equation for potential scattering, and was solved there by the introduction of a small imaginary
part. The same works here, and we replace the above formal expression by the following accurate
one,
S(x,y) =∫ d (^4) k
(2π)^4
eik·(x−y)
iγμkμ−m+iε
(23.41)
forε >0. The actual value of the integral is complicated, except form= 0, but will not be needed
explicitly here.
There is another way of expressing the free fermion propagator, which will be very useful in
carrying out perturbative calculations. It is based on the fact that to obtain the inverse of the Dirac
operator, it suffices to “patch together” two solutions to thehomogeneous equation. This is done
by using the time-ordering operation that we had encountered when carrying out time-depenedent
perturbation theory. Consider the following object,
Tψα(x)ψ ̄β(y)≡θ(x^0 −y^0 )ψα(x)ψ ̄β(y)−θ(y^0 −x^0 )ψβ(y)ψ ̄α(x) (23.42)Note the minus sign used in the time-ordering of fermion operators. Applying now the Dirac
operator in the coordinatexto both sides, we see that when the operator lands on the fieldψa(x),
the corresponding contribution cancels, in view of the factthatψα(x) satisfies the Dirac equation.
So, the only non-zero contribution arises from when the time-derivative in the Dirac equation hits
theθ-function, and this contribution gives,
(γμ∂μ−m)γα(
Tψα(x)ψ ̄β(y))
= δ(x^0 −y^0 )(γ^0 )γα(γ^0 )δβ{ψα(x),ψ†δ(y)}
= (γ^0 )γα(γ^0 )δβδαδδ(4)(x−y)
= δγβδ(4)(x−y) (23.43)Of course,Tψα(x)ψ ̄β(y) is an operator, not a number, but we can easily extract from it a pure
number, so that we find,
Sαβ(x−y) =〈∅|Tψα(x)ψ ̄β(y)|∅〉 (23.44)where|∅〉is the free-fermion vacuum state.
23.7 The concept of vacuum polarization
We are now ready to study one of the most basic effects in interacting quantum field theory,
namely vacuum polarization. We all know what a regular polarized medium is. For example, water
molecules exhibit an asymmetrical geometry, with the two Hydrogen atoms being on one side, and
the Oxygen atom being at the other side of the molecule.