Fermions in a field theory of currents

Abstract

Equal-time commutators between vector and axial-vector currents with fermion fields\(\bar \psi \) are investigated under the assumption that\([Q_5^\alpha (x_0 ),\bar \psi (x)] \propto \psi (x)\gamma _5 \tau ^\alpha \) for the axial chargesQ 5 α (x0). It is shown that there must be Schwinger terms in the equal-time commutators between the time components of the currents and fermion fields\(\bar \psi \) introduced in models of the Sugawara type. The chiral transformation properties of both the non-Schwinger and Schwinger terms are derived in a model-independent fashion. Next, assuming the equal-time commutator between the time-time component of the symmetric energy-momentum tensor and the spinor field\(\bar \psi \) to be as in quantum electrodynamics it is shown that the number of Schwinger terms in\([V_k^\alpha (x),\bar \psi (y)]\) or\([A_k^\alpha (x),\bar \psi (y)]\) is less than or equal to their number in\([V_0^\alpha (x),\bar \psi (y)]\) or\([A_0^\alpha (x),\bar \psi (y)]\) respectively and a number of relations for the Schwinger terms is derived. In addition, the equation of motion for\(\bar \psi \) is also derived and discussed in the Sugawara model.