LIOUVILLE EQUATION OF RELATIVISTIC CHARGED FERMION
相对论带电费密子Liouville方程

As a form of density matrix, the Wigner function is the distribution in quantum phase space. It is a 2×2 matrix function when one uses it to describe the non-relativistic fermion. While describing the relativistic fermion, it is usually represented by 4×4 matrix function. In this paper we obtain a Wigner function for the relativistic fermion in the form of 2×2 matrix, and the Liouville equation satisfied by the Wigner function. This equation is equivalent to the Dirac equation of charged fermion in QED. Our equation is also equivalent to the Dirac equation in the Walecka model applied to the intermediate energy nuclear collision while the nucleon is coupled to the vector meson only (or taking mean field approximation for the scalar meson). We prove that our 2×2 Wigner function completely describes the quantum system just the same as the relativistic fermion wave function. All the information about the observables can be obtained with our Wigner function.