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Solving the gauged Dirac equation perturbatively

  1. Feb 25, 2013 #1
    1. The problem statement, all variables and given/known data

    Given the gauge invariant Dirac equation[tex] (i\hbar \gamma^\mu D_{\mu} - mc)\psi(x, A) = 0 [/tex]Show that the following holds: [tex] \psi(x, A - \frac{\hbar}{e} \partial\alpha) = e^{i\alpha}\psi(x, A) [/tex]
    2. Relevant equations

    The covariant derivative is [tex] D_\mu = \partial_{\mu} + i\frac{e}{\hbar} A [/tex] And the Dirac equation expanded gives [tex] (i\hbar \gamma^\mu \partial_{\mu} - mc)\psi(x, A) = e\gamma^\mu A_{\mu}(x)\psi(x, A)[/tex]
    The free field Feynman propagator [tex] S_{F}(x-x') [/tex] satisfies [tex] (i\hbar \gamma^\mu \partial_{\mu} - mc)S_{F}(x-x') = i\hbar\delta^{(4)}(x - x') [/tex]

    3. The attempt at a solution

    So i seperate the hamiltonian density of the system into the unperturbed and the interaction terms [tex]\mathcal{H} = \mathcal{H}_{0} + \mathcal{H}_{interaction}[/tex]
    giving [tex]\mathcal{H}_{0} = c\overline{\psi}(-i\hbar \gamma^\mu \partial_{\mu} + mc)\psi[/tex]
    and [tex]\mathcal{H}_{int} = e\overline{\psi}(\gamma^\mu A_{\mu})\psi[/tex]

    All i can quess now is that the interaction term should give a contribution to a perturbative series, but i fail to see and accomplish this. Also, where does the free field Feynman propagator come into play?
     
  2. jcsd
  3. Feb 27, 2013 #2
    Are you sure the question involves perturbation theory? :confused:

    Surely all you have to do is show that the wavefunction
    [tex]e^{i\alpha}\psi[/tex]
    satisfies the Dirac equation with the replacement
    [tex] A_{\mu} → A_{\mu} - \frac{\hbar}{e} \partial_{\mu}\alpha.[/tex]
    That's the "gauge symmetry" of the equation - a gauge transformation on the vector potential induces a change of phase in ψ.
     
  4. Feb 27, 2013 #3
    Good point, i think the word "iteratively" was used by my professor. But i discussed it with others and they were also confused about the perturbative/iterative part. As of now, i still have no real progress.

    Edit:
    I must thank you for brining this up. I ignored that part of the question and showed it by just applying the U(1) symmetry to the equation with the covariant derivative. However, i ended up with an opposite sign. I'm just going to take it as a sign mistake for now.
     
    Last edited: Feb 27, 2013
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