Anomalies, boundaries and the in–in formalism

In the context of quantum field theory, an anomaly exists when a theory has a classical symmetry which is not a symmetry of the quantum theory. This short exposition aims at introducing a new point of view, which is that the proper setting for anomaly calculations is the 'in–in', or closed-time path formulation of quantum field theory. There are also some new results for anomalies in the context of boundary value problems, and a new correction to the a5 heat-kernel coefficient. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. (paper)