Quantization in singular real polarizations: K\"ahler regularization, Maslov correction and pairings

TL;DR

This paper investigates the Maslov correction in semiclassical states using K"ahler regularization, demonstrating its effectiveness in reproducing correct phases at caustic points, especially for the harmonic oscillator.

Contribution

It introduces a K"ahler regularized BKS pairing approach to handle singular real polarizations and accurately compute phases in semiclassical analysis.

Findings

Successfully reproduces phases at caustic points for the harmonic oscillator

Establishes existence of K"ahler regularization for general semiclassical states

Connects degenerating K"ahler polarizations to singular real polarizations

Abstract

We study the Maslov correction to semiclassical states by using a K\"ahler regularized BKS pairing map from the energy representation to the Schr\"odinger representation. For general semiclassical states, the existence of this regularization is based on recently found families of K\"ahler polarizations degenerating to singular real polarizations and corresponding to special geodesic rays in the space of K\"ahler metrics. In the case of the one-dimensional harmonic oscillator, we show that the correct phases associated with caustic points of the projection of the Lagrangian curves to the configuration space are correctly reproduced.