Symmetry, Local Linearization, and Gauge Classification of the Doebner-Goldin Equation

For the family of nonlinear Schr\"odinger equations derived by H.-D.~Doebner and G.A.~Goldin (J.Phys.A 27, 1771) we calculate the complete set of Lie symmetries. For various subfamilies we find different finite and infinite dimensional Lie symmetry algebras. Two of the latter lead to a local transformation linearizing the particular subfamily. One type of these transformations leaves the whole family of equations invariant, giving rise to a gauge classification of the family. The Lie symmetry algebras and their corresponding subalgebras are finally characterized by gauge invariant parameters.Comment: 17 pages, LaTeX, 1 figure, to appear in Reports on Mathematical Physic

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

Invariants of nonlinear gauge transformations of a family of nonlinear Schr\"odinger equations proposed by Doebner and Goldin are used to characterize the behaviour of exact solutions of these equations.Comment: 12 pages, LaTeX, to appear in "Nonlinear, Deformed and Irreversible Quantum Systems", Proceedings of an International Symposium on Mathematical Physics, World Scientific, Singapore 199

Structure of nonlinear gauge transformations

Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT) defined in terms of a wave function $\psi(x)$ 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 $\psi(x)$ is not sufficient for a well defined NGT. NGT that are well defined in terms of $\psi(x)$ form a semigroup parametrized by a real number $\gamma$ and a nonzero $\lambda$ which is either an integer or $-1\leq \lambda\leq 1$. An extension of NGT to projectors and general density matrices leads to NGT with complex $\gamma$. Both linearity of evolution and Hermiticity of density matrices are gauge dependent properties.Comment: Final version, to be published in Phys.Rev.A (Rapid Communication), April 199

Nonlinear Quantum Mechanics and Locality

It is shown that, in order to avoid unacceptable nonlocal effects, the free parameters of the general Doebner-Goldin equation have to be chosen such that this nonlinear Schr\"odinger equation becomes Galilean covariant.Comment: 10 pages, no figures, also available on http://www.pt.tu-clausthal.de/preprints/asi-tpa/012-97.htm

Comprehensive study of the critical behavior in the diluted antiferromagnet in a field

We study the critical behavior of the Diluted Antiferromagnet in a Field with the Tethered Monte Carlo formalism. We compute the critical exponents (including the elusive hyperscaling violations exponent $\theta$). Our results provide a comprehensive description of the phase transition and clarify the inconsistencies between previous experimental and theoretical work. To do so, our method addresses the usual problems of numerical work (large tunneling barriers and self-averaging violations).Comment: 4 pages, 2 figure

Roughening Transition of Interfaces in Disordered Systems

The behavior of interfaces in the presence of both lattice pinning and random field (RF) or random bond (RB) disorder is studied using scaling arguments and functional renormalization techniques. For the first time we show that there is a continuous disorder driven roughening transition from a flat to a rough state for internal interface dimensions 2<D<4. The critical exponents are calculated in an \epsilon-expansion. At the transition the interface shows a superuniversal logarithmic roughness for both RF and RB systems. A transition does not exist at the upper critical dimension D_c=4. The transition is expected to be observable in systems with dipolar interactions by tuning the temperature.Comment: 4 pages, RevTeX, 1 postscript figur

Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schr\"odinger Equations

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 Schr\"odinger 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 Schr\"odinger 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.Comment: RevTeX, 20 pages, no figures. also available on http://www.pt.tu-clausthal.de/preprints/asi-tpa/021-96.htm

Nonlocal looking equations can make nonlinear quantum dynamics local

A general method for extending a non-dissipative nonlinear Schr\"odinger and Liouville-von Neumann 1-particle dynamics to an arbitrary number of particles is described. It is shown at a general level that the dynamics so obtained is completely separable, which is the strongest condition one can impose on dynamics of composite systems. It requires that for all initial states (entangled or not) a subsystem not only cannot be influenced by any action undertaken by an observer in a separated system (strong separability), but additionally that the self-consistency condition $Tr_2\circ \phi^t_{1+2}=\phi^t_{1}\circ Tr_2$ is fulfilled. It is shown that a correct extension to $N$ particles involves integro-differential equations which, in spite of their nonlocal appearance, make the theory fully local. As a consequence a much larger class of nonlinearities satisfying the complete separability condition is allowed than has been assumed so far. In particular all nonlinearities of the form $F(|\psi(x)|)$ are acceptable. This shows that the locality condition does not single out logarithmic or 1-homeogeneous nonlinearities.Comment: revtex, final version, accepted in Phys.Rev.A (June 1998

On Integrable Doebner-Goldin Equations

We suggest a method for integrating sub-families of a family of nonlinear {\sc Schr\"odinger} equations proposed by {\sc H.-D.~Doebner} and {\sc G.A.~Goldin} in the 1+1 dimensional case which have exceptional {\sc Lie} symmetries. Since the method of integration involves non-local transformations of dependent and independent variables, general solutions obtained include implicitly determined functions. By properly specifying one of the arbitrary functions contained in these solutions, we obtain broad classes of explicit square integrable solutions. The physical significance and some analytical properties of the solutions obtained are briefly discussed.Comment: 23 pages, revtex, 1 figure, uses epsfig.sty and amssymb.st