ABSTRACT
As we mentioned in the Preface, one can say that QFT was ”born” in 1927, when P.A.M. Dirac succeeded in quantizing the electromagnetic field.
As well-known, the fundamental equations of the electromagnetic field (i.e. Maxwell’s equations) are obtained starting from of the Lagrangian density
L = − 1 4 FikFik, (4.1.1)
where
Fik = ∂Ak ∂xi
− ∂Ai ∂xk
= Ak,i −Ai,k (4.1.2)
is the electromagnetic field tensor, and Ai is the potential four-vector. It is also known that Ai is not uniquely determined by (4.1.2). Indeed, if we replace Ai by
A′i = Ai + ∂f
∂xi , (4.1.3)
we obtain the same Fik, i.e. the same field. The transformation (4.1.3) is known as a gauge condition. To uniquely determine the four-potential Ai, one introduces a supplementary condition, known as the Lorentz gauge condition
= Ai,i = 0. (4.1.4)
Ai = Ai,kk = 0 (4.1.5)
which is the wave equation for Ai. A straight quantum transposition of this formalism meets with
some difficulties, and here are three of them: a) The role of the field variables U(r) is played in our case by
the four-potential components Ai, which satisfy the wave equations (4.1.5). Unfortunately, these equations do not result directly by means of the Lagrangian density (4.1.1), but only through Maxwell’s equations and Lorentz’ condition (4.1.4).
