The Vierbein Formalism and Energy-Momentum Tensor of Spinors

To study the coupling system of space-time and Fermions, we need the explicit form of the energy-momentum tensor of spinors. The energy-momentum tensor is closely related to the tetrad frames which cannot be uniquely determined by the metric. This flexibility increases difficulties to derive the exact expression and easily leads to ambiguous results. In this paper, we give a detailed derivation for the energy-momentum tensor of Weyl and Dirac spinors. From the results we find that, besides the usual kinetic energy momentum term, there are three kinds of other additional terms. One is the nonlinear self-interactive potential, which acts like negative pressure. The other reflects the interaction of momentum $p^\mu$ with tetrad. The third is the spin-gravity coupling term which is a higher order infinitesimal in weak field, but may be important in a neutron star. This term is also closely related with magnetic field of a celestial body. These results are based on the decomposition of usual spin connection into geometrical part and dynamical part, which not only makes calculation simpler, but also highlights their different physical meanings. In addition, we get a new tensor $S^{\mu\nu}_{ab}$ in calculation of tetrad formalism, which plays an important role in the interaction of spinor with gravity.