11 research outputs found
On gauge transformations of B\"acklund type and higher order nonlinear Schr\"odinger equations
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 , where . 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
On Galilean invariance and nonlinearity in electrodynamics and quantum mechanics
Recent experimental results on slow light heighten interest in nonlinear
Maxwell theories. We obtain Galilei covariant equations for electromagnetism by
allowing special nonlinearities in the constitutive equations only, keeping
Maxwell's equations unchanged. Combining these with linear or nonlinear
Schroedinger equations, e.g. as proposed by Doebner and Goldin, yields a
Galilean quantum electrodynamics.Comment: 12 pages, added e-mail addresses of the authors, and corrected a
misprint in formula (2.10
Generalizations of Yang-Mills Theory with Nonlinear Constitutive Equations
We generalize classical Yang-Mills theory by extending nonlinear constitutive
equations for Maxwell fields to non-Abelian gauge groups. Such theories may or
may not be Lagrangian. We obtain conditions on the constitutive equations
specifying the Lagrangian case, of which recently-discussed non-Abelian
Born-Infeld theories are particular examples. Some models in our class possess
nontrivial Galilean (c goes to infinity) limits; we determine when such limits
exist, and obtain them explicitly.Comment: Submitted to the Proceedings of the 3rd Symposium on Quantum Theory
and Symmetries (QTS3) 10-14 September 2003. Preprint 9 pages including
reference
Generalizations of Nonlinear and Supersymmetric Classical Electrodynamics
We first write down a very general description of nonlinear classical
electrodynamics, making use of generalized constitutive equations and
constitutive tensors. Our approach includes non-Lagrangian as well as
Lagrangian theories, allows for electromagnetic fields in the widest possible
variety of media (anisotropic, piroelectric, chiral and ferromagnetic), and
accommodates the incorporation of nonlocal effects. We formulate
electric-magnetic duality in terms of the constitutive tensors. We then propose
a supersymmetric version of the general constitutive equations, in a superfield
approach.Comment: 15 pages, based on the presentation by G. A. Goldin at QTS