The splitting of the massless (\(1+1\))-dimensional Dirac field into right- and left-handed components. The quantization of the right- and left-handed fields. Construction of the Hamiltonian and of the U(1) current operator. The non-conservation of the U(1) current in the presence of an external Abelian gauge field. Derivation of the U(1) anomaly equation. Cancelation of anomalies. Non-invariance of the fermionic path integral measure under the axial U(1) transformations. Derivation of the U(1) anomaly equation in the path integral formulation of the quantum theory of the massless Dirac Dirac field in four-dimensional Euclidean space. The index of the Dirac operator.