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Minimal Subsitution from Lorentz Invariance

  1. Feb 16, 2013 #1

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

    [tex](i\not{\partial -m })\psi=0[/tex]

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

    [tex]\partial ^\mu \rightarrow \partial ^\mu+ieA^\mu\equiv D^\mu[/tex]

    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!

  2. jcsd
  3. Feb 16, 2013 #2


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    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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