134 The very early universe
and Lagrangian (4.2), when we substituteDμfor∂μ, is invariant under local gauge
transformations.
The gauge fieldAμcan be a dynamical field. To find the Lagrangian describing
its dynamics, we have to build a gauge-invariant Lorentz scalar out of the field
strengthAμand its derivatives. As follows from (4.5),
Fμν≡DμAν−DνAμ=∂μAν−∂νAμ (4.6)does not change under gauge transformations and therefore the Lorentz scalar
FμνFμνis the simplest Lagrangian we can construct. The scalarAμAμ,which
would give mass to the field, is not allowed because it would spoil gauge invariance.
In the resulting full Lagrangian,
L=iψγ ̄ μ∂μψ−mψψ ̄ −^14 FμνFμν−e(ψγ ̄ μψ)Aμ, (4.7)in which the reader will immediately recognize electrodynamics with the coupling
constant proportional to the electric chargee. Because the fine structure constant
α=e^2 / 4 π 1 /137 is small, one can consider the interaction term as a small
correction and hence develop perturbation theory.
It is convenient to represent this perturbation theory by Feynman diagrams,
where the interaction terme(ψγ ̄ μψ)Aμcorresponds to a vertex where electron
linesψ,ψ ̄ meet photon lineA. The incoming solid line corresponds toψand the
outgoing toψ ̄. Assuming that time runs “horizontally to the right,” Figure 4.1(a)
is read as follows: the electron enters the vertex, emits (or absorbs) the photon,
and goes on. A rule, which is justified in quantum field theory, is the following:
an electron “running backward in time” on the same diagram, but reoriented as in
Figure 4.1(b), is interpreted as its antiparticle, a positron, running forward in time.
timee−e−e−
e−e− e−
e+γ e−γγ(a) (b) (c)Fig. 4.1.