Gauge Symmetries,Topology and Quantisation

TL;DR

This paper reviews gauge invariance, boundary conditions, edge states in quantum Hall effect, and topological quantisation methods, including a proof of the spin-statistics theorem that avoids quantum field theory and relativity.

Contribution

It provides a topological approach to quantisation and gauge theories, including a novel proof of the spin-statistics theorem using fibre bundles and covering spaces.

Findings

Edge states derived from Chern-Simons action

Topological proof of spin-statistics theorem

Clarification of gauge invariance and boundary conditions

Abstract

The following two loosely connected sets of topics are reviewed in these lecture notes: 1) Gauge invariance, its treatment in field theories and its implications for internal symmetries and edge states such as those in the quantum Hall effect. 2) Quantisation on multiply connected spaces and a topological proof the spin-statistics theorem which avoids quantum field theory and relativity. Under 1), after explaining the meaning of gauge invariance and the theory of constraints, we discuss boundary conditions on gauge transformations and the definition of internal symmetries in gauge field theories. We then show how the edge states in the quantum Hall effect can be derived from the Chern-Simons action using the preceding ideas. Under 2), after explaining the significance of fibre bundles for quantum physics, we review quantisation on multiply connected spaces in detail, explaining also mathematical ideas such as those of the universal covering space and the fundamental group. These ideas are then used to prove the aforementioned topological spin-statistics theorem.e of the universal covering space and the fundamental group.