Contribution of vacuum fluctuation to the self-energy of the electron

[yztsz]

In the book" The Quantum Theory of Radiation", Heitler derived the transverse self-energy of the electron(Chapter III, Section18, Eq.(23))

$$\frac{{{e^2}}}{{\pi m}}\int_{\text{0}}^\infty {kdk} $$
which is the energy of the electron under the action of the vacuum fluctuation of the
radiation field. But it is one half of Weisskopf's result( (Phys. Rev. 56, 72,1939: page 81). Why? I can't figure it out.

 

[azntoon]

Heitler's result is only one half of Weisskopf's result because Heitler's calculation only accounts for the transverse components of the electromagnetic field, while Weisskopf also considers the longitudinal components. Heitler's result comes from calculating the energy of the electron due to the transverse components of the electromagnetic field, while Weisskopf's result comes from calculating the energy of the electron due to both the transverse and longitudinal components of the electromagnetic field.