EXACTNESS IN THE WKB APPROXIMATION FOR SOME HOMOGENEOUS SPACES

TL;DR

This paper investigates the conditions under which the WKB approximation yields exact results in certain homogeneous spaces, specifically $CP^N$ and $D_{N,1}$, using path integral methods and coherent states.

Contribution

It demonstrates that the WKB approximation is exact at leading order for Hamiltonians bilinear in creation and annihilation operators on these spaces.

Findings

WKB approximation terminates at leading order for specific Hamiltonians.

Path integral expressions for trace of evolution operators are derived using coherent states.

The approach suggests potential generalizations to other cases.

Abstract

Analysis of the WKB exactness in some homogeneous spaces is attempted. $CP^N$ as well as its noncompact counterpart $D_{N,1}$ is studied. $U(N+1)$ or U(N,1) based on the Schwinger bosons leads us to $CP^N$ or $D_{N,1}$ path integral expression for the quantity, ${\rm tr} e^{-iHT}$, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.