Covariant quantum fields on noncommutative spacetimes

A spinless quantum covariant field. on Minkowski spacetime Md+1 obeys the relation U(a, Lambda)phi(x)U(a, Lambda)(-1) = phi(Lambda x + a) where (a, Lambda) is an element of the Poincare group P and U : (a, Lambda) -> U(a, Lambda) is an unitary representation on quantum vector states. It expresses the fact that Poincare transformations are being unitarily implemented. It has a classical analogy where field covariance shows that Poincare transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no *-covariant Voros fields, a result we found earlier. The notion of Drinfel'd twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative.