The mechanism which guarantees the consistency of the angular-momentum conservation and commutation rules of a Lorentz-invariant theory with the requirement that the vacuum expectation of a vector operator be nonvanishing is examined in detail. A theory originally proposed by Bjorken which reproduces ordinary electrodynamics is presented in a manner which allows the calculation of the parameters of the theory. In particular the "consistency condition" is displayed and found to be quadratically, not cubically, divergent. It is shown that the original Bjorken solution occurs when the cutoff condition of the theory is taken literally. This attitude results in difficulties with current conservation and leads to transitions between the standard vacuum and anomalous degenerate states. These transitions alone, and not the ones directly involving the massless vector particles induced by the broken symmetry, are responsible for the ultimate consistency of the theory. An alternative formulation of the theory which does not take the cutoff so seriously, and hence places emphasis on the underlying operator structure rather than the perturbation Green's functions of the theory, is proposed. This presentation is essentially equivalent to the original formulation since it differs only by gauge terms. However, in this case no difficulty is encountered with current conservation and the theory is consistent in the manner required by normal formulations of the Goldstone theorem.

• Received 22 July 1964

DOI:https://doi.org/10.1103/PhysRev.136.B1404