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For a non-relativistic scale invariant system in two spatial dimensions, the quantum scattering amplitude $f(\theta)$ is given as a dispersion relation, with a simple closed form for ${\rm Im}(f(\theta)$) as well as the integrated cross-section $\sigma \propto {\rm Im}(f(\theta=0))$. For fixed $\theta \neq 0$, the classical limit is straightforward to obtain.

We consider some elementary features of Newtonian gravity, or electrostatics, as defined on an N-sphere. In particular, we present and discuss "the shell theorem" for this system., Comment: This work resulted from participation in the Bahamas Advanced Study Institute & Conferences (BASIC) 2022, 10 pages, 4 figures

Concise methods are used to compute the propagator for a non-relativistic particle subject to a potential with x^2 and 1/x^2 terms.

We discuss some equivalence relations between the non-relativistic quantum mechanics for particles subjected to potentials and for particles moving freely on background geometries. In particular, we illustrate how selected geometries can be used to regularize singular potentials.

We fantasize about an alternative history for theories of electromagnetism and gravitation., Comment: In tribute to Murray Gell-Mann

The volume charge density for a conducting ellipsoid is expressed in simple geometrical terms, and then used to obtain the known surface charge density as well as the uniform charge per length along any principal axis. Corresponding results are presented for conducting hyperellipsoids in any number of spatial dimensions.

We describe a field theory for "massive dual gravity" in N spacetime dimensions. We obtain a Lagrangian that gives the lowest order coupling of the field to the N-dimensional curl of its own energy-momentum tensor. We then briefly discuss classical solutions. Finally, we show the theory is the exact dual of the Ogievetsky-Polubarinov model generalized to any N., Comment: In tribute to Peter Freund, with additional references

We reconsider a massive dual spin 2 field theory in four spacetime dimensions. We obtain the Lagrangian that describes the lowest order coupling of the field to the four-dimensional curl of its own energy-momentum tensor. We then find some static solutions for the dual field produced by other energy-momentum sources and we compare these to similar static solutions for non-dual "finite range" gravity. Finally, through use of a nonlinear field redefinition, we show the theory is the exact dual of the Ogievetsky-Polubarinov model for a massive spin 2 field., Comment: Modified title

We discuss the grounded, equipotential ellipse in two-dimensional electrostatics to illustrate different ways of extending the domain of the potential and placing image charges such that homogeneous boundary conditions are satisfied. In particular, we compare and contrast the Kelvin and Sommerfeld image methods., Comment: One example and two Appendices added

We compute exterior Green functions for equipotential, grounded hyperspheres in N-dimensional electrostatics by squashing Riemannian wormholes, where an image charge is placed in the branch of the wormhole opposite the branch containing the source charge, thereby providing a vivid geometrical approach to a method first suggested in 1897 by Sommerfeld. We compare and contrast the strength and location of the image charge in the wormhole approach with that of the conventional Euclidean solution where an image charge of reduced magnitude is located inside the hypersphere. While the two approaches give mathematically equivalent Green functions, we believe they provide strikingly different physics perspectives., Comment: Companion paper for arXiv:1805.11147 with minor typos corrected in version 2. To appear in J Math Phys

We compute the exterior Green function for a grounded equi-potential circular ring in two-dimensional electrostatics by treating the system geometrically as a "squashed wormhole" with an image charge located in a novel but obvious position, thereby implementing a method first suggested in 1897 by Sommerfeld. We compare and contrast the strength and location of the image charge in the wormhole picture with that of the conventional point of view where an image charge is located inside the circular ring. While the two viewpoints give mathematically equivalent Green functions, we believe they provide strikingly different physics perspectives. We also comment on earlier Green function results by Hobson in 1900, and by Davis and Reitz in 1971, who applied Sommerfeld's method to analyze a grounded conducting circular disk in three-dimensional electrostatics., Comment: 13 figures

