5,743 research outputs found
Coherent states and the quantization of 1+1-dimensional Yang-Mills theory
This paper discusses the canonical quantization of 1+1-dimensional Yang-Mills
theory on a spacetime cylinder, from the point of view of coherent states, or
equivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed,
the coherent states are simply ordinary coherent states labeled by points in an
infinite-dimensional linear phase space. Gauge symmetry is imposed by
projecting the original coherent states onto the gauge-invariant subspace,
using a suitable regularization procedure. We obtain in this way a new family
of "reduced" coherent states labeled by points in the reduced phase space,
which in this case is simply the cotangent bundle of the structure group K.
The main result explained here, obtained originally in a joint work of the
author with B. Driver, is this: The reduced coherent states are precisely those
associated to the generalized Segal-Bargmann transform for K, as introduced by
the author from a different point of view. This result agrees with that of K.
Wren, who uses a different method of implementing the gauge symmetry. The
coherent states also provide a rigorous way of making sense out of the quantum
Hamiltonian for the unreduced system.
Various related issues are discussed, including the complex structure on the
reduced phase space and the question of whether quantization commutes with
reduction
The relationship between the Wigner-Weyl kinetic formalism and the complex geometrical optics method
The relationship between two different asymptotic techniques developed in
order to describe the propagation of waves beyond the standard geometrical
optics approximation, namely, the Wigner-Weyl kinetic formalism and the complex
geometrical optics method, is addressed. More specifically, a solution of the
wave kinetic equation, relevant to the Wigner-Weyl formalism, is obtained which
yields the same wavefield intensity as the complex geometrical optics method.
Such a relationship is also discussed on the basis of the analytical solution
of the wave kinetic equation specific to Gaussian beams of electromagnetic
waves propagating in a ``lens-like'' medium for which the complex geometrical
optics solution is already available.Comment: Extended version comprising two new section
Non-commutative integrable systems on -symplectic manifolds
In this paper we study non-commutative integrable systems on -Poisson
manifolds. One important source of examples (and motivation) of such systems
comes from considering non-commutative systems on manifolds with boundary
having the right asymptotics on the boundary. In this paper we describe this
and other examples and we prove an action-angle theorem for non-commutative
integrable systems on a -symplectic manifold in a neighbourhood of a
Liouville torus inside the critical set of the Poisson structure associated to
the -symplectic structure
Manifolds associated with -colored regular graphs
In this article we describe a canonical way to expand a certain kind of
-colored regular graphs into closed -manifolds by
adding cells determined by the edge-colorings inductively. We show that every
closed combinatorial -manifold can be obtained in this way. When ,
we give simple equivalent conditions for a colored graph to admit an expansion.
In addition, we show that if a -colored regular graph
admits an -skeletal expansion, then it is realizable as the moment graph of
an -dimensional closed -manifold.Comment: 20 pages with 9 figures, in AMS-LaTex, v4 added a new section on
reconstructing a space with a -action for which its moment graph is
a given colored grap
Critical sets of the total variance of state detect all SLOCC entanglement classes
We present a general algorithm for finding all classes of pure multiparticle
states equivalent under Stochastic Local Operations and Classsical
Communication (SLOCC). We parametrize all SLOCC classes by the critical sets of
the total variance function. Our method works for arbitrary systems of
distinguishable and indistinguishable particles. We also discuss the Morse
indices of critical points which have the interpretation of the number of
independent non-local perturbations increasing the variance and hence
entanglement of a state. We illustrate our method by two examples.Comment: 4 page
Syzygies in equivariant cohomology for non-abelian Lie groups
We extend the work of Allday-Franz-Puppe on syzygies in equivariant
cohomology from tori to arbitrary compact connected Lie groups G. In
particular, we show that for a compact orientable G-manifold X the analogue of
the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology
of X is reflexive, if and only if the equivariant Poincare pairing for X is
perfect. Along the way we establish that the equivariant cohomology modules
arising from the orbit filtration of X are Cohen-Macaulay. We allow singular
spaces and introduce a Cartan model for their equivariant cohomology. We also
develop a criterion for the finiteness of the number of infinitesimal orbit
types of a G-manifold.Comment: 28 pages; minor change
Semitoric integrable systems on symplectic 4-manifolds
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a
pair of real-valued smooth functions J, H on M for which J generates a
Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall
introduce new global symplectic invariants for these systems; some of these
invariants encode topological or geometric aspects, while others encode
analytical information about the singularities and how they stand with respect
to the system. Our goal is to prove that a semitoric system is completely
determined by the invariants we introduce
- …