Multicomponent WKB and Quantization

TL;DR

This paper develops a geometric approach to WKB approximation for multicomponent quantum systems with matrix-valued Hamiltonians, utilizing Fedosov's star-product construction to generalize scalar methods to complex bundles over symplectic manifolds.

Contribution

It introduces a geometric framework for multicomponent WKB and quantization using Fedosov's star-product, extending scalar techniques to general bundles on symplectic manifolds.

Findings

Provides a geometric WKB method for matrix Hamiltonians

Uses Fedosov's star-product to handle general symplectic bundles

Reduces multicomponent problems to scalar cases where possible

Abstract

Hamiltonians whose symbols are not simply real valued, but matrix or, more generally, endomorphism valued functions appear in many places in physics, examples being the Dirac equation, multicomponent wave equations like electrodynamics in media, and Yang-Mills theories, and the Born-Oppenheimer approximation in molecular physics. The aim of this paper is to give a completely geometric approach to the WKB approximation od such systems, and to reduce the problem ``as far as possible'' to the scalar case. A star-product formulation of quantum mechanics proves to be particularly useful in this context. As opposed to other approaches in the literature which restrict themselves to the use of the Moyal product and thus to the study of trivial bundles (or local trivializations) over $\real^{2n}$, we will consider general bundles over arbitrary symplectic manifolds. Here, Fedosov's construction \cite{fedosov} will be the adequate tool, since it gives an explicit construction for star products in this general setting.