Canonical Transformations in Quantum Mechanics

Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can, in principle, be realized quantum mechanically as a product of these transformations. It is found that the intertwining of two super-Hamiltonians is equivalent to there being a canonical transformation between them. A consequence is that the procedure for solving a differential equation can be viewed as a sequence of elementary canonical transformations trivializing the super-Hamiltonian associated to the equation. It is proposed that the quantum integrability of a system is equivalent to the existence of such a sequence.Comment: 27 pages, McGill 92-29 (revised version--several typos fixed in examples

An elegant solution of the n-body Toda problem

The solution of the classical open-chain n-body Toda problem is derived from an ansatz and is found to have a highly symmetric form. The proof requires an unusual identity involving Vandermonde determinants. The explicit transformation to action-angle variables is exhibited.Comment: LaTeX, 13 p

Clocks and Time

A general definition of a clock is proposed, and the role of clocks in establishing temporal pre-conditions in quantum mechanical questions is critically discussed. The different status of clocks as used by theorists external to a system and as used by participant-observers within a system is emphasized. It is shown that the foliation of spacetime into instants of time is necessary to correctly interpret the readings of clocks and that clocks are thus insufficient to reconstruct time in the absence of such a foliation.Comment: LaTeX, 19 p

Special functions from quantum canonical transformations

Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body Toda equation are also found. The derivation of these representations motivate the form of a two-dimensional generalized hypergeometric equation which contains the non-periodic Toda equation as a special case and whose solutions may be obtained by quantum canonical transformation.Comment: LaTeX, 24 pp., Imperial-TP-93-94-5 (revision: two sections added on the three-body Toda problem and a two-dimensional generalization of the hypergeometric equation

Fixing Einstein's equations

Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations. The physical and geometrical content of the original theory remain unchanged, but dynamical evolution is made sound. Here we present a new hyperbolic formulation in terms of $g_{ij}$, $K_{ij}$, and \bGam_{kij} that is strikingly close to the space-plus-time (``3+1'') form of Einstein's original equations. Indeed, the familiarity of its constituents make the existence of this formulation all the more unexpected. This is the most economical first-order symmetrizable hyperbolic formulation presently known to us that has only physical characteristic speeds, either zero or the speed of light, for all (non-matter) variables. This system clarifies the relationships between Einstein's original equations and the Einstein-Ricci and Frittelli-Reula hyperbolic formulations of general relativity and establishes links to other hyperbolic formulations.Comment: 8 pages, revte

Evaluation of light-gauge metal diaphragm behavior and the diaphragms interaction with post frames

Light-gauge steel sheeted diaphragms on wood frames may be used to transfer in-plane shear forces to the end walls in post frame structures. The amount of shear force transferred is dependent upon the in-plane stiffness of the diaphragm and the frame stiffness;The results of 31 full-scale cantilever diaphragm tests are presented. Two different sheet profiles and three fastener patterns were used. The results are compared with the predicted stiffness from a plane truss computer analog. Test variables include openings in the sheeting, recessing of purlins in from the sheet ends, seams in the length of the sheet and placing the purlins flat;An analytical method and an empirical equation to be used in conjunction with testing are presented. The analytical method predicts the stiffness of the control diaphragms within 1.5% of that predicted by the plane truss computer analog. The empirical equation was fitted to steel diaphragms on steel frames and can be used to adjust for different diaphragm lengths, purlin spacings, and sheet thicknesses;It was found that the location and size of an opening does influence diaphragm stiffness. Openings with sheeting on two sides only will reduce diaphragm stiffness approximately twice as much as openings with sheeting on three or four sides. Fastener stiffness and location have the largest impact on diaphragm stiffness. Increasing the number of fasteners at seams in the length of the diaphragm will off-set the effect of the discontinuous sheet length. Fasteners near the edge of the sheet have a much larger impact on stiffness than those near the center of the sheet. The plane truss computer analog predicted the test diaphragm stiffness reasonably well;Several methods of modeling the diaphragm frame interaction are reviewed. It was found that the plane frame/truss model will model frame-diaphragm interaction more accurately for complex structural systems if the diaphragms are represented by two spring elements. Also varying frame and diaphragm stiffness can be included easily