165 research outputs found
Generalized Sums over Histories for Quantum Gravity II. Simplicial Conifolds
This paper examines the issues involved with concretely implementing a sum
over conifolds in the formulation of Euclidean sums over histories for gravity.
The first step in precisely formulating any sum over topological spaces is that
one must have an algorithmically implementable method of generating a list of
all spaces in the set to be summed over. This requirement causes well known
problems in the formulation of sums over manifolds in four or more dimensions;
there is no algorithmic method of determining whether or not a topological
space is an n-manifold in five or more dimensions and the issue of whether or
not such an algorithm exists is open in four. However, as this paper shows,
conifolds are algorithmically decidable in four dimensions. Thus the set of
4-conifolds provides a starting point for a concrete implementation of
Euclidean sums over histories in four dimensions. Explicit algorithms for
summing over various sets of 4-conifolds are presented in the context of Regge
calculus. Postscript figures available via anonymous ftp at
black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps.Comment: 82pp., plain TeX, To appear in Nucl. Phys. B,FF-92-
Thompson's group is the orientation-preserving automorphism group of a cellular complex
We consider a planar surface ∑ of in nite type which has Thompson's group T as asymptotic mapping class group. We construct the asymptotic pants complex C of ∑ and prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the Thompson group T by Z=2Z
Algebraic Topology
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd
Dynamics for holographic codes
We describe how to introduce dynamics for the holographic states and codes
introduced by Pastawski, Yoshida, Harlow and Preskill. This task requires the
definition of a continuous limit of the kinematical Hilbert space which we
argue may be achieved via the semicontinuous limit of Jones. Dynamics is then
introduced by building a unitary representation of a group known as Thompson's
group T, which is closely related to the conformal group in 1+1 dimensions. The
bulk Hilbert space is realised as a special subspace of the semicontinuous
limit Hilbert space spanned by a class of distinguished states which can be
assigned a discrete bulk geometry. The analogue of the group of large bulk
diffeomorphisms is given by a unitary representation of the Ptolemy group Pt,
on the bulk Hilbert space thus realising a toy model of the AdS/CFT
correspondence which we call the Pt/T correspondence.Comment: 40 pages (revised version submitted to journal). See video of related
talk: https://www.youtube.com/watch?v=xc2KIa2LDF
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