Non-Abelian gauge field theories on a lattice and spontaneous breaking of chiral symmetry

The perturbation theory for a color-quark gauge theory on a spatial lattice is developed in time-dependent form. Graphical rules are suggested. These methods are then applied to study the nature of the ${\gamma }_5$ symmetry exhibited when the bare-quark-mass term is zero. Investigation of the quark proper self-energy leads to the conclusion that the ${\gamma }_5$ symmetry is realized in the Nambu-Goldstone mode. A Bethe-Salpeter equation for the proper ${\Gamma }^5$ vertex is solved in the ladder approximation, and shown to exhibit zero-mass poles.