Mathematical Physics
Org: Richard Froese (UBC) and Tom Osborn (Manitoba) [PDF]
 TWAREQUE ALI, Concordia University, 1455 de Maisonneuve Blvd. West,
Montreal, Quebec, Canada, H3G 1M8
BerezinToeplitz operators and vector coherent states
related to supersymmetric quantum mechanics
[PDF] 
In this talk we intend to discuss a set of newly constructed vector
coherent states and associated BerezinToeplitz operators, which
arise in supersymmetric quantum mechanics. Assuming that
supersymmetry is unbroken, these coherent states are related to the
two almost isospectral (i.e., except for the ground state)
Hamiltonians of the theory. The analytic representation of the system
lives on a Hilbert space consisting of two mutually orthogonal
subspaces of analytic and antianalytic functions, corresponding to
the bosonic and fermionic sectors, respectively. The representation
can also be transferred to a supersymmetric Hilbert space. We discuss
the associated quantization problem and the ensuing BerezinToeplitz
operators.
 RAINER DICK, University of Saskatchewan, Department of Physics
String and brane techniques for surface science and
nanotechnology
[PDF] 
String theory is a wellknown catalyst for seminal interactions
between mathematics and physics. Theoretical physicists became aware
of powerful mathematical tools like Lie algebra cohomology or the
theory of complex manifolds when they had to use these methods to
understand the quantum dynamics of relativistic strings.
It is well known that some of the mathematical techniques used in
string theory also apply to twodimensional critical systems, but it
is not so widely known that some of the mathematical techniques used
in brane world theories can also be adapted for calculations in
lowdimensional electron systems.
The dynamics of electrons at surfaces, interfaces, and quantum wires
is important for materials science and for the investigation of
magnetic and thermodynamic properties of lowdimensional systems.
The talk will specifically discuss the mapping between halforder
differentials and spinors for the description of fermions in
lowdimensional systems, and the use of brane world techniques to
describe the transition between surface and bulk properties of
electrons in the presence of a surface or interface.
 RICHARD FROESE, University of British Columbia
Absolutely continuous spectrum for discrete Schrödinger
operators on tree like graphs
[PDF] 
I will dicuss results obtained with David Hasler and Wolfgang Spitzer
on absolutely continuous spectrum on graphs, including our alternative
proof of Klein's result on extended states for the Anderson model on
the Bethe Lattice.
 STEPHEN GUSTAFSON, University of British Columbia
Schroedinger Flow and LandauLifshitz Dynamics
[PDF] 
The Schroedinger flow into the 2sphere is a geometric generalization
of the linear Schroedinger equation, and a particular case of the
LandauLifshitz equation for ferromagnets. Understanding the fate of
solutions (blowup? asymptotics?) is a challenge. I will describe
some recent work on these questions, part of a joint project with
K. Kang and T.P. Tsai.
 NANKUO HO, National ChengKung University, Department of Mathematics,
Tainan City, 701, Taiwan
YangMills connections on 2manifolds
[PDF] 
In "The YangMills equations over Riemann surfaces", Atiyah and
Bott studied the YangMills functional over a Riemann surface from
the point of view of Morse theory. We generalize their study to all
closed, compact, connected, possibly nonorientable surfaces. We
introduce the notion of "super central extension" of the fundamental
group of a surface. It is the central extension when the surface is
orientable. We establish a precise correspondence between YangMills
connections and representations of super central extension. Knowing
this exact correspondence, we work mainly at the level of
representation varieties which are finite dimensional instead of the
level of strata which are infinite dimensional.
I will explain the YangMills functional and YangMills connections
over a Riemann surface and compare with the nonorientable surface case
that we studied.
This is a joint work with C.C. Melissa Liu.
 EUGENE KRITCHEVSKI, McGill University, Montreal
Hierarchical Anderson Model
[PDF] 
The hierarchical Anderson model is the random selfadjoint operator
where L is a hierarchical Laplacian, V is a random potential and
c > 0 is a coupling constant measuring the strength of the disorder.
In this talk, I will first review the basic properties of L and the
associated spectral dimension d. Then I will present the following
results about the spectral behavior of H.
(1) If d < 4 then, with probability one, the spectrum of H
is pure point at all energies and for all c.
(2) If d < 1 then, in a natural scaling limit, the eigenvalues
of finite volume approximations to H converge to a Poisson point
process.
 KARLPETER MARZLIN, Institute for Quantum Information Science, University of
Calgary
Criteria for the Existence of Decoherencefree Subspaces
[PDF] 
Decoherencefree subspaces (DFS) are spanned by such states of an open
quantum system that are insensitive to the decoherence induced by the
reservoir to which the system is coupled. DFS are immune to this
coupling because of different physical effects, including destructive
interference between different transition amplitudes or energy
conservation.
We compare different definitions of DFS and explore rigorous criteria
for the existence of DFS in finitedimensional systems coupled to
Markovian reservoirs. The advantages and disadvantages of various
approaches are compared and a geometrical interpretation for DFS in
qubitsystems is given.
 MARCO MERKLI, Memorial University
Infinite Products of Random Matrices and Repeated
Interaction Dynamics
[PDF] 
Let Y_{n} be a product of n independent, identically distributed
random matrices M, with the properties that Y_{n} is bounded in
n, and that M has a deterministic (constant) invariant vector.
Assuming that the probability of M having only the simple
eigenvalue 1 on the unit circle does not vanish, we show that Y_{n}
is the sum of a fluctuating and a decaying process. The latter
converges to zero almost surely, exponentially fast as n ®¥. The fluctuating part converges in Cesaro mean to a limit
that is characterized explicitly by the deterministic invariant vector
and the spectral data of E[M] associated to 1. No
additional assumptions are made on the matrices M; they may have
complex entries and not be invertible.
We apply our general results to two classes of dynamical systems:
inhomogeneous Markov chains with random transition matrices
(stochastic matrices), and random repeated interaction quantum
systems. In both cases, we prove ergodic theorems for the dynamics,
and we obtain the form of the limit states.
 ROB MILSON, Dalhousie University
Curvature homogeneous geometries
[PDF] 
A pseudoRiemannian manifold M is curvature homogeneous of order
s if the components of the curvature tensor and its first s
covariant derivatives are constant relative to some local frame. If
M is locally homogeneous, then it is curvature homogeneous.
Remarkably, the converse is also true, in some fashion: if M is
curvature homogeneous of order s and if s is greater than a
certain bound, called the Singer index, then M is locally
homogeneous. We establish that the Singer index for 4dimensional
CH, Lorentzian manifolds is equal to 2. Our approach is to
formulate the field equations for a CH geometry as an involutive EDS
on the secondorder frame bundle of M.
 DAVID ROWE, Department of Physics. University of Toronto, Toronto, ON,
M5S 1A7
Quasisymmetry, critical phenomena, and embedded
representations
[PDF] 
The use of symmetry in the description of physical systems has turned
up a novel type of representation in group theory that has deep
implications for understanding critical phenomena. Loosely speaking,
if one is given a unitary representation of a group on a Hilbert
space, it can turn out that the projection of this representation onto
a subspace may be another unitary representation that is neither a
subrepresentation nor a subquotient of the original; such a
representation is called an embedded representation. This
concept provides a natural framework for understanding why transitions
between phases of systems, associated with different symmetries,
frequently exhibit critical phenomena. It is observed that a system
in one phase, appears to hold onto the symmetry associated with that
phase until a breaking point is reached at which a rapid transition
occurs to a new phase associated with a different symmetry. In fact,
it appears that such an apparent symmetry, which we call a
quasisymmetry, is appropriately associated with an embedded
representation that can change continuously as a system approaches a
critical point.
 CRISTINA STOICA, Wilfrid Laurier University
Variational principles for systems with configuration space
isotropy
[PDF] 
This presentation considers Lagrangian systems on tangent bundles,
with lifted symmetries and configuration space isotropy. We use a
twisted parametrisation of the phase space corresponding to phase
space slices based at zero points of tangent fibres. Using
Hamilton's variational principle with appropriate constraints, we
deduce the Lagrangian bundle equations in the twisted coordinates.
This complements earlier work describing the dynamics on the cotangent
side.
