Investigations in non-Abelian gauge theory

We consider two problems in this work. In the first problem, we show that in a non-Abelian SU(N) Yang-Mills Chern-Simons theory, broken from SU(3) to SU(2), the renormalized ratio of Chern-Simons coupling to the gauge coupling is 1/4π times an integer, appropriate for the unbroken N = 2 case. In the second problem, we construct states that implement Gauss\u27s law, and construct gauge-invariant operator-valued gauge and spinor fields. We show that there is a unitary operator that transforms the Gauss\u27s law operator (including quarks) for a non-Abelian gauge theory to the “pure glue” Gauss\u27s law operator. We use the unitary operator that effects this transformation to also transform the Hamiltonian for QCD in the temporal (A 0 = 0) gauge into a representation in which all fields (quark and gauge fields) are gauge-invariant. In this representation, the interaction between quarks and gluons is described completely by a nonlocal interaction involving quark color charge density that is the non-Abelian analog of the Coulomb interaction in QED, and another interaction between the transverse gauge-invariant gauge field and the transverse quark color-current density.