1,272 research outputs found
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 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 is not sufficient for a well defined NGT. NGT that are well
defined in terms of form a semigroup parametrized by a real number
and a nonzero which is either an integer or . An extension of NGT to projectors and general density matrices
leads to NGT with complex . 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 ). 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 is fulfilled. It is shown that a correct
extension to 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 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
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