem by interpreting wave-function collapse as the scattering of quantum states, thereby providing an account of the apparent loss of information. Finally, we investigate the origin of Planck's constant, proposing that it arises as the conserved charge of the Lorentz symmetry group. In this formulation,
ℏ is an invariant that relates Lie algebra generators to observable quantities and fixes the scale of quantum action. This framework offers a conceptually unified perspective on the foundations of quantum mechanics, connecting its postulates, measurement theory, and the emergence of the quantum of action to a common symmetry-based origin while identifying conditions under which the conventional quantum formalism is recovered.
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.
In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto
c-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.