521 research outputs found
Wigner functions, contact interactions, and matching
Quantum mechanics in phase space (or deformation quantization) appears to
fail as an autonomous quantum method when infinite potential walls are present.
The stationary physical Wigner functions do not satisfy the normal eigen
equations, the *-eigen equations, unless an ad hoc boundary potential is added
[Dias-Prata]. Alternatively, they satisfy a different, higher-order,
``*-eigen-* equation'', locally, i.e. away from the walls [Kryukov-Walton].
Here we show that this substitute equation can be written in a very simple
form, even in the presence of an additional, arbitrary, but regular potential.
The more general applicability of the -eigen- equation is then demonstrated.
First, using an idea from [Fairlie-Manogue], we extend it to a dynamical
equation describing time evolution. We then show that also for general contact
interactions, the -eigen- equation is satisfied locally. Specifically, we treat
the most general possible (Robin) boundary conditions at an infinite wall,
general one-dimensional point interactions, and a finite potential jump.
Finally, we examine a smooth potential, that has simple but different
expressions for x positive and negative. We find that the -eigen- equation is
again satisfied locally. It seems, therefore, that the -eigen- equation is
generally relevant to the matching of Wigner functions; it can be solved
piece-wise and its solutions then matched.Comment: 20 pages, no figure
Admissible states in quantum phase space
We address the question of which phase space functionals might represent a
quantum state. We derive necessary and sufficient conditions for both pure and
mixed phase space quantum states. From the pure state quantum condition we
obtain a formula for the momentum correlations of arbitrary order and derive
explicit expressions for the wavefunctions in terms of time dependent and
independent Wigner functions. We show that the pure state quantum condition is
preserved by the Moyal (but not by the classical Liouville) time evolution and
is consistent with a generic stargenvalue equation. As a by-product Baker's
converse construction is generalized both to an arbitrary stargenvalue
equation, associated to a generic phase space symbol, as well as to the time
dependent case. These results are properly extended to the mixed state quantum
condition, which is proved to imply the Heisenberg uncertainty relations.
Globally, this formalism yields the complete characterization of the
kinematical structure of Wigner quantum mechanics. The previous results are
then succinctly generalized for various quasi-distributions. Finally, the
formalism is illustrated through the simple examples of the harmonic oscillator
and the free Gaussian wave packet. As a by-product, we obtain in the former
example an integral representation of the Hermite polynomials.Comment: 34 pages, Latex fil
A Superconnection for Riemannian Gravity as Spontaneously Broken SL(4,R) Gauge Theory
A superconnection is a supermatrix whose even part contains the
gauge-potential one-forms of a local gauge group, while the odd parts contain
the (0-form) Higgs fields; the combined grading is thus odd everywhere. We
demonstrate that the simple supergroup (rank=3) in Kac'
classification (even subgroup ) prverline {SL}(4,R)\bar{SL}(4,R)\bar{SO}(1,3)$ unbroken. As a result, post-Riemannian SKY
gravity yields Einstein's theory as a low-energy (longer range) effective
theory. The theory is renormalizable and may be unitary.Comment: 11 pages, late
Wigner Functions with Boundaries
We consider the general Wigner function for a particle confined to a finite
interval and subject to Dirichlet boundary conditions. We derive the boundary
corrections to the "star-genvalue" equation and to the time evolution equation.
These corrections can be cast in the form of a boundary potential contributing
to the total Hamiltonian which together with a subsidiary boundary condition is
responsible for the discretization of the energy levels. We show that a
completely analogous formulation (in terms of boundary potentials) is also
possible in standard operator quantum mechanics and that the Wigner and the
operator formulations are also in one-to-one correspondence in the confined
case. In particular, we extend Baker's converse construction to bounded
systems. Finally, we elaborate on the applications of the formalism to the
subject of Wigner trajectories, namely in the context of collision processes
and quantum systems displaying chaotic behavior in the classical limit.Comment: 29 pages, 7 figures. Completely revised version, to appear in the J.
Math. Phy
On infinite walls in deformation quantization
We examine the deformation quantization of a single particle moving in one
dimension (i) in the presence of an infinite potential wall, (ii) confined by
an infinite square well, and (iii) bound by a delta function potential energy.
