The space of labels characterizing the elements of Schwinger’s basis for unitary quantum operators is endowed with a structure of symplectic type. This structure is embodied in a certain algebraic cocycle, whose main features are inherited by the symplectic form of classical phase space. In consequence, the label space may be taken as the quantum phase space: It plays, in the quantum case, the same role played by phase space in classical mechanics, some differences coming inevitably from its nonlinear character.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
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.
We develop a geometrical approach to Schrödinger quantum mechanics, alternative to the usual one, which is based on linear and algebraic structures such as Hilbert spaces, operator algebras, etc. The starting point of this approach is the Kähler structure possessed by the set of the pure states of a quantum system. The Kähler manifold of the pure states is regarded as a “quantum phase space”, conceptually analogous to the phase space of a classical hamiltonian system, and all the constituents of the conventional formulation, in particular the algebraic structure of the observables, are reproduced using a suitable “Kähler formalism”. We also show that the probabilistic character of the measurement process in quantum mechanics and the uncertainty principle are contained in the geometrical structure of the quantum phase space. Finally, we obtain a characterization for quantum phase spaces which can be interpreted as a statement of uniqueness for Schrödinger quantum mechanics.
Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space. For that purpose we consider Kähler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kähler potential.