Minimal Subsitution from Lorentz Invariance

[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

 

[Bill_K]

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

D_μψ = (∂_μ + ieA_μ)ψ → (∂_μψ + ieA_μ - ieψλ_,μ + ieψλ_,μ) exp(-ieλ(x)) = (D_μψ) exp(-ieλ(x)) is.