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Journal of Physics Conference Series

Diffeomorphism groups and nonlinear quantum mechanics

Gerald A. Goldin

Published Feb 8, 2012• 3 citations
Publisher

Abstract

This talk is dedicated to my friend and collaborator, Prof. Dr. Heinz-Dietrich Doebner, on the occasion of his 80th birthday. I shall review some highlights of the approach we have taken in deriving and interpreting an interesting class of nonlinear time-evolution equations for quantum-mechanical wave functions, with few equations; more detail may be found in the references. Then I shall comment on the corresponding hydrodynamical description.

DOI
10.1088/1742-6596/343/1/012006

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This talk is dedicated to my friend and collaborator, Prof. Dr. Heinz-Dietrich Doebner, on the occasion of his 80th birthday. I shall review some highlights of the approach we have taken in deriving and interpreting an interesting class of nonlinear time-evolution equations for quantum-mechanical wave functions, with few equations; more detail may be found in the references. Then I shall comment on the corresponding hydrodynamical description.
About 15 years ago, we (Heinz-Dietrich Doebner and I) proposed a special type of nonlinear modification of the usual Schrödinger time-evolution equation in quantum mechanics. Our equation was motivated by certain unitary representations of the group of diffeomorphisms of physical space, in the framework of either nonrelativistic local current algebra or quantum Borel kinematics. Subsequently, we developed this and related approaches to nonlinearity in quantum mechanics considerably further, to incorporate theories of measurement, groups of nonlinear gauge transformations, symmetry and invariance properties, unification of a large family of nonlinear perturbations, and possible physical contexts for quantum nonlinearity. Some of our results and highlights of some open questions are summarized.
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 Schrödinger equation different from the sort usually studied. Our equation follows from including a diffusion current in the equation of continuity.
An enlarged group G of nonlinear transformations, modeled on the general linear group GL(2,R), leads to a beautiful, apparently unremarked symmetry between the wave function's phase and the logarithm of its amplitude. Equations Doebner and I earlier proposed are embedded in a wider, natural family of nonlinear time-evolution equations, on which G acts as a gauge group (leaving physical observations invariant). There exist G-invariant quantities that reduce to the usual probability density and flux for linearizable quantum theories in a particular gauge.
In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.
The authors recently derived a family of nonlinear Schrodinger equations on R 3 from fundamental considerations of generalized symmetry: ih(cross) delta t psi =-(h(cross) 2 /2m) Del 2 psi +F( psi , psi ) psi +ih(cross)D( Del 2 psi +( mod Del psi mod 2 / mod psi mod 2 ) psi ), where F is an arbitrary real functional and D a real, continuous quantum number. These equations, descriptive of a quantum mechanical current that includes a diffusive term, correspond to unitary representations of the group Diff(M) parametrized by D, where M=R 3 is the physical space. In the present paper we explore the most natural ansatz for F, which is labelled by five real coefficients. We discuss the invariance properties, describe the stationary states and some non-stationary solutions, and determine the extra, dissipative terms that occur in the Ehrenfest theorem. We identify an interesting, Galilean-invariant subfamily whose properties we investigate, including the case where the dissipative terms vanish.