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Mandl & Shaw Problem 4.3

  1. Jun 21, 2011 #1
    1. The problem statement, all variables and given/known data

    The charge current density operator for the Dirac equation is defined as : [itex]s^\mu = - ec \bar{\psi}\gamma^\mu\psi[/itex].

    2. Relevant equations

    I need to show that the current density operator commutes when measured at two spacelike separated points :

    [itex][s^\mu(x),s^\nu(y)] = 0 [/itex] for [itex](x-y)^2 < 0[/itex].

    3. The attempt at a solution

    First, I inspired myself with the following post :

    https://www.physicsforums.com/showthread.php?t=234580"

    However, I admit that I don't understand how this thread has been tagged "solved" since there is nothing there to help anyone... (or I don't see how the second post is a hint good enough to end the thread).

    So, I get that by microcausality, the field must anti-commute :

    [itex]\{ \psi(x), \bar{\psi}(y) \} = 0[/itex].

    The commutator can be obtained by a simple substitution :

    [itex][s^\mu(x),s^\nu(y)] = e^2 c^2( \bar{\psi}(x) \gamma^\mu \psi (x) \bar{\psi}(y) \gamma^\nu \psi(y) - \bar{\psi}(y) \gamma^\nu \psi(y) \bar{\psi}(x) \gamma^\mu \psi (x) )[/itex]

    My problem here is how to deal with the [itex]\gamma^\mu[/itex]. I know that [itex]s^\mu[/itex] is basically 4 numbers ([itex]\mu[/itex] = 0, 1, 2, 3 or 4). So I deduce that [itex]- ec \bar{\psi}\gamma^\mu\psi[/itex] must also be 4 numbers.

    I tried expressing this quantity using the matrix indices. I got the following expression:

    [itex]s^\mu = - ec \bar{\psi}_\alpha (\gamma^\mu)_{\alpha \beta} \psi_{\beta}[/itex] (The Einstein summation convention on repeated indices is used here).

    By using this formalism, I wrote the commutator and I got the following expression :

    [itex][s^\mu(x),s^\nu(y)] = e^2 c^2( \bar{\psi}_\alpha (x) \psi_\beta (x) \bar{\psi}_\delta (y) \psi_\epsilon (y) (\gamma^\mu)_{\alpha \beta} (\gamma^\nu)_{\delta \epsilon} - \bar{\psi}_\alpha (y) \psi_\beta (y) \bar{\psi}_\delta (x) \psi_\epsilon (x) (\gamma^\nu)_{\alpha \beta} (\gamma^\mu)_{\delta \epsilon} )[/itex]

    This is the point where I am stuck. I notice that if I just exchange the indices [itex]\alpha \leftrightarrow \delta[/itex] and [itex]\beta \leftrightarrow \epsilon[/itex] in the second term, the commutator goes to zero without any problem.

    My questions are :

    #1 Can I do this ? It seems to easy to be true...

    #2 I put the [itex]\gamma[/itex] at the end of my expression... Can I move them in this way? Since, they are numbers, I took the liberty of moveing them around. However, can I do this with the fields ?

    #3 I did not use the microcausality condition to get this result. Is it required or not ? And if so, ca I have a hint as to where I should include it ?

    Thanks for your help (I spent a whole day on this thing and I am really going nowhere)!
     
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
  3. Jul 4, 2011 #2
    Nobody can help me ? :frown:
     
  4. Nov 17, 2011 #3

    lalo_u

    User Avatar
    Gold Member

    You can't do that. If you change the order of the summations you are changing the order of the matrices.
    I think should demonstrate that the commutator is a number and not a matrix using the definition of the 4-density current and gamma properties.
     
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