355 research outputs found
Infinite-Dimensional Representations of 2-Groups
A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory. We classify irreducible and indecomposable representations and intertwiners. We also classify "irretractable" representations--another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered "separable 2-Hilbert spaces", and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras
Smart Devices in Criminal Investigations: How Section 8 of the Canadian Charter of Rights and Freedoms Can Better Protect Privacy in the Search of Technology and Seizure of Information
This thesis examines the jurisprudence from the Supreme Court of Canada (SCC) on informational privacy under section 8 of the Canadian Charter of Rights and Freedoms as it relates to searches of technology in the context of criminal investigations. The development and use of technology in criminal investigations will be detailed along with an overview of the current state of the law in this area. Challenges with the interpretation of section 8 demonstrate a prevalent uncertainty. This thesis proposes a new approach for the SCC to apply to cases where technology intersects with section 8 of the Charter. The proposal rests on a clearer and broader understanding of privacy along with measurable categories for more predictable outcomes
Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector
In our companion paper, we focussed on the quantisation of the
Rarita-Schwinger sector of Supergravity theories in various dimensions by using
an extension of Loop Quantum Gravity to all spacetime dimensions. In this
paper, we extend this analysis by considering the quantisation of additional
bosonic fields necessary to obtain a complete SUSY multiplet next to graviton
and gravitino in various dimensions. As a generic example, we study concretely
the quantisation of the 3-index photon of 11d SUGRA, but our methods easily
extend to more general p-form fields. Due to the presence of a Chern-Simons
term for the 3-index photon, which is due to local SUSY, the theory is
self-interacting and its quantisation far from straightforward. Nevertheless,
we show that a reduced phase space quantisation with respect to the 3-index
photon Gauss constraint is possible. Specifically, the Weyl algebra of
observables, which deviates from the usual CCR Weyl algebras by an interesting
twist contribution proportional to the level of the Chern-Simons theory, admits
a background independent state of the Narnhofer-Thirring type.Comment: 12 pages. v2: Journal version. Minor clarifications and correction
2-Group Representations for Spin Foams
Just as 3d state sum models, including 3d quantum gravity, can be built using
categories of group representations, "2-categories of 2-group representations"
may provide interesting state sum models for 4d quantum topology, if not
quantum gravity. Here we focus on the "Euclidean 2-group", built from the
rotation group SO(4) and its action on the group of translations of 4d
Euclidean space. We explain its infinite-dimensional unitary representations,
and construct a model based on the resulting representation 2-category. This
model, with clear geometric content and explicit "metric data" on triangulation
edges, shows up naturally in an attempt to write the amplitudes of ordinary
quantum field theory in a background independent way.Comment: 8 pages; to appear in proceedings of the XXV Max Born Symposium: "The
Planck Scale", Wroclaw, Polan
On the representations of 2-groups in {Baez-Crans} 2-vector spaces
We prove that the theory of representations of a finite 2-group
in Baez-Crans 2-vector spaces over a field of characteristic zero
essentially reduces to the theory of -linear representations of the group of
isomorphism classes of objects of , the remaining homotopy
invariants of playing no role. It is also argued that a similar
result is expected to hold for topological representations of compact
topological 2-groups in suitable topological Baez-Crans 2-vector spaces.Comment: 9 page
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