A New Variable in General Relativity and Its Applications for Classic and Quantum Gravity

A new variable in the Riemannian geometry is introduced by the tetrad and the Ricci's coefficients of rotation, the characters of curve of the Riemannian geometry are determined completely by the new variable; for general relativity, all the Einstein-Hilbert action, the Einstein equation in general relativity and the Dirac equation in curved spacetime can be expressed by the new variable, and, further, as well the action of the theory on the interaction of gravitational, electromagnetic and spinor field (TGESF). All the characters of transformations of the new variable, the Einstein-Hilbert action and the action of TGESF under the general coordinate transformations and the local Lorentz transformations are discussed, respectively. After presenting the method of introduction of gravitational field in terms of the principle of gauge invariance based on the Dirac equation, and the ten constraint conditions for the tetrad are given, the vacuum-vacuum transition amplitude with the Faddeev-Popov ghost and the terms of external sources of the pure gravitational field is presented; finally, as well that of TGESF.