Gerald A. Goldin

Representational systems, learning, and problem solving in mathematics

Abstract This article explores aspects of a unified psychological model for mathematical learning and problem solving, based on several different types of representational systems and their stages of development. The goal is to arrive at a scientifically adequate theoretical framework, complex enough to account for diverse empirical results but sufficiently simple to be accessible and useful in mathematics education practice. Some perspectives on representational systems are discussed, and components of the model are described in relation to these ideas—including constructs related to imagistic thinking, heuristics and strategies, affect, and the fundamental role of ambiguity.

On a general nonlinear Schrödinger equation admitting diffusion currents

Abstract Some fundamental considerations of quantum theory suggest a general, complex nonlinear Schrodinger equation outside the classes most often studied. The equation follows from admitting quantum diffusion currents, so that the probability density satisfies a Fokker-Planck equation; the real diffusion coefficient D yields observable effects dependent on a dimensionless constant Γ=Dm/ℏ. The diffusion currents stem from unitary representations of an infinite-dimensional Lie algebra of vector fields and group of diffeomorphisms, earlier proposed to describe nonrelativistic quantum kinematics.

Children’s representation and structural development of the counting sequence 1–100

Abstract In an exploratory study, we interviewed 172 children from Grades K to 6, and an additional 92 high ability children from Grades 3 to 6, seeking to infer aspects of their internal imagistic representations from their drawings and explanations of the numbers 1–100. We interpret our observations with respect to developing theoretical models for mathematical learning and problem solving, based on characteristics of internal systems of representation. Focusing on children’s understandings of the conventional base ten system of numeration, we explore how internal representational systems for numbers may change through a period of structural development, to become eventually powerful, autonomous systems.

THE DIFFEOMORPHISM GROUP APPROACH TO NONLINEAR QUANTUM SYSTEMS

Unitary representations of diffeomorphism groups predict some unusual possibilities in quantum theory, including non-standard statistics and certain nonlinear effects. Many of the fundamental physical properties of “anyons” were first derived from their study by R. Menikoff, D.H. Sharp, and the author. This paper surveys new applications in two other domains: first (with Menikoff and Sharp) some surprising conclusions about the nature of quantum vortex configurations in ideal, incompressible fluids; second (with H.-D. Doebner) a natural description of dissipative quantum mechanics by means of a nonlinear Schrodinger equation different from the sort usually studied. Our equation follows from including a diffusion current in the equation of continuity.

On Galilean invariance and nonlinearity in electrodynamics and quantum mechanics

Abstract Recent experimental results such as those 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 Maxwells equations unchanged. Combining these with linear or nonlinear Schrodinger equations, e.g., as proposed by Doebner and Goldin, yields a consistent, nonlinear, Galilean Schrodinger–Maxwell electrodynamics.

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, and clarifies how discrete (lattice) anyon fields relate to anyon fields in the continuum. We also see how essential physical information is encoded. In particular, we show how the algebraic structure of the anyonic Fock space leads to a natural anyonic exclusion principle related to intermediate occupation number statistics, and obtain the partition function for an idealized gas of fixed anyonic vortices.

Nonlinear Schrödinger equations and the separation property

Hierarchies of nonlinear Schrodinger equations were investigated for multiparticle systems, satisfying the separation property, i.e., where product wave functions evolve by the separate evolution of each factor. Such a hierarchy defines a nonlinear derivation on tensor products of the single‐particle wave‐function space, and satisfies a certain homogeneity property characterized by two new universal physical constants. A canonical construction of hierarchies is derived that allows the introduction, at any particular ‘‘threshold’’ number of particles, of truly new physical effects absent in systems having fewer particles. In particular, if single quantum particles satisfy the usual (linear) Schrodinger equation, a system of two particles can evolve by means of a fairly simple nonlinear Schrodinger equation without violating the separation property. Examples of Galilean‐invariant hierarchies are given.

The PME Working Group on Representations

Abstract During the several years that the PME Working Group on Representations met, we discussed and refined various concepts related to representation and explored their applicability to our understanding of mathematical cognition. Some highlights of these discussions are described, and participants in the Working Group are acknowledged.

Generalizations of Yang-Mills theory with nonlinear constitutive equations

We generalize classical Yang–Mills theory by extending nonlinear constitutive equations for Maxwell fields to non-Abelian gauge groups. Such theories may or may not be Lagrangian. We obtain conditions on the constitutive equations specifying the Lagrangian case, of which recently discussed non-Abelian Born–Infeld theories are particular examples. Some models in our class possess nontrivial Galilean (c → ∞) limits; we determine when such limits exist and obtain them explicitly.

THE DIFFEOMORPHISM GROUP APPROACH TO ANYONS

We explain an approach to quantum mechanics, based on diffeomorphism groups and local current algebras, that predicts many of the fundamental properties of anyons. Our formulation also yields further insight into the meaning of particle statistics, and permits the unified description of a wide variety of quantum systems.