Local Currents for a Deformed Algebra of Quantum Mechanics with a Fundamental Length Scale

We explore some explicit representations of a certain stable deformed algebra of quantum mechanics, considered by R. Vilela Mendes, having a fundamental length scale. The relation of the irreducible representations of the deformed algebra to those of the (limiting) Heisenberg algebra is discussed, and we construct the generalized harmonic oscillator Hamiltonian in this framework. To obtain local currents for this algebra, we extend the usual nonrelativistic local current algebra of vector fields and the corresponding group of diffeomorphisms, modeling the quantum configuration space as a commutative spatial manifold with one additional dimension.Comment: 10 pages REVTex, no figure

On the Fock space for nonrelativistic anyon fields and braided tensor products

We realize the physical N-anyon Hilbert spaces, introduced previously via unitary representations of the group of diffeomorphisms of the plane, as N-fold braided-symmetric tensor products of the 1-particle Hilbert space. This perspective provides a convenient Fock space construction for nonrelativistic anyon quantum fields along the more usual lines of boson and fermion fields, but in a braided category. We see how essential physical information is thus encoded. In particular we show how the algebraic structure of our anyonic Fock space leads to a natural anyonic exclusion principle related to intermediate occupation number statistics, and obtain the partition function for an idealised gas of fixed anyonic vortices.Comment: Added some references, more explicit formulae for the discrete case and remark on partition function. 25 pages latex, no figure

Conformal inversion and Maxwell field invariants in four- and six-dimensional spacetimes

Conformally compactified (3+1)-dimensional Minkowski spacetime may be identified with the projective light cone in (4+2)-dimensional spacetime. In the latter spacetime the special conformal group acts via rotations and boosts, and conformal inversion acts via reflection in a single coordinate. Hexaspherical coordinates facilitate dimensional reduction of Maxwell electromagnetic field strength tensors to (3+1) from (4 + 2) dimensions. Here we focus on the operation of conformal inversion in different coordinatizations, and write some useful equations. We then write a conformal invariant and a pseudo-invariant in terms of field strengths; the pseudo-invariant in (4+2) dimensions takes a new form. Our results advance the study of general nonlinear conformal-invariant electrodynamics based on nonlinear constitutive equations.Comment: 10 pages, birkjour.cls, submitted for the Proceedings of the XXXIInd Workshop on Geometric Methods in Physics, (Bialowieza, Poland, July 2013), v2: minor improvement

The Prediction of Anyons: Its History and Wider Implications

Prediction of ``anyons'', often attributed exclusively to Wilczek, came first from Leinaas & Myrheim in 1977, and independently from Goldin, Menikoff, & Sharp in 1980-81. In 2020, experimentalists successfully created anyonic excitations. This paper discusses why the possibility of quantum particles in two-dimensional space with intermediate exchange statistics eluded physicists for so long after bosons and fermions were understood. The history suggests ideas for the preparation of future researchers. I conclude by addressing failures to attribute scientific achievements accurately. Such practices disproportionately hurt women and minorities in physics, and are harmful to science.Comment: Based on a presentation in the 50th Anniversary Special Session of the 34th International Colloquium on Group Theoretical Methods in Physics, Strasbourg, France, July 202

On gauge transformations of B\"acklund type and higher order nonlinear Schr\"odinger equations

We introduce a new, more general type of nonlinear gauge transformation in nonrelativistic quantum mechanics that involves derivatives of the wave function and belongs to the class of B\"acklund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schr\"odinger equation results in higher order partial differential equations. As an example, we derive a general family of 6th-order nonlinear Schr\"odinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current ${\bf \sigma}=\rho {\bf \nabla}\triangle \ln \rho$, where $\rho=\bar\psi \psi$. We derive gauge invariant quantities, and characterize the subclass of the 6th-order equations that is gauge equivalent to the free Schr\"odinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz

On Galilean invariance and nonlinearity in electrodynamics and quantum mechanics

Recent experimental results on slow light heighten interest in nonlinear Maxwell theories. We obtain Galilei covariant equations for electromagnetism by allowing special nonlinearities in the constitutive equations only, keeping Maxwell's equations unchanged. Combining these with linear or nonlinear Schroedinger equations, e.g. as proposed by Doebner and Goldin, yields a Galilean quantum electrodynamics.Comment: 12 pages, added e-mail addresses of the authors, and corrected a misprint in formula (2.10

Generalizations of Nonlinear and Supersymmetric Classical Electrodynamics

We first write down a very general description of nonlinear classical electrodynamics, making use of generalized constitutive equations and constitutive tensors. Our approach includes non-Lagrangian as well as Lagrangian theories, allows for electromagnetic fields in the widest possible variety of media (anisotropic, piroelectric, chiral and ferromagnetic), and accommodates the incorporation of nonlocal effects. We formulate electric-magnetic duality in terms of the constitutive tensors. We then propose a supersymmetric version of the general constitutive equations, in a superfield approach.Comment: 15 pages, based on the presentation by G. A. Goldin at QTS

Attitudes, Beliefs, Motivation and Identity in Mathematics Education: An Overview of the Field and Future Directions

Mathematics Education; Educational Psycholog