Photon-Field Operators: Wave-Packet Photons
In this chapter, we take up the central question of how to relate the quantum electrodynamic theory to photon wave mechanics as this is formulated in the energy wave function approach. Since the photon in the first-quantized description is related to the positive-frequency part of the electromagnetic field we start by dividing the transverse electric and magnetic free-field operators into their positive- and negative-frequency parts. In the plane-wave decomposition of the positive (negative)-frequency field part appears operators which annihilate (create) single quanta in the two helicity eigenstates. Two so-called free photon-field operators are then defined in analogy to the classical definition of the positive-frequency Riemann–Silberstein vectors. The bridge to photon wave mechanics is made by taking the matrix elements of these operators between a general one-photon state and the photon vacuum state. It is shown that the two matrix elements can be identified with the upper and lower components of the free-photon spinor. A single-photon state is always an eigenstate for the global photon number operator (with eigenvalue 1), but in general it is not an eigenstate of the global photon Hamilton operator, nor of the global photon momentum operator.