Solve the Dirac Equation: Unraveling Anticommutator Mystery

[raintrek]

[SOLVED] The Dirac Equation

I'm trying to understand the following property of the Dirac equation:

(i \gamma^{\mu}\partial_{\mu} - m)\Psi(x) = 0

Acting twice with (i \gamma^{\mu}\partial_{\mu} - m):

(i \gamma^{\mu}\partial_{\mu} - m)^{2} \Psi(x) = 0

= [ - \gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} - 2im\gamma^{\mu}\partial_{\mu} + m^{2}]\Psi = 0

But then somehow the book jumps to this step:

= [ 1/2 \left{\{\gamma^{\mu}, \gamma^{\nu}\right}\} \partial_{\mu}\partial_{\nu} + m^{2}]\Psi = 0

And I have no idea how it got there! I understand the { } denote an anticommutator, but I just can't see how the factor of 1/2 has appeared, where the minus has gone and where the middle term has gone. Help!

 

[Dick]

\gamma^\mu \gamma^\nu \partial_\mu \partial _\nu=\gamma^\nu \gamma^\mu \partial_\mu \partial _\nu. Add those and divide by two, to get the anticommutator. Let the middle term act on \Psi and use the dirac equation to see where it has gone. The minus will become an overall minus once you do all of this.

 

[raintrek]

Ah, I understand now why the commutator arises, thanks Dick. However, I'm still confused about the middle term business. If I understand you right, you're saying that:

(-2im\gamma^{\mu}\partial_{\mu})\Psi

is analagous to:

(i\gamma^{\mu}\partial_{\mu} - m)\Psi = 0

and hence should disappear? I don't quite understand that part...

 

[Dick]

It doesn't disappear. It's equal to -2m^2*psi. Do you see why?

 

[raintrek]

^ Ah! Got it, i\gamma^{\mu}\partial_{\mu} = m from the original eq. Yes, I see now! Thanks so much, Dick!