Chern-Simons "ground state" from the path integral

Abstract

We consider a path integral representation of the time evolution $\exp(-\frac{i}{\hbar}tH)$ for Lagrangians of the variable $A$ which can be represented in the form (quadratic in $Q$) ${\cal L}(A)=\frac{1}{2}Q(A){\cal M}Q(A)+\partial_{\mu}L^{\mu}$. We show that $\exp(-\frac{i}{\hbar}tH)\exp(\frac{i}{\hbar}\int d{\bf x}L^{0}) =\exp(\frac{i}{\hbar}\int d{\bf x}L^{0})$ up to an $A$-independent factor. We discuss examples of the states $\exp(\frac{i}{\hbar}\int d{\bf x}L^{0})$ 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 Schr\"odinger equation can be expressed by the solution of the self-duality equation in the leading order of $\hbar$ expansion. We discuss applications to gauge theory on a Lorentzian manifold and gauge theories of gravity.