Photon Propagarors in Quantum Electrodynamics

Abstract

Quantum electrodynamics in the Landau gauge is investigated in a different mann~r from Nakanishi's theory. A dipole ghost field is introduced in another simple way. It is shown that the Landau gauge representation can be obtained by a unitary transformation from the usual Feynman gauge representation. The two representations are equivalent to each other, though photon propagators take different forms superficially in each case. Supplementary conditions to eliminate the ghost states are discussed with the Lorentz condition. It is asserted that the interaction picture in the Landau gauge is also possible. l ) has proposed a formalism of quantum electrodynam­ ICS which includes the case of the quantization in the Landau gauge as well as that in the Feynman gauge. In Nakanishi:s theory the longitudinal or scalar photon is represented by a dipole ghost, the role' of which is essential to the form of the photon propagator.*) While his theory is skilfully constructed, there is another simple way of obtaining a formalism in the Landau gauge. The pres­ ent note aims at showing this fact. . In the present formalism we deal with a dipole ghost field· similarly to Naka­ nishi's theory but we introduce it in another sense. We show that the Landau gauge representation is related with the Feynman gauge representation (that is, the Gupta and Bleuler formalismS») by a unitary transformation in our interaction picture for a conserving current. In this sense, the two representations are equivalent to each other, though photon propagators take different forms super­ ficially in each case. In order to eliminate the ghost particles in the initial and final states of scattering and to obtain the Maxwell equations in the sense of expecta­ tion values of operators for physical states, necessary supplementary conditions are discussed. Their forms appear in part as Gupta's condition 4 ) for the Stueckel­ berg formalism of the vector meson field. As far as we deal with conserving . currents, the physical S-matrix' is unitary and therefore the theory is probabili­ stically interpretable. Juse) investigated quantum electrodynamics in Lehmann's spectral representation. He asserts that only the interacting electromagnetic field can be quantized in the Landau gauge. Certainly we cannot obtain bare photon *) Gota and Obara2) investigated a canonical quantization method of the free electromagnetic field in the Landau gauge starting with the same field equations as those in Nakanishi's case. In their quantization the dipole ghost states are not introduced, though the longitudinal and scalar photons have peculiar characters.