Topology of graph configuration spaces and quantum statistics
Abstract
In this thesis we develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs. Moreover we present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions, which have a nice physical interpretation as two-body potentials constructed from one-body potentials. We also give a brief introduction to discrete Morse theory.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2014
- DOI:
- 10.48550/arXiv.1408.7002
- arXiv:
- arXiv:1408.7002
- Bibcode:
- 2014arXiv1408.7002S
- Keywords:
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- Mathematical Physics;
- Mathematics - Algebraic Topology;
- Quantum Physics
- E-Print:
- PhD thesis, Bristol 2014