4.1 Basics 1334.1.1 Local gauge invariance
Particles are interpreted as elementary excitations of fields. The field describing
free fermions of spin one half (for instance, electrons) obeys the Dirac equation
iγμ∂μψ−mψ= 0 , (4.1)whereψis the four-component Dirac spinor andγμare the 4×4 Dirac matrices.
This equation can be derived from the Lorentz-invariant Lagrangian density
L=iψγ ̄ μ∂μψ−mψψ, ̄ (4.2)whereψ ̄≡ψ†γ^0 .This Lagrangian is also invariant underglobalgauge transfor-
mations: that is, it does not change when we multiplyψby an arbitrary complex
number with unit norm, for example exp(−iθ), whereθis constant in space and
time. What happens, however, if we allowθto vary from point to point, taking
θ=eλ(xα)to be an arbitrary function of space and time? Will the Lagrangian still
remain invariant under suchlocalgauge transformation? Obviouslynot. Acting on
λ(xα), the derivative∂μgenerates an extra term,
∂μψ→∂μ(e−ieλψ)=e−ieλ(∂μ−ie(∂μλ))ψ, (4.3)and the invariance of the original Lagrangian (4.2) can only be preserved if we
modify it by introducing an extra field. Under gauge transformations this field
should change in such a way as to cancel the extra term in (4.3). Let us consider
avector gauge field Aμand replace the derivative∂μin (4.3) with the “covariant
derivative”
Dμ≡∂μ+ieAμ. (4.4)If we assume that under gauge transformationsAμ→A ̃μ,then
Dμψ→D ̃μ(e−ieλψ)=e−ieλ(∂μ+ieA ̃μ−ie(∂μλ))ψ.Therefore wepostulatethe transformation law
Aμ→A ̃μ=Aμ+∂μλ, (4.5)and find
D ̃μ(e−ieλψ)=e−ieλDμψ.Hence
ψγ ̄ μDμψ→(ψ ̄eieλ)γμD ̃μ(e−ieλψ)=ψγ ̄ μDμψ.