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Quantization on submanifolds

H. D. Doebner, J. Tolar

Published Jan 1, 1984• 0 citations
OpenAlex

Abstract

Extrinsic position and momentum operators are defined on a Riemannian manifold (M,g) via an isometric embedding i: (M,g)----» (R i8 n )-An intrinsic Hamiltonian Hdescribing free dynamics on (M,g) is constructed by an associated Riemannian submer3ion satisfying a physically well justified condition.

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This thesis is the outcome of research conducted into the quantization of observables defined over a Riemannian configuration manifold, and is naturally divided into three chapters: \n \nThe first chapter is essentially concerned with the development from physical bases, of the concepts of classical and quantum “global measurability”, which, when combined with the requirement that all quantizable momenta shall be either classically or quantum mechanically globally measurable, result in the exclusion form the class of quantizable momenta of all those observables which are not quantizable in accordance with G. W. Mackey’s natural procedure. The refinement of exact global measurability is then introduced and the physically important class of “killing” momenta are found to be exactly classically and quantum mechanically globally measurable and moreover quantizable by means of Mackey’s scheme. \n \nThe second chapter seeks to analyse the algebraic structure of the Mackey-quantizable momenta, so as to compare and contrast his geometric scheme with the various algebraic schemes which have frequently been proposed. As an extension of this work, a natural geometric quantization is proposed for the more general class of observables “linear in momentum”, the set of quantizable such entities determines, and its algebraic properties discussed. It is concluded that, if Mackey’s procedure is, as we argue, correct and exhaustive of the quantizable momenta, then algebraic rules of quantization do not, in general, obtain among the momenta defined over a Riemannian space, ‘though in the case of momenta, such as the Killing momenta, reflecting the symmetry of the configuration manifold, such laws not only exist but are moreover of great practical importance. \n \nThe third chapter is concerned with the quantization of the observables “multilinear in momentum” and in particular with the delimitation of the possible operators of formal quantization as circumscribed by the requirement of operator symmetry, and the question of the essential self-adjointness of local observables associated with the multilinear momenta. Finally some possible means of determining and exact quantization are discussed and contrasted and a tentative selection made of a particular scheme which is then illustrated by concrete example on the real line with the usual metric. \n \nWe conclude this preface with a brief indication of the layout of the thesis: Each chapter or associated group of appendices has been treated for the purpose of all textual numbering and referencing systems, as a separate unit, and is prefixed by a detailed list of contents, and concluded by the appropriate group of reference, so that all pointers of a chapter or group of appendices refer, unless otherwise indicated, to the text of that chapter or group of appendices. Finally note that a bibliography has been included which will serve as a collected list of references to papers and books employed.
We show that any rule for quantization of functions on the phase space of a dynamical system is equivalent to a deformation of the Poisson bracket, in the sense of Lie algebras. All Schrödinger-type quantizations are equivalent mathematically but not physically. We examine whether a quantization rule exists in which the Poisson bracket of the free Hamiltonian and any function not more than quadratic in momentum gets quantized into the commutator of the quantized versions of these two observables. We show that if the (Riemannian) configuration manifold is irreducible, this condition uniquely specifies the quantization of these functions, in the cases when it can be satisfied. However, it can only be satisfied for systems whose configuration manifolds are one-dimensional, or have vanishing Ricci tensor, or are spaces of constant curvature. Thus the condition will not serve as a general physical principle which could be invoked to fix a quantization rule.