It is shown that it is possible to have a theory of currents without parity doubling if the algebra-of-fields commutation relations are changed by the introduction of operator Schwinger terms. In addition, it is necessary to impose a relationship among the currents generating the theory. The nonlinear formulation of the $\sigma$ model involving only the pion field is shown to be an example with these properties.

• Received 6 May 1969

DOI:https://doi.org/10.1103/PhysRev.184.1803