Article,
Hopf-algebra description of noncommutative-spacetime symmetries
(2003)cite arxiv:hep-th/0306013 Comment: 20 pages, no figures, LaTex. This version has exactly the same technical content as version 1, but the observation reported in Section VII is discussed more pedagogically.
Abstract
In the study of certain noncommutative versions of Minkowski spacetime there
is still a large ambiguity concerning the characterization of their symmetries.
Adopting as our case study the kappaMinkowski noncommutative space-time, on
which a large literature is already available, we propose a line of analysis of
noncommutative-spacetime symmetries that relies on the introduction of a Weyl
map (connecting a given function in the noncommutative Minkowski with a
corresponding function in commutative Minkowski) and of a compatible notion of
integration in the noncommutative spacetime. We confirm (and we establish more
robustly) previous suggestions that the commutative-spacetime notion of
Lie-algebra symmetries must be replaced, in the noncommutative-spacetime
context, by the one of Hopf-algebra symmetries. We prove that in kappaMinkowski
it is possible to construct an action which is invariant under a Poincare-like
Hopf algebra of symmetries with 10 generators, in which the noncommutativity
length scale has the role of relativistic invariant. The approach here adopted
does leave one residual ambiguity, which pertains to the description of the
translation generators, but our results, independently of this ambiguity, are
sufficient to clarify that some recent studies (gr-qc/0212128 and
hep-th/0301061), which argued for an operational indistiguishability between
theories with and without a length-scale relativistic invariant, implicitly
assumed that the underlying spacetime would be classical.
