140 research outputs found
Areal Theory
New features are described for models with multi-particle area-dependent
potentials, in any number of dimensions. The corresponding many-body field
theories are investigated for classical configurations. Some explicit solutions
are given, and some conjectures are made about chaos in such field theories.Comment: December 2001 "Coral Gables" conference contribution; Scientific
Workplace Late
Potentials Unbounded Below
Continuous interpolates are described for classical dynamical systems defined
by discrete time-steps. Functional conjugation methods play a central role in
obtaining the interpolations. The interpolates correspond to particle motion in
an underlying potential, . Typically, has no lower bound and can exhibit
switchbacks wherein changes form when turning points are encountered by the
particle. The Beverton-Holt and Skellam models of population dynamics, and
particular cases of the logistic map are used to illustrate these features.Comment: Based on a talk given 29 July 2010, at the workshop on Supersymmetric
Quantum Mechanics and Spectral Design, Centro de Ciencias de Benasque Pedro
Pascual. This version incorporates modifications to conform to the published
paper: Additional references and discussion; New section 3.2 on the Skellam
exponential model; Appendix change
Negative Probability and Uncertainty Relations
A concise derivation of all uncertainty relations is given entirely within
the context of phase-space quantization, without recourse to operator methods,
to the direct use of Weyl's correspondence, or to marginal distributions of x
and p.Comment: RevTeX, 4 page
More on Rotations as Spin Matrix Polynomials
Any nonsingular function of spin j matrices always reduces to a matrix
polynomial of order 2j. The challenge is to find a convenient form for the
coefficients of the matrix polynomial. The theory of biorthogonal systems is a
useful framework to meet this challenge. Central factorial numbers play a key
role in the theoretical development. Explicit polynomial coefficients for
rotations expressed either as exponentials or as rational Cayley transforms are
considered here. Structural features of the results are discussed and compared,
and large j limits of the coefficients are examined.Comment: Additional references, simplified derivation of Cayley transform
polynomial coefficients, resolvent and exponential related by Laplace
transform. Other minor changes to conform to published version to appear in J
Math Phy
Logistic Map Potentials
We develop and illustrate methods to compute all single particle potentials
that underlie the logistic map, x --> sx(1-x) for 0<s<=4. We show that the
switchback potentials can be obtained from the primary potential through
functional transformations. We are thereby able to produce the various branches
of the corresponding analytic potential functions, which have an infinite
number of branch points for generic s>2. We illustrate the methods numerically
for the cases s=5/2 and s=10/3
Branes, Quantum Nambu Brackets, and the Hydrogen Atom
The Nambu Bracket quantization of the Hydrogen atom is worked out as an
illustration of the general method. The dynamics of topological open branes is
controlled classically by Nambu Brackets. Such branes then may be quantized
through the consistent quantization of the underlying Nambu brackets: properly
defined, the Quantum Nambu Brackets comprise an associative structure, although
the naive derivation property is mooted through operator entwinement. For
superintegrable systems, such as the Hydrogen atom, the results coincide with
those furnished by Hamiltonian quantization--but the method is not limited to
Hamiltonian systems.Comment: 6 pages, LateX2e. Invited talk by CZ at the XIII International
Colloquium on Integrable Systems and Quantum Groups, Prague, June 18, 200
Morphing quantum mechanics and fluid dynamics
We investigate the effects of given pressure gradients on hydrodynamic flow
equations. We obtain results in terms of implicit solutions and also in the
framework of an extra-dimensional formalism involving the diffusion/Schrodinger
equation.Comment: Examples involving shocks and references adde
Phase-space Quantization of Field Theory
In this lecture, a limited introduction of gauge invariance in phase-space is
provided, predicated on canonical transformations in quantum phase-space. Exact
characteristic trajectories are also specified for the time-propagating Wigner
phase-space distribution function: they are especially simple - indeed,
classical - for the quantized simple harmonic oscillator. This serves as the
underpinning of the field theoretic Wigner functional formulation introduced.
Scalar field theory is thus reformulated in terms of distributions in field
phase-space. This is a pedagogical selection from work published in J Phys A32
(1999) 771 and Phys Rev D58 (1998) 025002, reported at the Yukawa Institute
Workshop "Gauge Theory and Integrable Models", 26-29 January, 1999.Comment: 14 pages, LaTeX, 1 eps figure, epsf.sty, ptptex.sty, ptp-text.sty
Reported at the YITP Workshop "Gauge Theory and Integrable Models", 26-29
January, 1999. References added and graphics update
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