We argue that duality symmetries can be manifestly realized when theories with these symmetries are quantized using phase space quantum theory. In particular, using background fields and phase space quantum theory, we quantize the bosonic string and show that it has SO(26,26) symmetry, even when the string is not compactified on a torus.

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 discuss the regions forbidden to classical scattering trajectories by repulsive potentials. We give explicit results for the asymptotic form of these regions, far from the scattering center, in terms of the scattering angle function.

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.

Scattering from a scale invariant potential in two spatial dimensions leads to a class of novel identities involving the sinc function.

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.

Non-relativistic quantum mechanical scattering from an inverse square potential in two spatial dimensions leads to a novel representation of the Bernoulli numbers.

We revisit an old subject to discuss relationships between the dynamics for particles subjected to potentials and the dynamics for particles moving freely on background geometries, in the context of non-relativistic quantum mechanics. In particular, we illustrate how selected geometries can be used to regularize singular potentials. We also compute scattering amplitudes for quanta incident on a static non-relativistic wormhole.

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.

In the holographic picture, the Brout-Englert-Higgs (BEH) mechanism in [Formula: see text]-dimensional Yang–Mills theories is conjectured to provide a Higgs-like mechanism for gravity in [Formula: see text] dimensions, resulting in massive (or massless) gravitons in IR (or UV) completions. Accordingly, one could imagine dual (magnetic-type) fields of massive gravitons in the IR (low-energy) limit that are coupled to the curl of their own energy–momentum, as well as to the rotation of matter fields on large scales. This hypothesis, which might solve cosmological issues currently ascribed to dark matter and dark energy, needs to be examined by the future Laser Interferometer Space Antenna mission using observations of gravitational waves emitted from extragalactic sources.

We fantasize about an alternative history for theories of electromagnetism and gravitation.

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. The presentation is at a level suitable for use in graduate courses on electrostatics, as a supplement to more traditional material.

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.

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.

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.

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. In tribute to Richard Feynman (1918-1988) and Arnold Sommerfeld (1868-1951)

Massive dual spin zero fields are reconsidered in four spacetime dimensions. A closed-form Lagrangian is presented that describes a field coupled to the curl of its own energy-momentum tensor.

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 that encourage students to think more broadly about the subject. We present our discussion at a level suitable for use in advanced undergraduate and introductory graduate courses on electrostatics, or for use as a supplement to introduce the methods of Riemannian geometry in the context of general relativity.

We discuss a set of local fields which provide the on-shell massless states expected for the fundamental linear representation of supersymmetry in twelve dimensions. We recover the states of the anticipated N=16, S=4, massless supermultiplet upon reduction to four dimensions. Using algebraic methods, we also construct ghost arrays which permit the covariant quantization of each of the gauge fields in the set.