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160
Progress of Theoretical physics, Vol. 40, No.1, July 1968
Photon Propagators in Quantum Electrodynamics
Kan-ichi YOKOYAMA
Research Institute for Theoretical Physics
Hiroshima University, Takehara, Hiroshima-ken
(Received February 22, 1968) Quantum electrodynamics in the Landau gauge is investigated in a different 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.1. Recently Nakanishi
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 Nakanishi'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 Obara 2) 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 photonshave peculiar characters. Downloaded from https://academic.oup.com/ptp/article/40/1/160/1852665 by guest on 17 May 2023
Photon Propagators in Quantum Electrodynamics 161
propagators in the Landau gauge In the interaction picture, unless we introduce any auxiliary ghost field at the beginning of the formulation. However, we can introduce desirable auxiliary fields for our purpose. They enable us to have the photon propagator in the Landau gauge, while their effects are restricted only to virtual states and also do not destroy the unitarity of the physical S-matrix. Therefore, we can still use the interaction picture from which the conventional perturbation theory starts. Just's case seems to us that the spectral representation is treated without any explicit use of virtually playing ghosts but with their implicit contributions.2. We start from the following formal Lagrangian:
L = Lusual + Lghost ,
Lghost = 0" B 0 "Bo -t A,
2B o2, where LeI denotes the free Lagrangian part of the electron field and A, IS a real constant. The metric IS chosen as goo = -gii=l, i=l, 2,3, (2· 2) The current conservation law 0 "j" = 0 is always assumed. Apart from Lghost, the Lagrangian is nothing else but that of the usual theory of Gupta and Bleuler.3) The parts Lusual and Lghost are completely isolated from each other. Therefore, it is a matter of course that we can obtain the Gupta and Bleuler formalism itself leaving Lghost as a nonsense part. However, as will be seen in the next section, the existence of Lghost is important in the Landau gauge representation. Band Bo represent two auxiliary scalar fields as a massless case of Froissart's dipole ghost field. 6) B corresponds to the dipole ghost field to its pair field Bo .The field equations for Band Bo become
OB= _A,
2B o,OBo=O, (0=0"0,,) (2·3)
From the usual procedure, we have the following commutation relations: where [B(x), Bo(x')] = [Bo(x) , B(x')] =iD(x-x'), [B (x), B (x')] = iA,2jj (x -x'), [Bo (x) , Bo(x')] =0,(2·4) Downloaded from https://academic.oup.com/ptp/article/40/1/160/1852665 by guest on 17 May 2023