Axial anomaly and Schwinger terms in two-dimensional general quantum field theory

We compute the vacuum expectation value ({Ω},j^μ(x)j^ν (y){Ω}){≡}J^μν(x-y) for a relativistic, Hermitian, not necessarily local vector field in two-dimensional space-time when ∂_μ{J}^{{μ}{ν}}$ =${{∂}}_{{μ}}J^νμ=∂_μ{{ε}}_{{α}}^{{μ}}J^αν =∂_μ{{ε}}_{{α}}^{{μ}}J^να=0 holds in S'(openR²), the space of tempered distributions. As the space-time symmetry group of the model we take the inhomogeneous proper orthochronous Lorentz group P₊^↑. ({Ω},Tj^μ(x)j^ν(y){Ω}) is verified to have anomalous (axial-)vector Ward identities due to nonvanishing equal-time current-current commutators. Additional conditions on j^μ are specified which imply that j^μ is a free zero-mass Wightman field with ∂_μ{j}^{{μ}}$=${{∂}}_{{μ}}ε_ν^μ{j}^{{ν}}$=0.