Anomaly for noncommutative differentials and the origin of time
Seminar Room 1, Newton Institute
We take a closer look at a phenomenon whereby many noncommutative coordinate algebras of interest do not in fact admit associative differential calculi that deform classical differentials while preserving expected symmetries. In physical terms there is an anomaly and we show that it is expressed by the curvature of a certain Poisson-compatible preconnection. We show that the anomaly is present for all standard quantum groups C_q(G) and for all enveloping algebras U(g) of a semisimple Lie algebra g (viewed as a quantisation of g^*). Due to this anomaly, if one wishes to preserve classical dimensions one must have non-associative exterior algebras and we construct these. Alternatively, one must neutralise the anomaly by adding `extra dimensions' in the cotangent bundle. We show how this works for C_q(SU_2) and U(su_2) where the extra dimension can be viewed as a spontaneously generated time with respect to which the original fields naturally obey Schroedinger's equation. We argue that this is a fairly general phenomenon whereby any sufficiently noncommutative differential geometry induces its own `time evolution'. We also argue that these induced extra dimensions in another context are intimately bound up with the renormalisation group.