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Journal of Physics A Mathematical and General

The quantisation and measurement of momentum observables. II

Kay-Kong Wan, Keith McFarlane

Published Oct 1, 1981• 5 citations
Publisher

Abstract

For pt.I see ibid., vol.13, no.8, p.2673-88 (1980). Concepts of classical and quantum global measurability introduced in an earlier paper are discussed in greater detail and in the context of an n-dimensional Riemannian manifold. The ideas are illustrated with diagrams and examples. A notion of exact measurability is introduced and is shown to imply quantum global measurability. The physically important Killing momenta are shown to be exactly, and hence quantum globally, measurable.

DOI
10.1088/0305-4470/14/10/015

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For pt.I see ibid., vol.13, no.8, p.2673-88 (1980). Concepts of classical and quantum global measurability introduced in an earlier paper are discussed in greater detail and in the context of an n-dimensional Riemannian manifold. The ideas are illustrated with diagrams and examples. A notion of exact measurability is introduced and is shown to imply quantum global measurability. The physically important Killing momenta are shown to be exactly, and hence quantum globally, measurable.
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.
Mackey's scheme (1963) for the quantisation of classical momenta generating complete vector fields (complete momenta) is introduced, the differential operators corresponding to these momenta are introduced and discussed, and an isomorphism is shown to exist between the subclass of first-order self-adjoint differential operators, whose symmetric restrictions are essentially self-adjoint, and the complete classical momenta. Difficulties in the quantisation of incomplete momenta are discussed, and a critique given. Finally, in an attempt to relate the concept of completeness to measurability, concepts of classical and quantum global measurability are introduced, and shown to require completeness. These results afford strong physical insight into the nature of complete momenta, and lead us to suggest a quantisability condition based upon global measurability.
The problem of physical reality and measurability of observables is discussed on the example of phenomena revealed by the author. The essence of the phenomena is as follows. After measurement of the coordinate, the variance of the momentum can be smaller than the initial one and depends on the result of the coordinate measurement even if the coordinate and the momentum are uncorrelated. This refers to any pair of conjugate observables, in particular, to the number of quanta and the phase. In a state with a minimum product of uncertainties of the coordinate and the momentum, the mean value of the observable $${{\left\{ {\hat x^2 ,\hat p^2 } \right\}} \mathord{\left/ {\vphantom {{\left\{ {\hat x^2 ,\hat p^2 } \right\}} 2}} \right. \kern-\nulldelimiterspace} 2}$$ , which corresponds to the classical product of the squares of the coordinate and the momentum, is negative.
For pt.I see ibid. vol.22, p.4877 (1989). Previously (I) the author considered a second-quantised Hamiltonian which can be used to model a large number of important strongly interacting many-body systems. He has shown that the dynamics of such systems can be described exactly using quantum fields. The equations of motion derived in this way are highly nonlinear partial differential equations (PDE). He now uses a standard field theoretical approach where initially the fields are treated as classical functions. He studies the solutions of these equations which remarkably, may be obtained making full use of very recent mathematical discoveries (the symmetry reduction method). He finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper. These cases may correspond to different physical situations and the last, the second-order case, corresponds to the most general physical situation. Subsequently he outlines possible quantisation procedures for these fields. He then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise.