7 - Trace anomalies from ordinary and susy quantum mechanics

Summary

We now turn to a second class of anomalies, namely the trace anomalies. These are anomalies in the local scale invariance of actions for scalar fields, spin-½ fields and certain vector and antisymmetric tensor fields (vectors in n = 4, antisymmetric tensors with two indices in n = 6, etc.). One needs gauge-fixing terms and ghosts, but the trace anomalies of these vector and antisymmetric tensor fields are independent of the gauge chosen. From a technical point of view, these anomalies are very interesting, because one needs higher loop graphs on the worldline to compute them. In fact, due to the β dependence of the measure of the quantum mechanical path integrals, A = (2πħβ)^−n/2, one needs (½n + 1)-loop calculations in quantum mechanics for the one-loop trace anomalies of n-dimensional quantum field theories. Already in two dimensions one needs two-loop graphs and in four dimensions three-loop graphs. Another interesting technical point regards the fermions. In the path integral they now have antiperiodic boundary conditions. Originally we devised a path integral approach in which fermions were still treated by an operator formalism and in which actions were operator-valued. Here we shall instead present a complete path integral approach, with ordinary actions, in which the fermions are described in the path integral by Grassmann fields. Contrary to the case of chiral anomalies, we shall not use a background field formalism for the fermions because the background fermions are constant and thus cannot accomodate anti-periodic boundary conditions. Instead we shall directly use fermionic quantum fields with antiperiodic boundary conditions. The results we find agree with the results in the literature for trace anomalies obtained by different methods (see, for example).