Chern-Simons "ground state" from the path integral

Abstract

We consider a path integral representation of the time evolution $\exp (-\frac{i}{\hslash }tH)$ for Lagrangians of the variable $A$ which can be represented in the form (quadratic in $Q$) $\mathcal{L}(A)=\frac{1}{2}Q(A)\mathcal{M}Q(A)+{\partial }_{\hat{I}}¼L^{\hat{I}}¼$. We show that $\exp (-\frac{i}{\hslash }tH)\exp (\frac{i}{\hslash }\int d\mathbf{x}{L}^0)=\exp (\frac{i}{\hslash }\int d\mathbf{x}{L}^0)$ up to an $A$-independent factor. We discuss examples of the states $\exp (\frac{i}{\hslash }\int d\mathbf{x}{L}^0)$ in quantum mechanics and in quantum field theory (the Chern-Simons states in Yang-Mills theory, Kodama states in quantum gravity). We show the relevance of these states for a determination of the dynamics in terms of stochastic perturbations of self-duality equations. The solution of the Schrödinger equation can be expressed by the solution of the self-duality equation in the leading order of $\hslash$ expansion. We discuss applications to gauge theory on a Lorentzian manifold and gauge theories of gravity.

14 pages