Geometry of the transport equation in multicomponent WKB approximations

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 * The research of both authors was partially supported by DOE Contract DE-FG03-93ER25177. We would like to thank Hans Duistermaat, Mikhail Karasev, Robert Littlejohn, and Jim Morehead for helpful discussions.1 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.

Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols

“…which is the expression found in [EW96], who explain how to relate it also to [LF91]. The scalar Hamiltonian h has the property that its quantization approximates the action of the full operator H on the range of the projection Π ε up to terms of order ε 2 ,…”

The Geometry of Radiative Transfer

“…[41]. Preliminary work in the literature that considers matrix-valued quantization from a geometric perspective is [25,12,8], although to our knowledge no complete picture exists at the moment. An interesting open question is also the transition from Maxwell's equations to radiative transfer in spacetime, the natural setting of electromagnetic theory.…”

The time-dependent Born-Oppenheimer approximation

“…The construction of the projection P ε follows a general scheme outlined by Emmerich and Weinstein [8], refined in [5] and [23], and finally applied to the Born-Oppenheimer approximation by Martinez and Sordoni [21]. The basic idea is to determine the coefficients P j in the expansion P ε (n) = P + n j=1 ε j P j order by order such that P ε (n) is approximately a projection and commutes with H ε up to terms of order ε n+1 .…”

“…The justification of the first-order Born-Oppenheimer approximation as in [29] is based on generalizing the standard adiabatic theorem of quantum mechanics to a space-adiabatic theorem. The higher order corrections require a systematic scheme developed in [5,8,21,23,25,26,28] heavily based on pseudo-differential calculus with operator valued symbols, a method which traces back to the pioneering work of Sjöstrand [27]. We therefore refrain from explaining the general scheme in detail, but instead present the main ideas and a new elementary derivation of the correct second order Born-Oppenheimer Hamiltonian.…”