9 research outputs found
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
Thermodynamics of a non-commutative fermion gas
Building on the recent solution for the spectrum of the non-commutative well
in two dimensions, the thermodynamics that follows from it is computed. In
particular the focus is put on an ideal fermion gas confined to such a well. At
low densities the thermodynamics is the same as for the commutative gas.
However, at high densities the thermodynamics deviate strongly from the
commutative gas due to the implied excluded area resulting from the
non-commutativity. In particular there are extremal macroscopic states,
characterized by area, number of particles and angular momentum, that
correspond to a single microscopic state and thus have vanishing entropy. When
the system size and excluded area are comparable, thermodynamic quantities,
such as entropy, exhibit non-extensive features.Comment: 18 pages, 11 figure
The N=1 Supersymmetric Landau Problem and its Supersymmetric Landau Level Projections: the N=1 Supersymmetric Moyal-Voros Superplane
The N=1 supersymmetric invariant Landau problem is constructed and solved. By
considering Landau level projections remaining non trivial under N=1
supersymmetry transformations, the algebraic structures of the N=1
supersymmetric covariant non(anti)commutative superplane analogue of the
ordinary N=0 noncommutative Moyal-Voros plane are identified
Bound state energies and phase shifts of a non-commutative well
Non-commutative quantum mechanics can be viewed as a quantum system
represented in the space of Hilbert-Schmidt operators acting on non-commutative
configuration space. Within this framework an unambiguous definition can be
given for the non-commutative well. Using this approach we compute the bound
state energies, phase shifts and scattering cross sections of the non-
commutative well. As expected the results are very close to the commutative
results when the well is large or the non-commutative parameter is small.
However, the convergence is not uniform and phase shifts at certain energies
exhibit a much stronger then expected dependence on the non-commutative
parameter even at small values.Comment: 12 pages, 8 figure
Variations on the Planar Landau Problem: Canonical Transformations, A Purely Linear Potential and the Half-Plane
The ordinary Landau problem of a charged particle in a plane subjected to a
perpendicular homogeneous and static magnetic field is reconsidered from
different points of view. The role of phase space canonical transformations and
their relation to a choice of gauge in the solution of the problem is
addressed. The Landau problem is then extended to different contexts, in
particular the singular situation of a purely linear potential term being added
as an interaction, for which a complete purely algebraic solution is presented.
This solution is then exploited to solve this same singular Landau problem in
the half-plane, with as motivation the potential relevance of such a geometry
for quantum Hall measurements in the presence of an electric field or a
gravitational quantum well
A (p,q)-deformed Landau problem in a spherical harmonic well: spectrum and noncommuting coordinates
A (p,q)-deformation of the Landau problem in a spherically symmetric harmonic
potential is considered. The quantum spectrum as well as space noncommutativity
are established, whether for the full Landau problem or its quantum Hall
projections. The well known noncommutative geometry in each Landau level is
recovered in the appropriate limit p,q=1. However, a novel noncommutative
algebra for space coordinates is obtained in the (p,q)-deformed case, which
could also be of interest to collective phenomena in condensed matter systems.Comment: 9 pages, no figures; updated reference