The continuity of the gauge fixing condition n ⋠ ∂n ⋠ A = 0

To investigate the continuity of the gauge condition n⋅∂n⋅A=0, the continuity of generators of Wilson lines is studied here in the frame of non-Abelian gauge theory in the space R⨂S1⨂S1⨂S1, which guarantees the continuity of the gauge condition n⋅∂n⋅A=0 as proved in our previous work. Starting from SU(2) theory, it is proved that the gauge fixing condition is continuous in a compact subspace of R⨂S1⨂S1⨂S1 given that gauge potentials are differentiable with continuous derivatives. For continuous SU(2) gauge potentials in the space R⨂S1⨂S1⨂S1 that satisfy suitable boundary conditions, which is topologically equivalent to continuous gauge potentials in a compact space, the gauge fixing condition n⋅∂n⋅A=0 is proved to be continuous given that gauge potentials are differentiable with continuous derivatives. The conclusion is extended to the case that lengths along all directions of the space tend to ∞. Lorentz invariance is restored in this limit for quantities which are multiplicative renormalized and free from possible singularities in functional integrals. The same conclusion is proved for general SU(Nc) theory.