Quantization of the Yang-Mills Field

Abstract

Recent efforts in quantum field theory have given rise to increased interest in nonlinear, gauge-type field theories. In this paper, we examine the Yang-Mills field, which is a theory of this type, which lies between electrodynamics and general relativity in complexity. The quantum Yang-Mills field is introduced to satisfy the requirement of invariance under isotopic phase transformations of the second kind. The theory is then put into its first-order form, so as to make it amenable to quantization by the methods of the Schwinger action principle. Quantization is then carried out for the two different gauge conditions; the first (which is the analog of the radiation gauge in electrodynamics) leads to a perturbation treatment of the constraints of the theory and the second to a rigorous solution of them. Two conditions for a consistent quantization are investigated. These are (1) the requirement that the Lagrange equations be identical to the Heisenberg equations of motion (the latter being evaluated using the field commutation relations) and (2) the requirement that the constraint equations be conserved in time (which leads to a condition on double commutators in the $q$-number theory). In the radiation gauge, these conditions are shown to be satisfied in lowest-order perturbation theory. In the second gauge used, they are rigorously satisfied. It is also shown for that gauge that the field Hamiltonian is positive definite when coupling to Fermion isotopic currents is ignored. No investigation of consistency conditions arising from Lorentz covariance requirements are made.