Differential Calculus and Gauge Transformations on a Deformed Space

Abstract

Deformed gauge transformations on deformed coordinate spaces are considered for any Lie algebra. The representation theory of this gauge group forces us to work in a deformed Lie algebra as well. This deformation rests on a twisted Hopf algebra, thus we can represent a twisted Hopf algebra on deformed spaces. That leads to the construction of Lagrangian invariant under a twisted Lie algebra.This article is based on common work with

Symmetries in noncommutative field theories: Hopf versus Lie

“…In other words, the twist operator is not covariant. According to a more moderate point of view [43] it is enough to have proper commutation relation between the symmetry generators and a coassociative coproduct, so that one has a Hopf algebra based on a proper Lie algebra. Gauge covariance of twist operator is not required.…”

Deformed Gauge Theories

“…In general this will not be the case. Star differentiation and star differential operators have been thoroughly discussed [4], [5].…”

Towards Noncommutative Quantum Reality

“…4 Further Discussions and Conclusions Wess (2007) had remarked that "That a change in the concept of space for very short distances might be necessary was already anticipated in 1854 by Riemann in his famous inaugural lecture." as he also quoted from Riemann (1854) "Now it seems that the empirical notions on which the metric determinations of space are founded, the notion of a solid body and a light ray, cease to be valid for the infinitely small.…”

“…One cannot even be sure that such questions are sensible ones. Physicists have however been working on noncommutative spacetime models essentially based on the idea of the noncommutative position, and time, coordinate variables (Doplicher, Fredenhagen, & Roberts, 1995;Wess & Zumino, 1990;Wess, 2007), commonly believed to be necessary for Planck scale physics. To keep a good control on the otherwise speculative nature of such physical theories, the nature of the physics for the noncommutative coordinate variables of quantum mechanics may be a very useful guideline, especially if that also gives a noncommutative model of the physical space.…”