Coherent states, Path integral, and Semiclassical approximation

TL;DR

This paper demonstrates that path integrals for certain groups are exactly solvable using semiclassical methods when expressed with generalized coherent states, emphasizing the importance of discretization for correct results.

Contribution

It shows WKB exactness of path integrals for $SU(2)$ and $SU(1,1)$ using generalized coherent states and highlights the necessity of discretized path integrals over continuum forms.

Findings

WKB approximation is exact in large spin limit for these groups.

Discretized path integral formulation is essential for correct results.

Continuum path integrals can lead to incorrect conclusions.

Abstract

Using the generalized coherent states we argue that the path integral formulae for $SU(2)$ and $SU(1,1)$ (in the discrete series) are WKB exact,if the starting point is expressed as the trace of $e^{-iT\hat H}$ with $\hat H$ being given by a linear combination of generators. In our case,WKB approximation is achieved by taking a large ``spin'' limit: $J,K\rightarrow \infty$. The result is obtained directly by knowing that the each coefficient vanishes under the $J^{-1}$($K^{-1}$) expansion and is examined by another method to be legitimated. We also point out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression leads us to a wrong result. Therefore a great care must be taken when some geometrical action would be adopted, even if it is so beautiful, as the starting ingredient of path integral.