Fermion Representations of Sugawara's Theory of Currents

Abstract

We continue the work of Coleman, Gross, and Jackiw who have argued that the products of currents occurring in Sugawara's model should be interpreted as a singular limit as the coordinate separations approach zero, if the currents are the usual $\overline{\ensuremath{\psi}}\frac{1}{2}{\ensuremath{\lambda}}^{a}{\ensuremath{\gamma}}_{\ensuremath{\mu}}\ensuremath{\psi}$. They found that ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$ of the free-fermion theory can be expressed in the Sugawara form. We now point out that their method works even in the presence of interactions; consequently, Sugawara's energy-momentum tensor can only describe the massless free part of any fermion theory. This can also be seen from the fact that Sugawara's equations of motion for fermion currents are equivalent to $\ensuremath{\gamma}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\partial}\ensuremath{\psi}=0$. We suggest that the most natural interaction term to be added to Sugawara's ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$ is the fully normal-ordered $\frac{1}{2}F:{{V}_{\ensuremath{\lambda}}}^{a}{V}^{a\ensuremath{\lambda}}$: occurring in Heisenberg's nonlinear spinor theory. The resulting equations of motion for the currents still have the Sugawara form (with an additional normal-ordered term). Finally, we attempt to derive the commutators of ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$ with itself and with the currents, using current algebra without any reference to the underlying fields. Here we do not know how to handle timelike separations, and with spacelike separations alone we do not find a consistent solution.