Field-Current Identities and Algebra of Fields

It is shown that the possible identity between the various hadronic current operators and the corresponding spin-1 meson field operators determines the general structure of the hadronic part of the total Lagrangian. In particular, the identity between the isovector electromagnetic current and the neutral $\rho$-meson field implies that the $\rho$ dependence of the strong interaction must be the same as that in the Yang-Mills theory, except for the mass term of the $\rho$ meson. The explicit form of the interaction Lagrangian makes possible a general study of the local equal-time commutators of the various hadronic current operators, including the effects of the electromagnetic interaction. Many of these electromagnetic correction terms depend only on the general requirement of gauge invariance, and are independent of whether the proposed field-current identities are valid or not. For example, the usual Schwinger term $\lambda \left(\frac{\partial }{\partial r_j}\right){\delta }^3(r-r^{\prime })$ in the commutator between the time component of any charged hadronic weak interaction current and the $j$th space component of its Hermitian conjugate should be replaced by $\lambda \left[\right(\frac{\partial }{\partial r_j})+ie{A}_j]{\delta }^3(r-r^{\prime })$, where $A_j$ is the electromagnetic field operator. The contribution of such a correction term, i.e., $\lambda \mathrm{ie}{A}_j{\delta }^3(r-r^{\prime })$, remains present in the integrated form of the commutator. In the usual current algebra, $\lambda$ is mathematically undefined. If field-current identities hold, then these current commutators are the same as the corresponding algebra of the field operators, and $\lambda$ becomes a well-defined $c$ number. Some speculative remarks concerning the possible extension of the algebra of fields to the lepton currents are presented.