Minimal Subsitution from Lorentz Invariance

1. Feb 16, 2013

Sekonda

Hello,

My question is on coupling the photons to our Dirac field for electrons, we have the Dirac equation:

$$(i\not{\partial -m })\psi=0$$

By Lorentz invariance we can change our space-time measure by:

$$\partial ^\mu \rightarrow \partial ^\mu+ieA^\mu\equiv D^\mu$$

Though I cannot see why Lorentz invariance implies that this change is invariant?

Sorry if I haven't explained my issue well, any help appreciated!

SK

2. Feb 16, 2013

Bill_K

It's not Lorentz invariance that's involved here, it's gauge invariance. Under an electromagnetic gauge transformation, Aμ → Aμ + λ and ψ → ψ exp(-ieλ(x)). So ∂μψ → (∂μψ - ieψλ) exp(-ieλ(x)), which by itself is not covariant, but the combination

Dμψ = (∂μ + ieAμ)ψ → (∂μψ + ieAμ - ieψλ + ieψλ) exp(-ieλ(x)) = (Dμψ) exp(-ieλ(x)) is.

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