In deformation quantization, considered as an autonomous formulation of quantum
mechanics, the Wigner function of stationary states must be found by solving
the so-called \*-genvalue (``stargenvalue'') equation for the Hamiltonian.
For the cases considered here, this pseudo-differential equation is difficult
to solve directly, without an ad hoc modification of the potential. Here we
treat the infinite wall as the limit of a solvable exponential potential.
Before the limit is taken, the corresponding \*-genvalue equation involves
the Wigner function at momenta translated by imaginary amounts. We show that it
can be converted to a partial differential equation, however, with a
well-defined limit. We demonstrate that the Wigner functions calculated from
the standard Schr\"odinger wave functions satisfy the resulting new equation.
Finally, we show how our results may be adapted to allow for the presence of
another, non-singular part in the potential.Comment: 22 pages, to appear in Annals of Physic
Measurement-induced decoherence and Gaussian smoothing of the Wigner distribution function
We study the problem of measurement-induced decoherence using the phase-space
approach employing the Gaussian-smoothed Wigner distribution function. Our
investigation is based on the notion that measurement-induced decoherence is
represented by the transition from the Wigner distribution to the
Gaussian-smoothed Wigner distribution with the widths of the smoothing function
identified as measurement errors. We also compare the smoothed Wigner
distribution with the corresponding distribution resulting from the classical
analysis. The distributions we computed are the phase-space distributions for
simple one-dimensional dynamical systems such as a particle in a square-well
potential and a particle moving under the influence of a step potential, and
the time-frequency distributions for high-harmonic radiation emitted from an
atom irradiated by short, intense laser pulses.Comment: Accepted in Annals of Physic
Formal solutions of stargenvalue equations
The formal solution of a general stargenvalue equation is presented, its
properties studied and a geometrical interpretation given in terms of
star-hypersurfaces in quantum phase space. Our approach deals with discrete and
continuous spectra in a unified fashion and includes a systematic treatment of
non-diagonal stargenfunctions. The formalism is used to obtain a complete
formal solution of Wigner quantum mechanics in the Heisenberg picture and to
write a general formula for the stargenfunctions of Hamiltonians quadratic in
the phase space variables in arbitrary dimension. A variety of systems is then
used to illustrate the former results.Comment: 29 pages. Revised version, to appear in Annals of Physic
Dynamical Gauge-Higgs Unification in the Electroweak Theory
doublet Higgs fields are unified with gauge fields in the model of Antoniadis, Benakli and Quir\'{o}s' on the orbifold
. The effective potential for the Higgs fields (the
Wilson line phases) is evaluated. The electroweak symmetry is dynamically
broken to by the Hosotani mechanism. There appear light Higgs
particles. There is a phase transition as the moduli parameter of the complex
structure of is varied.Comment: 14 pages, 3 figures, v.
Causal Interpretation and Quantum Phase Space
We show that the de Broglie-Bohm interpretation can be easily implemented in
quantum phase space through the method of quasi-distributions. This method
establishes a connection with the formalism of the Wigner function. As a
by-product, we obtain the rules for evaluating the expectation values and
probabilities associated with a general observable in the de Broglie-Bohm
formulation. Finally, we discuss some aspects of the dynamics.Comment: 13 pages, LaTe
Autonomous generation of all Wigner functions and marginal probability densities of Landau levels
Generation of Wigner functions of Landau levels and determination of their
symmetries and generic properties are achieved in the autonomous framework of
deformation quantization. Transformation properties of diagonal Wigner
functions under space inversion, time reversal and parity transformations are
specified and their invariance under a four-parameter subgroup of symplectic
transformations are established. A generating function for all Wigner functions
is developed and this has been identified as the phase-space coherent state for
Landau levels. Integrated forms of generating function are used in generating
explicit expressions of marginal probability densities on all possible two
dimensional phase-space planes. Phase-space realization of unitary similarity
and gauge transformations as well as some general implications for the Wigner
function theory are presented.Comment: 28 pages, 2 tables. Typos and a minor revision only section
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