Deformed Gauge Theories Research Articles

General Relativity and Deformed Gauge Theories

In this paper we report on recent results concerning gauge theories relative to deformed (or quantum) Lie groups. After giving a brief review of basic concepts, we construct consistent gauge theories relative to a one–parameter family of deformations of the Poincare group in 3 + 1 dimensions with a generalized Palatini action invariant under the local Lorentz (undeformed) subgroup. For any value of the deformation parameter the metric sector turns out to be ordinary Einstein gravity. While gauge fields are not c–numbers, metric tensor, Christoffel symbols and Riemann tensor all commute among themselves.

Comments on the non-commutative description of classical gravity

We find a one-parameter family of Lagrangian descriptions for classical general relativity in terms of tetrads which are not c-numbers. Rather, they obey exotic commutation relations. These noncommutative properties drop out in the metric sector of the theory, where the Christoffel symbols and the Riemann tensor are ordinary commuting objects and they are given by the usual expression in terms of the metric tensor. Although the metric tensor is not a c-number, we argue that all measurements one can make in this theory are associated with c-numbers, and thus that the common invariant sector of our one-parameter family of deformed gauge theories (for the case of zero torsion) is physically equivalent to Einstein's general relativity.

Deformed Chern-Simons theories

We construct a Chern-Simons action for q -deformed gauge theory which is a simple and straightforward generalization of the usual one. Space-time continues to be an ordinary (commuting) manifold, while the gauge potentials and the field strengths become q -commuting fields. Our approach, which is explicitly carried out for the case of ‘minimal’ deformations, has the advantage of leading naturally to a consistent Hamiltonian structure that has essentially all of the features of the undeformed case. For example, using the new Poisson brackets, the constraints form a closed algebra and generate q -deformed gauge transformations.