Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three-parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered.
Galilean space-time symmetries are discussed in the context of the Liouville space bra-ket formalism and the Wigner-Weyl representations. Also considered is Liouville conjugation (changing operators into their Hermitian conjugates), owing to which any symmetry transformation can be performed by both a unitary and an antiunitary superoperator. Special reference is made to the symmetries of relaxation superoperators for open systems, emphasis being on the practical significance of Wigner-Weyl as symmetry-adapted epresentations. As a side result, symplectic Fourier transforms are shown to be equivalent to inversions.
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.
Following a general procedure developed previously [Ann. Henri Poincaré 1, 685 (2000)], here we construct Wigner functions on a phase space related to the similitude group in two dimensions. Since the group space in this case is topologically homeomorphic to the phase space in question, the Wigner functions so constructed may also be considered as being functions on the group space itself. Previously the similitude group was used to construct wavelets for two-dimensional image analysis; we discuss here the connection between the wavelet transform and the Wigner function.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.
The trace of an arbitrary product of quantum operators with the density operator is rendered as a multiple phase space integral of the product of their Weyl symbolswith the Wigner function. Interspersing the factors with various evolution operators, one obtains an evolving correlation. The kernel for the matching multiple integral thatevolves within the Weyl representation is identified with the trace of a single compound unitary operator. Its evaluation within a semiclassical approximation then becomes asum over the periodic trajectories of the corresponding classical compound canonical transformation. The search for periodic trajectories can be bypassed by an exactly equivalent initialvalue scheme, which involves a change of integration variable and a reduced compound unitary operator. Restriction of all the operators to observables with smooth nonoscillatoryWeyl symbols reduces the evolving correlation to a single phase space integral. If each observable undergoes independent Heisenberg evolution, the overallcorrelation evolves classically. Otherwise, the kernel acquires a nonclassical phase factor, though it still depends on a purely classical compound trajectory: e.g. the fasefor a double return of the quantum Loschmidt echo does not coincide with twice the phase for a single echo.