5 - Introduction to anomalies

Summary

We now start the second part of this book, namely the computation of anomalies in higher-dimensional quantum field theories using quantum mechanical (QM) path integrals. Anomalies arise when the symmetries of a classical system cannot all be preserved by the quantization procedure. Those symmetries which turn out to be violated by the quantum corrections are called anomalous. The anomalous behavior is encoded in the quantum effective action which fails to be invariant: its nonvanishing variation is called the anomaly. As we shall see, the ordinary Dirac action for a chiral fermion in n dimensions has anomalies which can be computed by using an N = 1 supersymmetric (susy) nonlinear sigma model in one (timelike) dimension. Although this relation between a nonsusy quantum field theory (QFT) and a susy QM system may seem surprising at first sight, it becomes plausible if one notices that the Dirac operator γ^µD_µ contains hermitian Dirac matrices γ^m (where γ^µ = γ^me_m^µ, with e_m^µ being the inverse vielbein field) satisfying the same Clifford algebra {γ^l, γ^m} = 2δ^lm (with l, m = 1, …, n flat indices) as the equal-time anti-commutation rules of a real (Majorana) fermionic quantum mechanical point particle ψ^a(t) with a = 1, …, n, namely {ψ^a(t), ψ^b(t)} = ħδ^ab.