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Generalized Gauge Invariants for Certain Nonlinear Schrodinger Equations

Gerald A. Goldin

Published Jan 1, 2000• 1 citation
OpenAlex

Abstract

tions #N (x 1,. xN ,t), defined on the (pos=6Tfiflx# configurationsonfig in a way that is s trictly local ands atis#+ as eparation condition for products tates [6]. The jusT#6flx=Fj forcons6Fflx=F them to be gaugetrans##FWflx=Fj is as follows For all of the nonlinear quantumtheories under dis#+jflx=j we interpret # = = R 2as the probability densfi y in configurationsnfigu We adopt as a working hypothes6 the view (taken by many theorisx= that all meas++flx= ts in ordinary quantum mechanics can be regarded as asTFW6=fl of pos#666flx measx=++T ts made at di#erent times where externalfields exerting forces may be impos= on thesefl=fi between measjfi6flx ts [7, 8]. Then for any wave function # obeying aSchrodinger equation (linear or nonlinear) in our family, the wave function # #transflx==W by (1.1) obeys a trans=flx## Schrodinger equation,suat in the family, withgauge-transFTfls externalfields while theoutcomes of all phys#Ffi measF#Wflx ts remain invariant. To be explicit, with #

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tions #N (x 1,. xN ,t), defined on the (pos=6Tfiflx# configurationsonfig in a way that is s trictly local ands atis#+ as eparation condition for products tates [6]. The jusT#6flx=Fj forcons6Fflx=F them to be gaugetrans##FWflx=Fj is as follows For all of the nonlinear quantumtheories under dis#+jflx=j we interpret # = = R 2as the probability densfi y in configurationsnfigu We adopt as a working hypothes6 the view (taken by many theorisx= that all meas++flx= ts in ordinary quantum mechanics can be regarded as asTFW6=fl of pos#666flx measx=++T ts made at di#erent times where externalfields exerting forces may be impos= on thesefl=fi between measjfi6flx ts [7, 8]. Then for any wave function # obeying aSchrodinger equation (linear or nonlinear) in our family, the wave function # #transflx==W by (1.1) obeys a trans=flx## Schrodinger equation,suat in the family, withgauge-transFTfls externalfields while theoutcomes of all phys#Ffi measF#Wflx ts remain invariant. To be explicit, with #
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.
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.
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.
Nonlinear Dirac equations (NLDE) are derived through a group N^2 of nonlinear (gauge) transformation acting in the corresponding state space. The construction generalises a construction for nonlinear Schrödinger equations. To relate N^2 with physically motivated principles we assume: locality (i.e. it contains no explicit derivative and no derivatives of the wave function), separability (i.e. it acts on product states componentwise) and Poincaré invariance. Furthermore we want that a positional density is invariant under N^2. Such nonlinear transformations yield NLDE which describe physically equivalent systems. To get 'new' systems, we extend this NLDE (gauge extension) and present a family of NLDE which is a slight nonlinear generalisation of the Dirac equation. We discuss and comment the fact that nonlinear evolutions are not consistent with the usual framework of quantum theory. To develop a corresponding extended framework one needs models for nonlinear evolutions which also indicate possible physical consequences of nonlinearities.
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.