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arXiv:quant-ph/9711053 · Physical Review A

Structure of nonlinear gauge transformations

Marek Czachor

Published Apr 1, 1998• 3 citations
arXivHTMLPDF

Abstract

Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT's) defined in terms of a wave function $\ensuremath{\psi}(x)$ do not form a group. To get a group property one has to consider transformations that act differently on different branches of the complex argument function and the knowledge of the value of $\ensuremath{\psi}(x)$ is not sufficient for a well-defined NGT. NGT's that are well defined in terms of $\ensuremath{\psi}(x)$ form a semigroup parametrized by a real number \ensuremath{\gamma} and a nonzero \ensuremath{\lambda} that is either an integer or $\ensuremath{-}1<~\ensuremath{\lambda}<~1.$ An extension of NGT's to projectors and general density matrices leads to NGT's with complex \ensuremath{\gamma}. Both the linearity of evolution and Hermiticity of density matrices are gauge-dependent properties.

DOI
10.1103/physreva.57.r2263

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Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT's) defined in terms of a wave function \ensuremathψ(x)\ensuremath{\psi}(x)\ensuremathψ(x) do not form a group. To get a group property one has to consider transformations that act differently on different branches of the complex argument function and the knowledge of the value of \ensuremathψ(x)\ensuremath{\psi}(x)\ensuremathψ(x) is not sufficient for a well-defined NGT. NGT's that are well defined in terms of \ensuremathψ(x)\ensuremath{\psi}(x)\ensuremathψ(x) form a semigroup parametrized by a real number \ensuremath{\gamma} and a nonzero \ensuremath{\lambda} that is either an integer or \ensuremath−1< \ensuremathλ< 1.\ensuremath{-}1<~\ensuremath{\lambda}<~1.\ensuremath−1< \ensuremathλ< 1. An extension of NGT's to projectors and general density matrices leads to NGT's with complex \ensuremath{\gamma}. Both the linearity of evolution and Hermiticity of density matrices are gauge-dependent properties.
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.
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.
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.
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.