Relating quantization and time evolution in the presence of gauge symmetry

Up to now there is no definition or an example of a gauge-invariant quantum field theory in four-dimensional space-time, with mathematical rigor and in the absence of approximations. This is mainly due to three features of the theory: 1) the canonical quantization is not manifestly Lorentz invariant; 2) the Lagrangian is singular due to the gauge symmetry; and 3) the gaussian measure is not gauge invariant. At least one of these features becomes an obstacle to mathematical rigor in either the canonical or the path-integral quantization. Using the fact that there is a wave-function associated to any probability distribution, we study a class of statistical field theories in four-dimensional space-time where the (classical) canonical coordinates when modified by the unitary time evolution, verify the canonical commutation relations. We show that these statistical field theories have all the features of a gauge-invariant quantum field theory in four-dimensional space-time. Thus the features 1), 2) and 3) are not obstacles to define the theory in our formalism.20 p