Wigner distributions and quantum mechanics on Lie groups: The case of the regular representation (original) (raw)
Related papers
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.