Physical meaning of the photon wave function

Recently, the author [Phys. Rev. A 49, 2839 (1994)] proposed a quantum-mechanical theory of a photon, in which negative energy states can be dismissed from physical photon states without causing any difficulties. In this Brief Report the physical meaning of a photon wave function is investigated more precisely. The interpretation of a photon wave function as a probability amplitude is guaranteed by requiring the coarse-grained condition that the linear dimensions of a volume or a surface concerned in the configuration space are large compared to the photon wavelengths. The calculation made here is essentially based on that presented by Mandel [Phys. Rev. 144, 1071 (1966)] in his analysis of the photon number operator. But our calculation includes that of Mandel as the zeroth-order approximation. As a result, it is shown that although the position operator ◯ in the ordinary quantum mechanics be considered a correct photon operator strictly cannot, it becomes meaningful to some extent under the coarse-grained condition. In connection with ◯, we also examine the velocity operator $[\mathbf{◯},Ĥ]/iħ$ ($Ĥ$ is the photon Hamiltonian), and show that the probability current density is definable under the coarse-grained condition.