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Quantum anomalies: A few physics applications

Chapter 17 exhibits various examples where classical symmetries cannot be transferred to quantum theories. The obstructions are characterized by anomalies. The examples involve chiral symmetries combined with currents or gauge symmetries, leading to chiral anomalies. In particular, anomalies lead to obstruction in the construction of theories. In particular, the structure of the Standard Model of particle physics is constrained by the requirement of anomaly cancellation. Other applications, like the relation between electromagnetic pi0 decay and the axial anomaly, are described. Anomalies are related to the Dirac operator index, leading to relations between anomaly and topology. To prove anomaly cancellation beyond perturbation theory, one can use lattice regularization. However, the definition of lattice chiral transformations is non–trivial. It is based on solutions of the Ginsparg–Wilson relation.