Gauge Problem in Quantum Field Theory

Abstract

The difficulties connected with the quantization of the electromagnetic field are analyzed in the framework of axiomatic field theory. Under the assumptions: (1) existence of the vacuum, invariant under the Poincar\'e group, (2) existence of a representation of the Poincar\'e group such that the fields have tensor transformation properties, and (3) analyticity of the two-point function in the forward tube, it is proved that the Maxwell equations ${\ensuremath{\partial}}^{\ensuremath{\mu}}{F}_{\ensuremath{\mu}\ensuremath{\nu}}=0$, ${\ensuremath{\epsilon}}^{\ensuremath{\lambda}\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}{\ensuremath{\partial}}_{\ensuremath{\mu}}{F}_{\ensuremath{\nu}\ensuremath{\rho}}=0$ do not admit the classical solution ${F}_{\ensuremath{\mu}\ensuremath{\nu}}={\ensuremath{\partial}}_{\ensuremath{\mu}}{A}_{\ensuremath{\nu}}\ensuremath{-}{\ensuremath{\partial}}_{\ensuremath{\nu}}{A}_{\ensuremath{\mu}}$, where the potential transforms as a four-vector. The result is very general and can be obtained without assuming either local commutativity or the spectral condition; it is also independent of whether the metric is positive or not. Thus the difficulties connected with the gauge problem have very little to do with the Lorentz condition, indefinite metric, etc., but rather they arise at the very beginning with the introduction of the four-vector potential. As a corollary of the above result, the representations of the Poincar\'e group for massless spin-one particles in quantum field theory are shown to be essentially different from the corresponding ones of the classical case. The Fock representation in the Gupta-Bleuler approach is discussed in connection with the Lehmann-Symanzik-Zimmermann formulation of quantum electrodynamics. Some care must be used in the definition of the asymptotic Hilbert space since, by the previous result, unphysical photon states must be introduced as asymptotic states in order to have a local theory. The requirement that these states should not affect any physical result imposes definite restrictions on the $S$ matrix. These conditions guarantee that the definition of physical photon states in terms of equivalence classes still has significance in the presence of interactions.