# Expanding gamma matrices

1. Sep 23, 2012

$$\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0$$

How do I expand

$$i\hbar \gamma^0$$

the matrix in this term, I am a bit lost. All the help would be appreciated!

2. Sep 23, 2012

### zje2009

It's itself.Using Mathematica.

3. Sep 24, 2012

Expanding that, gives back that?

What is mathematica??

4. Sep 24, 2012

### Bill_K

The gamma matrices are 4 x 4 matrices whose values depend on the basis ("representation") you decide to use in spinor space. For a list of possibilities, see "gamma matrix" in Wikipedia.

5. Sep 25, 2012

so you don't know how to expand the terms I asked of?

6. Sep 25, 2012

### Bill_K

I thought the Wikipedia article explained it pretty clearly. It gives the explicit form of γ0 in the Dirac, Weyl and Majorana representations. Isn't that what you want?

But if that's not what you mean by "expanding" it, the only other thing to do is this...

iħγ0

7. Sep 26, 2012

Why have that when I started off with that... my equation I really thought was simple. HOW do you expand $$i \hbar \gamma^0$$

The answer I was looking for was not a go to wiki one! And no wiki does not explain it well for me, I am new at this stuff.

8. Sep 26, 2012

Can you show me, in plane mathematical language, in an equation, how to expand it please.

9. Sep 26, 2012

### kloptok

You will have to define what you mean by "expand". So far we have only been able to guess, and apparently this was not what you intended. So define "expand" please.

10. Sep 26, 2012

### PhilDSP

Do you mean this? $\qquad \gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \qquad \qquad$ (which is the Dirac representation)

so that $\ \ \qquad \qquad i \hbar \gamma^0 = \begin{pmatrix} i \hbar & 0 & 0 & 0\\ 0 & i \hbar & 0 & 0\\ 0 & 0 & -i \hbar & 0\\ 0 & 0 & 0 & -i \hbar \end{pmatrix} \qquad \qquad$

Last edited: Sep 26, 2012
11. Sep 28, 2012

Why is then when $$\gamma^{0}^2$$ is equal to 1?

12. Sep 28, 2012

### PhilDSP

$$\qquad (\gamma^0)^2 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$

13. Sep 28, 2012