Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0201004 · Journal of Mathematical Physics

On gauge transformations of Bäcklund type and higher order nonlinear Schrödinger equations

Gerald A. Goldin, Vladimir M. Shtelen

Published May 1, 2002• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.1465514

Related papers

On gauge transformations of B\\"acklund type and higher order nonlinear Schr\\"odinger equations

Goldin, G A, Shtelen, V M

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 B\\"acklund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schr\\"odinger equation results in higher order partial differential equations. As an example, we derive a general family of 6th-order nonlinear Schr\\"odinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current ${\\bf \\sigma}=\\rho {\\bf \\nabla}\\triangle \\ln \\rho $, where $\\rho=\\bar\\psi \\psi$. We derive gauge invariant quantities, and characterize the subclass of the 6th-order equations that is gauge equivalent to the free Schr\\"odinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz.

•Jan 2, 2002•4 citations

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

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

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.

arXiv:quant-ph/9709036•Journal of Mathematical Physics•Jan 1, 1999•61 citations

Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations

H. D. Doebner, Gerald A. Goldin

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.

Physical Review A•Nov 1, 1996•134 citations

Nonlinear Dirac equations and nonlinear gauge transformations

H. D. Doebner, Renat Zhdanov

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.

arXiv:quant-ph/0304167•Apr 25, 2003•6 citations

Nonlinear Gauge Transformations and Exact Solutions of the Doebner-Goldin Equation

P. Nattermann, Wolfgang Scherer

Invariants of nonlinear gauge transformations of a family of nonlinear Schrödinger equations proposed by Doebner and Goldin are used to characterize the behaviour of exact solutions of these equations.

arXiv:quant-ph/9506033•Jun 21, 1995•8 citations

Generalized Quantum Mechanics and Nonlinear Gauge Transformations

P. Nattermann

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.

arXiv:quant-ph/9703017•Mar 12, 1997•0 citations
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.
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 B\\"acklund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schr\\"odinger equation results in higher order partial differential equations. As an example, we derive a general family of 6th-order nonlinear Schr\\"odinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current ${\\bf \\sigma}=\\rho {\\bf \\nabla}\\triangle \\ln \\rho $, where rho=barpsipsi\\rho=\\bar\\psi \\psirho=barpsipsi. We derive gauge invariant quantities, and characterize the subclass of the 6th-order equations that is gauge equivalent to the free Schr\\"odinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz.
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.
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.
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.