Fermion Avatars of the Sugawara Model

We argue that the expression for the energy-momentum tensor as a quadratic function of currents given in the Sugawara model must be reinterpreted as the limit of the corresponding expression, with the currents given a small separation minus its vacuum expectation value as the separation goes to zero, if the expression is to have any chance at all of making sense in a quantum field theory. We calculate this limit for a wide class of free Dirac theories with the usual currents in both two and four dimensions. In many (though not all) cases, we find that the limit yields the usual energy-momentum tensor. In calculating the limit in four dimensions, we have to multiply by the square of the separation to ensure that the limit exists; this is a reflection of the divergence of the Schwinger terms in the equal-time commutators of currents.