van Vleck determinants: geodesic focussing and defocussing in Lorentzian spacetimes

TL;DR

This paper develops theoretical tools for computing the van Vleck determinant in Lorentzian spacetimes, which is crucial for understanding geodesic focusing and has broad applications in physics such as quantum mechanics and optics.

Contribution

It provides extensive theoretical methods specifically for calculating the van Vleck determinant related to geodesic flows in Lorentzian spacetimes, linking various physical applications.

Findings

Enhanced methods for computing the van Vleck determinant.

Implications for geodesic focusing and defocusing analysis.

Broader relevance to quantum and optical phenomena.

Abstract

The van Vleck determinant is an ubiquitous object, arising in many physically interesting situations such as: (1) WKB approximations to quantum time evolution operators and Green functions. (2) Adiabatic approximations to heat kernels. (3) One loop approximations to functional integrals. (4) The theory of caustics in geometrical optics and ultrasonics. (5) The focussing and defocussing of geodesic flows in Riemannian manifolds. While all of these topics are interrelated, the present paper is particularly concerned with the last case and presents extensive theoretical developments that aid in the computation of the van Vleck determinant associated with geodesic flows in Lorentzian spacetimes. {\sl A fortiori} these developments have important implications for the entire array of topics indicated. PACS: 04.20.-q, 04.20.Cv, 04.60.+n. To appear in Physical Review D47 (1993) 15 March.