This is a survey of the electrostatic potentials produced by charged straight-line segments, in various numbers of spatial dimensions, with comparisons between uniformly charged segments and those having non-uniform linear charge distributions that give rise to ellipsoidal equipotentials surrounding the segments. A uniform linear distribution of charge is compatible with ellipsoidal equipotentials only for three dimensions. In higher dimensions, the linear charge density giving rise to ellipsoidal equipotentials is counter-intuitive --- the charge distribution has a maximum at the center of the segment and vanishes at the ends of the segment. Only in two dimensions is the continuous charge distribution intuitive --- for that one case of ellipsoidal equipotentials, the charge is peaked at the ends of the segment and minimized at the center.

Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an alternate derivation. The central factorial numbers play key roles in both derivations., Comment: 6 Figures. References updated in v2, along with some editing of text

The dynamics of topological open branes is controlled by Nambu Brackets. Thus, they might be quantized through the consistent quantization of the underlying Nambu brackets, including odd ones: these are reachable systematically from even brackets, whose more tractable properties have been detailed before., Comment: 12 pp, 1 fig, LateX2e/WS-procs9x6 macros. Contribution to the proceedings of QTS3, 10-14 Sep 2003, Cincinnati, World Scientific (SPIRES conf C03/09/10)

We use time-independent canonical transformation methods to discuss the energy eigenfunctions for the simple linear potential, pedagogically setting the stage for some field theory calculations to follow. We then discuss the Schr\"odinger wave-functional method of calculating correlation functions for Liouville field theory. We compare this approach to earlier treatments, in particular we check against known weak-coupling results for the Liouville field defined on a cylinder. Finally, we further set the stage for future Liouville calculations on curved two-manifolds and briefly discuss simple quantum mechanical systems with time-dependent Hamiltonians., Comment: 14 pages, latex, no figures

This is a synopsis and extension of Phys.~Rev.~{\em D49} 5408 (1994). The Pseudodual Chiral Model illustrates 2-dimensional field theories which possess an infinite number of conservation laws but also allow particle production, at variance with naive expectations---a folk theorem of integrable models. We monitor the symmetries of the pseudodual model, both local and nonlocal, as transmutations of the symmetries of the (very different) usual Chiral Model. We refine the conventional algorithm to more efficiently produce the nonlocal symmetries of the model. We further find the Canonical transformation which connects the usual chiral model to its fully equivalent dual model, thus contradistinguishing the pseudodual theory., Comment: 10 pages, Latex , ANL-HEP-CP-94-33

We discuss the pseudodual chiral model to illustrate a class of two-dimensional theories which have an infinite number of conservation laws but allow particle production, at variance with naive expectations. We describe the symmetries of the pseudodual model, both local and nonlocal, as transmutations of the symmetries of the usual chiral model. We refine the conventional algorithm to more efficiently produce the nonlocal symmetries of the model, and we discuss the complete local current algebra for the pseudodual theory. We also exhibit the canonical transformation which connects the usual chiral model to its fully equivalent dual, further distinguishing the pseudodual theory., Comment: 15 pages, ANL-HEP-PR-93-85,Miami-TH-1-93,Revtex (references updated, format improved to Revtex)

In the quantized two-dimensional non-linear supersymmetric $\sigma$-model, the supercurrent supermultiplet, which contains the energy-momentum tensor, is transformed by the nonlocal symmetry of the model into the isospin current supermultiplet. This effect incorporates supersymmetry into the known infinite-dimensional Yangian deformation symmetry of plain $\sigma$-models, leads to precisely the same nontrivial extension of the two-dimensional super-Poincar\'e group as found previously for the Poincar\'e group, and thus determines the theory's mass spectrum. A generalization to all higher-order nonlocal charges is conjectured such that their generating function, the so-called ``master charge'', has a definite Lorentz spin which depends on the spectral parameter., Comment: 6 pages, Latex, (completion of refs and reformatting)