Spontaneous breaking of discrete symmetries and the semiclassical approximation

Abstract

The generating functional $Z(j)\ensuremath{\equiv}{〈0|0〉}_{j}$ is evaluated asymptotically as $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ with special attention to the existence of several extremal points of the action. We show that $Z(j)\ensuremath{\sim}N\ensuremath{\Sigma}{\ensuremath{\alpha}}^{}\mathrm{exp}[(\frac{i}{\ensuremath{\hbar}}){W}_{\ensuremath{\alpha}}(j)]$, where $\ensuremath{\alpha}$ labels the extremal points. Here each ${W}_{\ensuremath{\alpha}}$ has a Taylor expansion in $\ensuremath{\hbar}$, Some implications and interpretations of this new result are given.