We have made a study of the canonical quantization of non-Abelian gauge theories. We begin with the covariant formulation, in which all four components of the guage field are independent, and the Lorentz condition is imposed on the wavefunction at all times. This Lorentz condition is equivalent to the imposition of the Gauss law and the Lorentz condition on the initial wavefunction. The Lorentz condition enables us to separate out the variable A0, the time component of the gauge field, and leads to the Weyl (temporal) gauge formulation in which A0 is absent. The use of the Gauss law separates out additional components of the gauge field. This separation is done with the introduction of curvilinear coordinate. We show that the choice of the ”north pole” of the coordinate system cooresponds to the choice of gauge, which demostrates the equivalence of the quantum theory in different gauges. Unitarity and Lorentz covariance are easily proved in our formulation.We also derived the Hamiltonian in the Coulomb gauge and that in the axial gauge. By using the theorem of equivalence of Feynman rules, we show that the rules in the Coulomb gauge as well as those in the axial gauge are different from the ones conventionally adopted. This proves that the conventional rules are inconsistent.
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