Gauge Problem in Quantum Field Theory. III. Quantization of Maxwell Equations and Weak Local Commutativity

The problem of the quantization of the Maxwell equations is analyzed in connection with the basic assumptions of quantum field theory. It is shown that it is impossible to quantize the Maxwell equations by means of a potential $A_{\mu }\mathrm{}\left(x\right)\mathrm{}$ which is a weakly local field. Thus, a result which was known for the Coulomb gauge is shown to hold in general: The quantization of the Maxwell equations requires the use of a potential $A_{\mu }\mathrm{}\left(x\right)\mathrm{}$ which is both noncovariant and nonlocal. It is shown that a weakly local and/or covariant operator $A_{\mu }\mathrm{}\left(x\right)\mathrm{}$ can be introduced only in a Hilbert space in which the vectors corresponding to physical states do not form a dense set, and therefore unphysical states must be present. The connections with the Gupta-Bleuler formulation are discussed.