1,163 research outputs found
The Weyl-Heisenberg Group on the Noncommutative Two-Torus: A Zoo of Representations
In order to assess possible observable effects of noncommutativity in
deformations of quantum mechanics, all irreducible representations of the
noncommutative Heisenberg algebra and Weyl-Heisenberg group on the two-torus
are constructed. This analysis extends the well known situation for the
noncommutative torus based on the algebra of the noncommuting position
operators only. When considering the dynamics of a free particle for any of the
identified representations, no observable effect of noncommutativity is
implied.Comment: 24 pages, no figure
Harmonic oscillator in a background magnetic field in noncommutative quantum phase-space
We solve explicitly the two-dimensional harmonic oscillator and the harmonic
oscillator in a background magnetic field in noncommutative phase-space without
making use of any type of representation. A key observation that we make is
that for a specific choice of the noncommutative parameters, the time reversal
symmetry of the systems get restored since the energy spectrum becomes
degenerate. This is in contrast to the noncommutative configuration space where
the time reversal symmetry of the harmonic oscillator is always broken.Comment: 7 pages Late
Revisiting the Fradkin-Vilkovisky Theorem
The status of the usual statement of the Fradkin-Vilkovisky theorem, claiming
complete independence of the Batalin-Fradkin-Vilkovisky path integral on the
gauge fixing "fermion" even within a nonperturbative context, is critically
reassessed. Basic, but subtle reasons why this statement cannot apply as such
in a nonperturbative quantisation of gauge invariant theories are clearly
identified. A criterion for admissibility within a general class of gauge
fixing conditions is provided for a large ensemble of simple gauge invariant
systems. This criterion confirms the conclusions of previous counter-examples
to the usual statement of the Fradkin-Vilkovisky theorem.Comment: 21 pages, no figures, to appear in Jnl. Phys.
Spectrum of the non-commutative spherical well
We give precise meaning to piecewise constant potentials in non-commutative
quantum mechanics. In particular we discuss the infinite and finite
non-commutative spherical well in two dimensions. Using this, bound-states and
scattering can be discussed unambiguously. Here we focus on the infinite well
and solve for the eigenvalues and eigenfunctions. We find that time reversal
symmetry is broken by the non-commutativity. We show that in the commutative
and thermodynamic limits the eigenstates and eigenfunctions of the commutative
spherical well are recovered and time reversal symmetry is restored
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