Non-existence of natural states for Abelian Chern-Simons theory

TL;DR

This paper proves that Abelian Chern-Simons theory, modeled as a functor from surfaces to C*-algebras, cannot have a natural state, highlighting a fundamental limitation in the algebraic approach to topological quantum field theories.

Contribution

It provides an elementary proof demonstrating the non-existence of natural states in Abelian Chern-Simons theory within the algebraic framework.

Findings

No natural state exists for Abelian Chern-Simons theory.

The non-existence phenomenon extends from Lorentzian QFTs to topological QFTs.

The proof is elementary and accessible.

Abstract

We give an elementary proof that Abelian Chern-Simons theory, described as a functor from oriented surfaces to C*-algebras, does not admit a natural state. Non-existence of natural states is thus not only a phenomenon of quantum field theories on Lorentzian manifolds, but also of topological quantum field theories formulated in the algebraic approach.