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Journal of Physics A Mathematical and General

Properties of nonlinear Schrodinger equations associated with diffeomorphism group representations

H. D. Doebner, Gerald A. Goldin

Published Mar 7, 1994• 104 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/27/5/036

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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.
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.
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.
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.
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 Bäcklund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schrödinger equation results in higher order partial differential equations. As an example, we derive a general family of sixth-order nonlinear Schrödinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current σ=ρ∇Δ ln ρ, where ρ=ψ̄ψ. We derive gauge invariant quantities, and characterize the subclass of the sixth-order equations that is gauge equivalent to the free Schrödinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz.
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group ϕt\phi_tϕt​ of transformations on a Hilbert space, H\mathcal{H}H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schr\"{o}dinger equation to a submanifold invariant under the flow ϕt\phi_tϕt​. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow ϕt\phi_tϕt​. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.