The Master Field for QCD and q-Deformed Quantum Field Theory

The master fields for the large $N$ limit of matrix models and gauge theory are constructed. The master fields satisfy to standard equations of relativistic field theory but fields are quantized according to a new rule. To define the master field we use the Yang-Feldman equation with a free field quantized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a non-Abelian gauge symmetry. For the construction of the master field it is essential to work in Minkowski space-time and to use the Wightman correlation functions. The BRST quantization of the master field for gauge theory and a loop equation are considered.