Canonical transformations to action and angle variables and their representations in quantum mechanics

ANNALS OF PHYSICS 113, 481-482 (1978)

Abstracts

of Papers

Canonical Transformations to Action Mechanics. M. MOSHINSKY AND

to Appear

and

Angie

in Future

Variables

and

T. H. SELIGMAN, Instituto Apdo. Postal 20-364, Mtxico 20, D.F.

(UNAM),

Their

Issues

Representations

de Fisica, Universidad

in Quantum

de MBxico

We study the classical canonical transformations to action and angle variables for the repulsive and attractive oscillator and the free particle. These transformations turn out to be nonbijective (not one-to-one onto), and we introduce a sheet structure in phase space to restore bijectiveness. We find the “ambiguity group” for the three problems mentioned which connects points in one phase space that are mapped on a single point in the other. The different irreducible representations of this group can then be used to characterize different components of functions of canonically conjugate variables, thus providing us with an alternative procedure to recover bijectiveness. The above picturecharacterized by the “ambiguity spin” components-is readily translated into quantum mechanics. Indeed, by introducing wavefunctions with ambiguity spin, we are able to enlarge our Hilbert spaces in such a way that the spectra of the Hamiltonian and the action variable become the same, and thus a unitary representation of the classical canonical transformation becomes possible. These unitary representations are explicitly obtained for the three problems mentioned above and we also determine the correspondences between operators in the original and new Hilbert spaces due to the canonical transformation leading to action and angle variables. The suggestions on how to enlarge the Hilbert spaces seem to come entirely from the classical structure in phase space, thus allowing the surmise that some aspects of quantum mechanics (such as the spectra of simple operators) are already implicitly contained in classical mechanics. Production

and Signature

Physics, Harvard

of Intermediate

Bosons. MICHEL PERROTTET,Lyman Laboratory Massachusetts 02138.

Vector

University, Cambridge,

of

The front-back asymmetry of outgoing muons in the inclusive reaction pj -+Z (+p+p-) + anything is calculated in the framework of the Drell-Yan model. We get very general expressions for the front-back asymmetry and the differential cross section in terms of the weak coupling constants and the parton distributions. Results for W+ production and proton-proton collisions are also given. Numerical estimates are made using various parton distributions and the WeinbergSalam model for definiteness. We comment on the predictions of gauge theories that accommodate the absence of parity violating effects in atomic physics. We also give a new estimate of the inclusive cross sections pj or pp + Z (- p+p-) or W*(+ p*tyJ + anything. The effects of asymptotic freedom in the front-back asymmetries and the total cross sections are investigated. We reanalyze results for the purely leptonic reaction e+e- + y, Z + p+p-. Elastic

y-Proton

Scattering

at Low

and Intermediate

Energies.

RADESCU, Institute of Physics and Nuclear Engineering,

I. GUIASU,

C. POMPONN,

AND E. E.

P.O. Box 5206, Bucharest, Romania.

A new dispersive analysis of the elastic y-proton scattering at photon laboratory energies lower than 450 MeV, based on the use of Bardeen-Tung invariant amplitudes free of both kinematical singularities and zeros, is presented. The requirements of relativistic (and gauge) invariance are then incorporated from the beginning, no ad hoc subtractions being necessary any longer to satisfy the existing low-energy theorems. The adopted dispersion representation makes use as much as possible of fixed momentum transfer (t) dispersion relations, the subtraction functions (when unavoidable) being expressed as dispersion relations in t. The absorptive parts in various channels are evaluated within the two-particle unitarity approximation by means of recent tables of pion photoproduction

481 0003-4916/78/1132-0481$05.00/0 Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.