24 research outputs found
Fuzzy Nambu-Goldstone Physics
In spacetime dimensions larger than 2, whenever a global symmetry G is
spontaneously broken to a subgroup H, and G and H are Lie groups, there are
Nambu-Goldstone modes described by fields with values in G/H. In
two-dimensional spacetimes as well, models where fields take values in G/H are
of considerable interest even though in that case there is no spontaneous
breaking of continuous symmetries. We consider such models when the world sheet
is a two-sphere and describe their fuzzy analogues for G=SU(N+1),
H=S(U(N-1)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy
versions of continuum models on S^2 when the target spaces are Grassmannians
and flag manifolds described by (N+1)x(N+1) projectors of rank =< (N+1)/2.
These fuzzy models are finite-dimensional matrix models which nevertheless
retain all the essential continuum topological features like solitonic sectors.
They seem well-suited for numerical work.Comment: Latex, 18 pages; references added, typos correcte
On the structure of the space of generalized connections
We give a modern account of the construction and structure of the space of
generalized connections, an extension of the space of connections that plays a
central role in loop quantum gravity.Comment: 30 pages, added references, minor changes. To appear in International
Journal of Geometric Methods in Modern Physic
Full regularity for a C*-algebra of the Canonical Commutation Relations. (Erratum added)
The Weyl algebra,- the usual C*-algebra employed to model the canonical
commutation relations (CCRs), has a well-known defect in that it has a large
number of representations which are not regular and these cannot model physical
fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs
of a countably dimensional symplectic space (S,B) and such that its
representation set is exactly the full set of regular representations of the
CCRs. This construction uses Blackadar's version of infinite tensor products of
nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalised
group algebra, explained below) for the \sigma-representation theory of the
abelian group S where \sigma(.,.):=e^{iB(.,.)/2}.
As an easy application, it then follows that for every regular representation
of the Weyl algebra of (S,B) on a separable Hilbert space, there is a direct
integral decomposition of it into irreducible regular representations (a known
result).
An Erratum for this paper is added at the end.Comment: An erratum was added to the original pape
Barycentric decomposition of quantum measurements in finite dimensions
We analyze the convex structure of the set of positive operator valued
measures (POVMs) representing quantum measurements on a given finite
dimensional quantum system, with outcomes in a given locally compact Hausdorff
space. The extreme points of the convex set are operator valued measures
concentrated on a finite set of k \le d^2 points of the outcome space, d<
\infty being the dimension of the Hilbert space. We prove that for second
countable outcome spaces any POVM admits a Choquet representation as the
barycenter of the set of extreme points with respect to a suitable probability
measure. In the general case, Krein-Milman theorem is invoked to represent
POVMs as barycenters of a certain set of POVMs concentrated on k \le d^2 points
of the outcome space.Comment: !5 pages, no figure
Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles
We apply some methods of homology and K-theory to special classes of branes
wrapping homologically nontrivial cycles. We treat the classification of
four-geometries in terms of compact stabilizers (by analogy with Thurston's
classification of three-geometries) and derive the K-amenability of Lie groups
associated with locally symmetric spaces listed in this case. More complicated
examples of T-duality and topology change from fluxes are also considered. We
analyse D-branes and fluxes in type II string theory on with torsion flux and demonstrate in details
the conjectured T-duality to with no flux. In the
simple case of , T-dualizing the circles reduces to
duality between with
flux and with no flux.Comment: 27 pages, tex file, no figure
Monoids of intervals of simple refinement monoids and non-stable K-Theory of multiplier algebras
We show that the representation of the monoid of intervals of a simple refinement monoid in terms of affine semicontinuous functions, given by Perera in 2001, fails to be faithful in the case of strictly perforated monoids. We give some potential applications of this result in the context of monoids of intervals and K-Theory of multiplier rings