Anomalies in gauge theories

We discuss the scale and chiral anomalies in quantum chromodynamics and their implications for the theory of hadrons. In the first part the physical meaning of the anomaly is demonstrated. To this end the simplest gauge model is considered - the two-dimensional Schwinger model. In this model it is extremely easy to explain which properties of the theory are responsible for the quantum anomaly, and the physical nature of the phenomenon is elucidated. The second part is devoted to the derivation of the anomaly relations within QCD. The subtle question of the multiloop corrections is discussed. The third part presents applications. Starting from the chiral and scale anomalies we obtain low-energy theorems which lead, in turn, to predictions for observable processes. In particular, a proof based on first principles is given of the existence of massless pions, the amplitudes are calculated for the conversion of gluon operators into pions, which can be measured experimentally. Other practical problems are also considered, the solution of which turns out to be possible due to anomalies.