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, $V$. Typically, $V$ has no lower bound and can exhibit switchbacks wherein $V$ 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