Subsidiary Condition in Quantum Electrodynamics

The subsidiary condition $\frac{\partial {A_{\mu }}^{\mathrm{}(+)\mathrm{}}}{\partial x_{\mu }|n〉}=0\mathrm{}$, usually known as the "Gupta-Bleuler" condition, is shown to be inadequate as a criterion for defining physical states in quantum electrodynamics in the Lorentz gauge. The condition is shown not to be covariant and to fail to define state vectors that remain in the physical subspace. An alternative subsidiary condition, which is satisfactory, is discussed and is shown to require an extensively different formulation of the collision problem in quantum electrodynamics. Some possible physical consequences of the inadequacy of $\frac{\partial {A_{\mu }}^{\mathrm{}(+)\mathrm{}}}{\partial x_{\mu }|n〉}=0\mathrm{}$ are proposed; these include effects in the decays of short-lived particles, and the fact that in some types of strong interactions, acting simultaneously with electromagnetic ones, $S$-matrix elements may occur which predict transitions from the physical space into the part of space in which the subsidiary condition is violated. The solution to the collision problem for stable charged particles that have only electromagnetic interactions is shown to be identical to that obtainable from the present theory.