  Due to proton spin crisis it is necessary to understand the gauge invariant
definition of the spin and orbital angular momentum of the quark and gluon from
first principle. In this paper we derive the gauge invariant Noether's theorem
by using combined Lorentz transformation plus local gauge transformation. We
find that the notion of the gauge invariant definition of the spin (or orbital)
angular momentum of the electromagnetic field does not exist in Dirac-Maxwell
theory although the notion of the gauge invariant definition of the spin (or
orbital) angular momentum of the electron exists. We find that the gauge
invariant definition of the spin angular momentum of the electromagnetic field
in the literature is not correct because of the non-vanishing surface term in
Dirac-Maxwell theory although the corresponding surface term vanishes for
linear momentum. We also show that the Belinfante-Rosenfeld tensor is not
required to obtain symmetric and gauge invariant energy momentum tensor of the
electron and the electromagnetic field in Dirac-Maxwell theory.
