The Witten Hamiltonian and QSL Hamiltonian yield two qualitatively different quasilocal expressions of the energy-momentum and proofs of the positive energy quasilocalization in gravitational fields. However, the exploration by the two approaches for the angular momentum remains void. We engage in the losing part in this monograph. We are also motivated by the quest for the connection between the gravitating energy and angular momentum. It is believed the connection will offer a further norm for the quasilocal expression of the gravitational angular momentum, which has not yet had sufficiently confining criteria. After trials, we find all of our proposed Hamiltonians based on Witten's conception fail while the QSL Hamiltonian succeeds. This work takes off the slack investigation of the spinor angular momentum. Hopefully this takeoff can provide some insights for the communities of quantum gravity and classical gravitational theory to continue attempting at the possible candidates of the spinor Hamiltonian for the angular momentum by Witten's and, more importantly, unveiling the connection between the gravitational energy and angular momentum.

Siao-Jing Li, Bachelor of Science: Studied in Physics Department at Fu Jen Catholic University; Master of Science:Studied in Physics Department at National Central University. Teaching Assistant in Electrophysics Department at National Chiao Tung University; Research Assistant in the Research Center of Applied Sciences at Academia Sinica.