Quantum systems with finite Hilbert space where position x and momentum \np take values in Z(d) (integers modulo d) are considered. Symplectic tranformations \nS(2¿,Z(p)) in ¿-partite finite quantum systems are studied and \nconstructed explicitly. Examples of applying such simple method is given \nfor the case of bi-partite and tri-partite systems. The quantum correlations \nbetween the sub-systems after applying these transformations are discussed \nand quantified using various methods. An extended phase-space x¿p¿X¿P \nwhere X, P ¿ Z(d) are position increment and momentum increment, is introduced. \nIn this phase space the extended Wigner and Weyl functions are \ndefined and their marginal properties are studied. The fourth order interference \nin the extended phase space is studied and verified using the extended \nWigner function. It is seen that for both pure and mixed states the fourth \norder interference can be obtained.
An extended phase space x – p – X – P is introduced for one-mode systems, where x – p is the position–momentum plane and X – P is the quantum correlations plane. The Wigner functions W ( x , p ) and the Weyl functions ( X , P ) for arbitrary operators are studied in this space. Creation and annihilation operators for Wigner and Weyl functions are introduced and are used to perform displacements and squeezing in the x – p – X – P extended phase space. The formalism can be used for the construction of quantum states with desirable correlation properties.
The Wigner-Weyl (WW) phase-space formulation of quantum mechanics is discussed within the Liouville-space formalism, where quantum operators A^ are viewed as vectors, represented by L kets \ensuremath{\Vert}A^>>, on which act ``superoperators''; the scalar product is A^\ensuremath{\Vert}B^>>=TrA${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$B^. With every operator A^, we associate commutation and anticommutation superoperators A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and A${\mathrm{^}}^{+}$, defined by their actions on any operator B^ as A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$B^=${\mathrm{\ensuremath{\Elzxh}}}^{\mathrm{\ensuremath{-}}1}$[A^,B^], A${\mathrm{^}}^{+}$B^=1/2(A^B^+B^A^). The WW representation corresponds to the choice of a special basis in Liouville space, namely, the eigenbasis of the position and momentum anticommutation superoperators q${\mathrm{^}}^{+}$ and p${\mathrm{^}}^{+}$ (where [q^,p^]=i\ensuremath{\Elzxh}). These, together with the commutation superoperators q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$, form a canonical set of superoperators, [q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$]=[q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$,p${\mathrm{^}}^{+}$]=i (the other commutators vanishing), as functions of which all other super- operators can be expressed. Weyl ordering is expressed as f(q^,p^
)Weyl ordering=f(q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{+}$)1^. A generalization of Ehrenfest's theorem is obtained.
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.
Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU (2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.