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

Abstract

We compute the vacuum expectation value (\ensuremath{\Omega},${\mathit{j}}^{\mathrm{\ensuremath{\mu}}}$(x)${\mathit{j}}^{\ensuremath{\nu}}$ (y)\ensuremath{\Omega})\ensuremath{\equiv}${J}^{\ensuremath{\mu}\ensuremath{\nu}}$(x-y) for a relativistic, Hermitian, not necessarily local vector field in two-dimensional space-time when ${\ensuremath{\partial}}_{\ensuremath{\mu}}$${J}^{\ensuremath{\mu}\ensuremath{\nu}}$ =${\ensuremath{\partial}}_{\ensuremath{\mu}}$${J}^{\ensuremath{\nu}\ensuremath{\mu}}$=${\ensuremath{\partial}}_{\ensuremath{\mu}}$${\ensuremath{\epsilon}}_{\ensuremath{\alpha}}^{\ensuremath{\mu}}$${J}^{\ensuremath{\alpha}\ensuremath{\nu}}$ =${\ensuremath{\partial}}_{\ensuremath{\mu}}$${\ensuremath{\epsilon}}_{\ensuremath{\alpha}}^{\ensuremath{\mu}}$${J}^{\ensuremath{\nu}\ensuremath{\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 ${P}_{+}^{\ensuremath{\uparrow}}$. (\ensuremath{\Omega},${\mathrm{Tj}}^{\ensuremath{\mu}}$(x)${j}^{\ensuremath{\nu}}$(y)\ensuremath{\Omega}) is verified to have anomalous (axial-)vector Ward identities due to nonvanishing equal-time current-current commutators. Additional conditions on ${j}^{\ensuremath{\mu}}$ are specified which imply that ${j}^{\ensuremath{\mu}}$ is a free zero-mass Wightman field with ${\ensuremath{\partial}}_{\ensuremath{\mu}}$${j}^{\ensuremath{\mu}}$=${\ensuremath{\partial}}_{\ensuremath{\mu}}$${\ensuremath{\epsilon}}_{\ensuremath{\nu}}^{\ensuremath{\mu}}$${j}^{\ensuremath{\nu}}$=0.