Gamma matrices and how they operate

1. Sep 29, 2012

1. The problem statement, all variables and given/known data

Just a matter of convention (question)

2. Relevant equations

$$\gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}$$

3. The attempt at a solution

If then,

$$\gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}$$

and $$\gamma^0$$ is just $$\beta$$ and $$\beta \alpha^k = \gamma^k$$ is it true then that

$$\gamma^k = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\alpha^k$$

2. Oct 1, 2012

Can no one confirm I have done this right?

3. Oct 1, 2012

dextercioby

Isn't the particular form of a Dirac matrix representation dependent ? So then only the general relations will hold, i.e.

βrepαkrepkrep

where i/o <rep> one has the Dirac, Majorana or Weyl/chiral representations.

4. Oct 1, 2012

I think so. I think you have to work with $$D(\psi(x,t))$$ on the three matrices $$\gamma^1,\gamma^2,\gamma^3$$ to get back the matrix $$i\gamma^0 \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix}$$

which when squared gives you the chirality.

5. Oct 1, 2012

Now I am really confused: consider the matrix form of $$a^k$$ and calculate it all out we have

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \end{pmatrix}$$

A nullified matrix?

Have I got my $$a^k$$ matrix right... ?

6. Oct 1, 2012

$$a^k$$ is just a submatrix, right? of

0_2 sigma^k

\sigma^k 0_2

k=1,2,3

in my case, 1 and 3

7. Oct 1, 2012

I just don't understand why the relationship

$$\beta \alpha^k = \gamma^k$$

would be important if it spat out a zero matrix, which makes me wonder strongly whether I even have the right conditions down.

8. Oct 1, 2012

TSny

You will not get the null matrix. For example, check the element in the first row, third column of the resultant matrix.

9. Oct 1, 2012

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & -1 & 0 & 0 \\-1 & 0 & 0 & 0 \end{pmatrix}$$