Quantum Field Theory

line can be reached by a gauge transformation and are identified. To make progress,we pick a representative from each gauge orbit. It doesn’t matter which representativewe pick — after all, they’re all physically equivalent. But we should make sure that wepick a “good” gauge, in which we cut the orbits.Different representative configurations of a physical state are called different gauges.There are many possibilities, some of which will be more useful in different situations.Picking a gauge is rather like picking coordinates that are adapted to a particularproblem. Moreover, different gauges often reveal slightly different aspects of a problem.Here we’ll look at two different gauges:• Lorentz Gauge: ∂ µ A µ = 0To see that we can always pick a representative configuration satisfying ∂ µ A µ = 0,suppose that we’re handed a gauge field A ′ µ satisfying ∂ µ(A ′ ) µ = f(x). Then wechoose A µ = A ′ µ + ∂ µλ, where∂ µ ∂ µ λ = −f (6.14)This equation always has a solution. In fact this condition doesn’t pick a uniquerepresentative from the gauge orbit. We’re always free to make further gaugetransformations with ∂ µ ∂ µ λ = 0, which also has non-trivial solutions. As thename suggests, the Lorentz gauge 3 has the advantage that it is Lorentz invariant.• Coulomb Gauge: ∇ · ⃗A = 0We can make use of the residual gauge transformations in Lorentz gauge to pick∇ · ⃗A = 0. (The argument is the same as before). Since A 0 is fixed by (6.10), wehave as a consequenceA 0 = 0 (6.15)(This equation will no longer hold in Coulomb gauge in the presence of chargedmatter). Coulomb gauge breaks Lorentz invariance, so may not be ideal for somepurposes. However, it is very useful to exhibit the physical degrees of freedom:the 3 components of ⃗ A satisfy a single constraint: ∇ · ⃗A = 0, leaving behind just2 degrees of freedom. These will be identified with the two polarization states ofthe photon. Coulomb gauge is sometimes called radiation gauge.3 Named after Lorenz who had the misfortune to be one letter away from greatness.– 127 –