  The Witten spinor Hamiltonian formulation has previously been shown to be
able to yield a quasilocalization for the GR energy-momentum which leads to a
proof of the positive energy when the spinor satisfies the Witten equation. In
this work we investigate whether this formulation can also describe the GR
angular momentum. We conceive four candidates of the quadratic spinor
Hamiltonian for the angular momentum based on Witten's scheme. The first one is
acquired by substituting the spinor pseudovectorial parameterization for the
spinor vectorial parameterization of the displacement in the Witten
Hamiltonian. The other three each are composed of other quadratic spinor terms,
all with the displacement consisting of the spinor parameterization of an
antisymmetric 2-rank tensor and a position vector, one having 4 terms and the
others each having twice distinct 2 of the 4 terms. With possible field
perturbations around the at spacetime we find none of the four candidates can
give the angular momentum quasilocalization. The importance and prospects for
successfully including the angular momentum in Witten's formalism are
discussed.
