The Chern-Simons Source as a Conformal Family and Its Vertex Operators

TL;DR

This paper extends a canonical approach to quantum Chern-Simons theory by including sources, showing their states form conformal families, constructing vertex operators, and establishing the spin-statistics theorem through geometric methods.

Contribution

It introduces a formalism incorporating sources into quantum Chern-Simons theory, constructs vertex operators for abelian and nonabelian sources, and proves the spin-statistics relation geometrically.

Findings

Quantum states of sources form conformal families.

Regularized Wilson lines are vertex operators.

Spin-statistics theorem holds without relativistic fields.

Abstract

In a previous work, a straightforward canonical approach to the source-free quantum Chern-Simons dynamics was developed. It makes use of neither gauge conditions nor functional integrals and needs only ideas known from QCD and quantum gravity. It gives Witten's conformal edge states in a simple way when the spatial slice is a disc. Here we extend the formalism by including sources as well. The quantum states of a source with a fixed spatial location are shown to be those of a conformal family, a result also discovered first by Witten. The internal states of a source are not thus associated with just a single ray of a Hilbert space. Vertex operators for both abelian and nonabelian sources are constructed. The regularized abelian Wilson line is proved to be a vertex operator. We also argue in favor of a similar nonabelian result. The spin-statistics theorem is established for Chern-Simons dynamics even though the sources are not described by relativistic quantum fields. The proof employs geometrical methods which we find are strikingly transparent and pleasing. It is based on the research of European physicists about ``fields localized on cones.''