Pion Gauge Invariance and Low-Energy Theorems

Zero-mass pion theories invariant under $c$-number translations ("chiral transformations") of the pion field are studied in a general framework. The operator which induces the chiral transformation is defined in Fock space (in which it is not unitary) and in von Neumann's infinite-tensor-product space (in which it is unitary). The transformed (noninvariant) Fock-space vacuum is recognized as a coherent state in the tensor-product space. The generator of the chiral transformation—a constant of the motion in gauge-invariant theories—is diagonalized, and its eigenvectors, the "chiral states," are employed in one of two derivations of a low-energy theorem for zero-mass pion emission and absorption, assuming gauge invariance of the theory. The other method of derivation is also used to rederive the electromagnetic gauge conditions. Then Lagrangian models (gradient-coupling, $c$-number, and operator theory) are studied in which the invariance is realized provided the current is suitably restricted. Implications of the low-energy theorem are checked (exactly for the $c$-number theory, in lowest-order perturbation theory for the operator theory). A larger class of models is then considered in which, it is shown, the complicated set of transformations under which the Lagrangian is invariant reduce, by virtue of the field equations and the asymptotic condition, to a simple pion translation when expressed in terms of the asymptotic fields, and hence obey the supposition of our theorem, which we again check in lowest-order perturbation theory.