Gauge Invariant Noether's Theorem in Yang-Mills Theory

The gauge invariant definition of the spin dependent gluon distribution function from first principle is necessary to study the proton spin crisis at high energy colliders. In this paper we derive the gauge invariant Noether's theorem in Yang-Mills theory by using combined Lorentz transformation plus local non-abelian gauge transformation. We find that the definition of the gauge invariant spin (or orbital) angular momentum of the Yang-Mills field does not exist in Yang-Mills theory although the definition of the gauge invariant spin (or orbital) angular momentum of the quark exists. We show that the gauge invariant definition of the spin angular momentum of the Yang-Mills field in the literature is not correct because of the non-vanishing boundary surface term in Yang-Mills theory. We also find that the Belinfante-Rosenfeld tensor in Yang-Mills theory is not required to obtain the symmetric and gauge invariant energy-momentum tensor of the quark and the Yang-Mills field.