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arXiv:quant-ph/9709036 · Journal of Mathematical Physics

Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equations

H. D. Doebner, Gerald A. Goldin, P. Nattermann

Published Jan 1, 1999• 61 citations
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Abstract

Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schrödinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time evolutions in quantum mechanics.

DOI
10.1063/1.532786

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Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schrödinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time evolutions in quantum mechanics.
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.
Motivated by the problems of interpretation of a nonlinear evolution equation in quantum mechanics we discuss in this contribution the concept of nonlinear gauge transformations, that has recently been introduced in joint work with Doebner and Goldin, in the framework of Mielnik's Generalized Quantum Mechanics. Using these gauge transformations we construct linear quantum systems in a ``nonlinear disguise'', and a gauge generalization of these (in analogy to the minimal coupling of orthodox quantum mechanics) leads to a unification of Bialynicki-Birula--Mycielski and Doebner--Goldin evolution equations for the quantum system. The notion of nonlinear observables introduced by Lücke is finally discussed in the same framework.
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.
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.