A Note On The Chern-Simons And Kodama Wavefunctions

TL;DR

This paper investigates the properties of the Chern-Simons wavefunction in Yang-Mills theory, revealing its energy spectrum, relation to fermion vacua, and potential parallels in gravitational theories, despite its unphysical nature.

Contribution

It provides a detailed analysis of the Chern-Simons wavefunction's physical implications and its connection to supersymmetry and gravitational analogs.

Findings

Positive helicity gauge bosons have positive energy in this state

Negative helicity gauge bosons have negative energy and some negative norm states

The Chern-Simons state is a supersymmetric partner of the naive fermion vacuum

Abstract

Yang-Mills theory in four dimensions formally admits an exact Chern-Simons wavefunction. It is an eigenfunction of the quantum Hamiltonian with zero energy. It is known to be unphysical for a variety of reasons, but it is still interesting to understand what it describes. We show that in expanding around this state, positive helicity gauge bosons have positive energy and negative helicity ones have negative energy. Some of the negative energy states would have negative norm. We also show that the Chern-Simons state is the supersymmetric partner of the naive fermion vacuum in which one does not fill the fermi sea. Finally, we give a sort of explanation of ``why'' this state exists. Similar properties can be expected for the analogous Kodama wavefunction of gravity.