State Sums and (Quantum) Geometry: Frank Hellmann's PhD thesis

[marcus]

http://arxiv.org/abs/1102.1688
State Sums and Geometry
Frank Hellmann
PhD Thesis, 106 pages
(Submitted on 8 Feb 2011)
"In this thesis I review the definition of topological quantum field theories through state sums on triangulated manifolds. I describe the construction of state sum invariants of 3-manifolds from a graphical calculus and show how to evaluate the invariants as boundary amplitudes. I review how to define such a graphical calculus through SU(2) representation theory. I then review various geometricity results for the representation theory of SU(2), Spin(4) and SL(2,C), and define coherent boundary manifolds for state sums based on these representations. I derive the asymptotic geometry of the SU(2) based Ponzano-Regge invariant in three dimensions, and the SU(2) based Ooguri models amplitude in four dimensions. As a corollary to the latter results I derive the asymptotic behaviour of various recently proposed spin foam models motivated from the Plebanski formulation of general relativity. Finally the asymptotic geometry of the SL(2,C) based model is derived."

[marcus]

Some of the chapters could probably fill a much-needed pedagogical function.
At first sight, the exposition seems unusually clear. The diagrams are well-thought, and helpful.

Notice that Hellmann's PhD thesis advisor was John Barrett at Nottingham, and it is in part due to Barrett's vision and efforts that the themes of State Sum model and TQFT have become prominent in QG (which means quantum geometry as well as gravity :smile:)

Just in the past few days Barrett posted a thematic paper which shares some general ideas and outlook with this thesis. And many of these ideas will be reflected in the lineup of speakers at the June Zurich conference "Quantum Theory and Gravitation" (to be held at Einstein's alma mater).

So this very clearly written thesis is part of a bunch of things going on at the same time.