2.4 Gauge Invariance
However, we note that
(2.4.8) ∂μψ ̃=∂μ(eiθψ) =eiθ(∂μ+i∂μθ)ψ,
indicating that∂μis not covariant as defined by (2.4.7). In view of (2.4.8), to obtain a covari-
ant derivative, we have to add a termGμto∂μ. Namely, we need to defineDμby
(2.4.9) Dμ=∂μ+igGμ withgbeing a coupling constant,
whereGμis a 4-dimensional vector field and for (2.4.6)Gμtransforms as
(2.4.10) G ̃μ=Gμ−
1
g∂μθ.Thus, it is readily to check that
D ̃μψ ̃=(∂μ+igG ̃μ)(eiθψ) =eiθ(∂μ+igGμ)ψ=eiθDμψ.Hence, the derivative operator defined by (2.4.9) is covariant under the gauge transformation
(2.4.6) and (2.4.10):
(2.4.11) ψ ̃=eiθψ, G ̃=Gμ−
1
g∂μθ.In view of (2.4.11) and (2.4.5), the fieldGμand the coupling constantgin (2.4.9) are the
electromagnetic potentialAμand electric chargee:
Gμ=Aμ the electromagnetic potential,
g=e the electric charge.The above process illustrates that the electromagnetic potentialAμand the electric charge
eare the outcomes from theU( 1 )gauge invariance. In other words, from the phase angle
symmetry of particles (U( 1 )gauge symmetry) we deduce the following conclusions:
(2.4.12)
U( 1 )gauge symmetry ⇒ U( 1 )gauge fieldAμ,
U( 1 )gauge fieldAμ = electromagnetic potential,
U( 1 )gauge coupling constantg = electric charge.Recall that in the Einstein general theory of relativity, the principle of general relativity
not only leads to the equivalence of gravitational potential to the Riemannian metricgμ ν,
but also determines the field action—the Einstein-Hilbert action (2.3.35). Now, by theU( 1 )
gauge invariance, we are able to deduce both the electromagnetic potentialAμas (2.4.12) and
theU( 1 )gauge action as follows.
In fact, in the same fashion as used in (2.3.29) for gravity, by the covariance of (2.4.7) we
derive the covariant fieldFμ νas
(2.4.13)
[
DμDν−DνDμ]
ψ=ieFμ νψ,