Wigner distributions and quantum mechanics on Lie groups: The case of the regular representation (original) (raw)
Wigner distributions for finite dimensional quantum systems: An algebraic approach
Pramana, 2005
We discuss questions pertaining to the definition of 'momentum', 'momentum space', 'phase space', and 'Wigner distributions'; for finite dimensional quantum systems. For such systems, where traditional concepts of 'momenta' established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail.
2005
A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.
WignerWeyl isomorphism for quantum mechanics on Lie groups
Journal of mathematical …, 2005
The WignerWeyl isomorphism for quantum mechanics on a compact simple Lie group is developed in detail. Several features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a semiquantized phase space, a structure on ...
Wigner distributions for flnite dimensional quantum systems: An algebraic approach
2005
We discuss questions pertaining to the definition of 'momentum', 'momentum space', 'phase space' and 'Wigner distributions'; for finite dimensional quantum systems. For such systems, where traditional concepts of 'momenta' established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail.
A note on the distributions in quantum mechanical systems
Journal of Physics: Conference Series, 2021
In this paper, we study the distributions and the affine distributions of the quantum mechanical system. Also, we discuss the controllability of the quantum mechanical system with the related question concerning the minimum time needed to steer a quantum system from a unitary evolution U(0) = I of the unitary propagator to a desired unitary propagator Uf . Furthermore, the paper introduces a description of a k⊕p sub-Finsler manifold with its geodesics, which equivalents to the problem of driving the quantum mechanical system from an arbitrary initial state U(0) = I to the target state Uf , some illustrative examples are included. We prove that the Lie group G on a Finsler symmetric manifold G/K can be decomposed into KAK.
Connection between two Wigner functions for spin systems
Physical Review A, 2000
In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU͑2͒ that serves to analyze two-particle spin states on a sphere. Recent work by our group has included the definition of an apparently distinct Wigner function on generic Lie groups whose natural range has the dimension of the group and serves for all square-integrable representations; for the SO͑3͒ case this entails a three-dimensional ''metaphase'' space. Both have the fundamental properties covariance and completeness. Here we show how the former is obtained as a restriction of the latter.
Wigner Functions for Arbitrary Quantum Systems
Physical review letters, 2016
The possibility of constructing a complete, continuous Wigner function for any quantum system has been a subject of investigation for over 50 years. A key system that has served to illustrate the difficulties of this problem has been an ensemble of spins. Here we present a general and consistent framework for constructing Wigner functions exploiting the underlying symmetries in the physical system at hand. The Wigner function can be used to fully describe any quantum system of arbitrary dimension or ensemble size.
Alternative Hamiltonians and Wigner quantization
Journal of Optics B: Quantum and Semiclassical Optics, 2003
The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.
On Families of Wigner Functions for N-Level Quantum Systems
Symmetry, 2021
A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.
On the Moduli Space of Wigner Quasiprobability Distributions for N-Dimensional Quantum Systems
Journal of Mathematical Sciences, 2019
A mapping between operators on the Hilbert space of an N-dimensional quantum system and Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich-Weyl kernel. It is shown that the moduli space of Stratonovich-Weyl kernels is given by the intersection of the coadjoint orbit space of the group SU(N) and a unit (N − 2)dimensional sphere. The general considerations are exemplified by a detailed description of the moduli space of 2, 3, and 4-dimensional systems. Bibliography: 30 titles.
Quantum Fourier transform, Heisenberg groups and quasi-probability distributions
New Journal of Physics, 2011
This paper aims to explore the inherent connection among Heisenberg groups, quantum Fourier transform and (quasiprobability) distribution functions. Distribution functions for continuous and finite quantum systems are examined first as a semiclassical approach to quantum probability distribution. This leads to studying certain functionals of a pair of "conjugate" observables, connected via the quantum Fourier transform. The Heisenberg groups emerge naturally from this study and we take a rapid look at their representations. The quantum Fourier transform appears as the intertwining operator of two equivalent representation arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra.The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We also study the reconstruction of Wigner function from the marginal distributions via inverse Radon transform. We discuss relevance of our results and methods for computational and tomographic applications. ¦
Reality of the Wigner Functions and Quantization
Research Letters in Physics, 2009
Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.
Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics
International Journal of Modern Physics B, 1996
It is shown, that for quantum systems the vectorfield associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schr\"odinger and Heisenberg pictures. We illustrate these ambiguities in terms of simple examples.
Quantum control via Wigner measures and Wigner functions
2014
We develop an approach for analyzing open quantum systems which can be used to investigate quantum control problems, based on the use of both the Wigner functions and the so-called Wigner measures. We also propose an axiomatic definition of coherent quantum feedback control (see [1] and the collection of articles in [2]). While the results relating to the Wigner functions and measures are quite technical, the latter topic is more conceptual. The main advantage of using the Wigner functions and measures is the fact that their domains are the phase spaces, and hence the transition from the Wigner measure or the Wigner function of the composition of two subsystems to the Wigner measure or function of any of the subsystems, is quite similar to the transition from the usual probability on the product of two phase spaces to the probability on any of these spaces; the latter probability is just the projection of the probability on the Cartesian product. Actually, if the dimension of the ph...
The Wigner Function for General Lie Groups and the Wavelet Transform
Annales Henri Poincaré, 2000
We build Wigner maps, functions and operators on general phase spaces arising from a class of Lie groups, including non-unimodular groups (such as the affine group). The phase spaces are coadjoint orbits in the dual of the Lie algebra of these groups and they come equipped with natural symplectic structures and Liouville-type invariant measures. When the group admits square-integrable representations, we present a very general construction of a Wigner function which enjoys all the desirable properties, including full covariance and reconstruction formulae. We study in detail the case of the affine group on the line. In particular, we put into focus the close connection between the well-known wavelet transform and the Wigner function on such groups.
Phase space quantum mechanics via frame quantization on finite groups
The 5th Innovation and Analytics Conference & Exhibition (IACE 2021)
In this work, the version of Wigner transforms and Wigner functions on discrete systems are formulated. Using frame quantization, the star-product of phase space functions are also computed for a particular finite group. Specifically, the Wigner transform, Wigner function and star-product are computed for the dicyclic group. Frames are the discrete version of coherent states. In the literature, quasiprobability distributions are essential in the study of the phase space representations of quantum mechanics, and are naturally associated to frame quantizations. For this work, the unitary irreducible representations of a finite group play an essential role in the construction.
The Wigner distribution for classical systems
Physics Letters A, 2002
We present an explicit procedure for obtaining the equation of motion for the Wigner distribution when the underlying governing equation is a linear ordinary or partial differential equation. The cases of constant and variable coefficients are considered.
The action-angle Wigner function: a discrete, finite and algebraic phase space formalism
Journal of Physics A: Mathematical and General, 2000
The action-angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space. In this work, a discrete and finite-dimensional phase space formalism, in which the phase space variables are discrete and the time is continuous, is developed and the fundamental properties of the discrete Weyl-Wigner-Moyal quantization are derived. The action-angle Wigner function is shown to exist in the semi-discrete limit of this quantization scheme. A comparison with other formalisms which are not explicitly based on canonical discretization is made. Fundamental properties that an actionangle phase space distribution respects are derived. The dynamical properties of the action-angle Wigner function are analysed for discrete and finite-dimensional model Hamiltonians. The limit of the discrete and finite-dimensional formalism including a discrete analogue of the Gaussian wavefunction spread, viz. the binomial wavepacket, is examined and shown by examples that standard (continuum) quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity.