How to Show Commutativity of Charge Current Density Operator?

[jmlaniel]

Homework Statement

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

Homework Equations

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

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

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 :

$\{ \psi(x), \bar{\psi}(y) \} = 0$.

The commutator can be obtained by a simple substitution :

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

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

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

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

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

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

This is the point where I am stuck. I notice that if I just exchange the indices $\alpha \leftrightarrow \delta$ and $\beta \leftrightarrow \epsilon$ 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 $\gamma$ 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)!

 

[jmlaniel]

Nobody can help me ?

 

[lalo_u]

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.