We impose the symmetry principles to elaborate the invariance of the Abelian and non- Abelian gauges and evaluate its implications to the laws of physics. The application of the symmetry principle into the field which undergoes a rotation in a phase space implies the exixtence of an Abelian gauge field Meanwhile, the application of symmetry principle to the dirac field as a doublet of SU (2)group implies the existence of a non - Abelian gauge field with structure constant satisfies the Lie group algebra.