Geometry of the transport equation in multicomponent WKB approximations

TL;DR

This paper provides a comprehensive geometric derivation of the transport equation in multicomponent WKB approximations, revealing new tensorial structures and obstructions related to eigenbundles and phase space curvature.

Contribution

It introduces a geometric framework for the transport equation without relying on local sections or complete diagonalizability, and interprets the 'no-name' tensor in terms of curvature and second fundamental form.

Findings

The 'no-name' tensor is a curvature-related geometric object.

A simplified form of the tensor is obtained in the non-degenerate case.

Obstructions to WKB state existence are characterized geometrically.

Abstract

Although the WKB approximation for multicomponent systems has been intensively studied in the literature, its geometric and global aspects are much less well understood than in the scalar case. In this paper we give a completely geometric derivation of the transport equation, without using local sections and without assuming complete diagonalizability of the matrix valued principal symbol, or triviality of its eigenbundles. The term (called ``no-name term'' in some previous literature) appearing in the transport equation in addition to the covariant derivative with respect to a natural projected connection will be a tensor, independent of the choice of any sections. We give a geometric interpretation of this tensor, involving the contraction of the curvature of the eigenbundle and an analog of the second fundamental form with the Poisson tensor in phase space. In the non-degenerate case this term may be rewritten in an even simpler geometric form. Finally, we discuss obstructions to the existence of WKB states and give a geometric description of the quantization condition for WKB states for a non-degenerate eigenvalue-function.