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

We compute the vacuum expectation value (Ω,${\mathit{j}}^{\mathrm{\mu }}$(x)${\mathit{j}}^{\nu }$ (y)Ω)≡$J^{\mu \nu }$(x-y) for a relativistic, Hermitian, not necessarily local vector field in two-dimensional space-time when ${\partial }_{\mu }$$J^{\mu \nu }$ =${\partial }_{\mu }$$J^{\nu \mu }$=${\partial }_{\mu }$${\epsilon }_{\alpha }^{\mu }$$J^{\alpha \nu }$ =${\partial }_{\mu }$${\epsilon }_{\alpha }^{\mu }$$J^{\nu \alpha }$=0 holds in S’(${\mathrm{openR}}^2$), the space of tempered distributions. As the space-time symmetry group of the model we take the inhomogeneous proper orthochronous Lorentz group $P_{+}^{\uparrow }$. (Ω,${\mathrm{Tj}}^{\mu }$(x)$j^{\nu }$(y)Ω) is verified to have anomalous (axial-)vector Ward identities due to nonvanishing equal-time current-current commutators. Additional conditions on $j^{\mu }$ are specified which imply that $j^{\mu }$ is a free zero-mass Wightman field with ${\partial }_{\mu }$$j^{\mu }$=${\partial }_{\mu }$${\epsilon }_{\nu }^{\mu }$$j^{\nu }$=0.