Quantum-classical correspondence of the relativistic equations

According to the Heisenberg correspondence principle, in the classical limit, quantum matrix element of a Hermitian operator reduces to the coefficient of the Fourier expansion of the corresponding classical quantity. In this article, such a quantum-classical connection is generalized to the relativistic regime. For the relativistic free particle or the charged particle moving in a constant magnetic field, it is shown that matrix elements of quantum operators go to quantities in Einstein’s special relativity in the classical limit. Especially, matrix element of the standard velocity operator in the Dirac theory reduces to the classical velocity. Meanwhile, it is shown that the classical limit of quantum expectation value is the time average of the classical variable.