334 research outputs found
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 , , 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
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