919 research outputs found
Semiclassical quantization and spectral limits of h-pseudodifferential and Berezin-Toeplitz operators
We introduce a minimalistic notion of semiclassical quantization and use it
to prove that the convex hull of the semiclassical spectrum of a quantum system
given by a collection of commuting operators converges to the convex hull of
the spectrum of the associated classical system. This gives a quick alternative
solution to the isospectrality problem for quantum toric systems. If the
operators are uniformly bounded, the convergence is uniform. Analogous results
hold for non-commuting operators.Comment: 27 pages, 3 figure
Symplectic spectral geometry of semiclassical operators
In the past decade there has been a flurry of activity at the intersection of
spectral theory and symplectic geometry. In this paper we review recent results
on semiclassical spectral theory for commuting Berezin-Toeplitz and
h-pseudodifferential operators. The paper emphasizes the interplay between
spectral theory of operators (quantum theory) and symplectic geometry of
Hamiltonians (classical theory), with an eye towards recent developments on the
geometry of finite dimensional integrable systems.Comment: To appear in Bulletin of the Belgian Mathematical Society, 11 page
Survey on recent developments in semitoric systems
Semitoric systems are a special class of four-dimensional completely
integrable systems where one of the first integrals generates an
-action. They were classified by Pelayo & Vu Ngoc in terms of
five symplectic invariants about a decade ago. We give a survey over the recent
progress which has been mostly focused on the explicit computation of the
symplectic invariants for families of semitoric systems depending on several
parameters and the generation of new examples with certain properties, such as
a specific number of singularities of lowest rank.Comment: 15 pages, 5 figure
A family of compact semitoric systems with two focus-focus singularities
About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by
means of five invariants. Standard examples are the coupled spin oscillator on
and coupled angular momenta on , both having exactly one focus-focus singularity. But so
far there were no explicit examples of systems with more than one focus-focus
singularity which are semitoric in the sense of that classification. This paper
introduces a 6-parameter family of integrable systems on and proves that, for certain ranges of the parameters, it is a
compact semitoric system with precisely two focus-focus singularities. Since
the twisting index (one of the semitoric invariants) is related to the
relationship between different focus-focus points, this paper provides systems
for the future study of the twisting index.Comment: Update to most recent version: some typos removed; minor inaccuracies
corrected; better layou
Slow Schroedinger dynamics of gauged vortices
Multivortex dynamics in Manton's Schroedinger--Chern--Simons variant of the
Landau-Ginzburg model of thin superconductors is studied within a moduli space
approximation. It is shown that the reduced flow on M_N, the N vortex moduli
space, is hamiltonian with respect to \omega_{L^2}, the L^2 Kaehler form on
\M_N. A purely hamiltonian discussion of the conserved momenta associated with
the euclidean symmetry of the model is given, and it is shown that the
euclidean action on (M_N,\omega_{L^2}) is not hamiltonian. It is argued that
the N=3 flow is integrable in the sense of Liouville. Asymptotic formulae for
\omega_{L^2} and the reduced Hamiltonian for large intervortex separation are
conjectured. Using these, a qualitative analysis of internal 3-vortex dynamics
is given and a spectral stability analysis of certain rotating vortex polygons
is performed. Comparison is made with the dynamics of classical fluid point
vortices and geostrophic vortices.Comment: 22 pages, 2 figure
Adiabatically coupled systems and fractional monodromy
We present a 1-parameter family of systems with fractional monodromy and
adiabatic separation of motion. We relate the presence of monodromy to a
redistribution of states both in the quantum and semi-quantum spectrum. We show
how the fractional monodromy arises from the non diagonal action of the
dynamical symmetry of the system and manifests itself as a generic property of
an important subclass of adiabatically coupled systems
The Cheshire Cap
A key role in black hole dynamics is played by the inner horizon; most of the
entropy of a slightly nonextremal charged or rotating black hole is carried
there, and the covariant entropy bound suggests that the rest lies in the
region between the inner and outer horizon. An attempt to match this onto
results of the microstate geometries program suggests that a `Higgs branch' of
underlying long string states of the configuration space realizes the degrees
of freedom on the inner horizon, while the `Coulomb branch' describes the
inter-horizon region and beyond. Support for this proposal comes from an
analysis of the way singularities develop in microstate geometries, and their
close analogy to corresponding structures in fivebrane dynamics. These
singularities signal the opening up of the long string degrees of freedom of
the theory, which are partly visible from the geometry side. A conjectural
picture of the black hole interior is proposed, wherein the long string degrees
of freedom resolve the geometrical singularity on the inner horizon, yet are
sufficiently nonlocal to communicate information to the outer horizon and
beyond.Comment: 64 pages, 8 figures. Version 2: References added, together with
substantial elaborations and clarification
Anomalous dimension in semiclassical gravity
The description of the phase space of relativistic particles coupled to
three-dimensional Einstein gravity requires momenta which are coordinates on a
group manifold rather than on ordinary Minkowski space. The corresponding field
theory turns out to be a non-commutative field theory on configuration space
and a group field theory on momentum space. Using basic non-commutative Fourier
transform tools we introduce the notion of non-commutative heat-kernel
associated with the Laplacian on the non-commutative configuration space. We
show that the spectral dimension associated to the non-commutative heat kernel
varies with the scale reaching a non-integer value smaller than three for
Planckian diffusion scales.Comment: RevTeX, 9 pages, 1 figure; v2: typos corrected and reference adde
Nonrelativistic anyons in external electromagnetic field
The first-order, infinite-component field equations we proposed before for
non-relativistic anyons (identified with particles in the plane with
noncommuting coordinates) are generalized to accommodate arbitrary background
electromagnetic fields. Consistent coupling of the underlying classical system
to arbitrary fields is introduced; at a critical value of the magnetic field,
the particle follows a Hall-like law of motion. The corresponding quantized
system reveals a hidden nonlocality if the magnetic field is inhomogeneous. In
the quantum Landau problem spectral as well as state structure (finite vs.
infinite) asymmetry is found. The bound and scattering states, separated by the
critical magnetic field phase, behave as further, distinct phases.Comment: 19 pages, typos corrected; to appear in Nucl. Phys.
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