1
$\begingroup$

I see different definitions in different places so here is my question.

why is $\gamma^{0}$ sometimes defined as a 2 by 2 matrix and sometimes as a 4 by 4 matrix? shouldn't a definition be something that does not change?

$\endgroup$
  • 1
    $\begingroup$ Do you mean perhaps a $2\times 2$ matrix of a $2 \times 2$ matrix (making it actually $4\times 4$)? Sometimes that's the way it is displayed in textbooks. $\endgroup$ – JamalS Apr 11 '18 at 12:37
  • $\begingroup$ I have 1 in the first element of the first row and -1 on the last element of the last row. It dosen't say that it is something different than 1. however, do you mean it is definitely always as a 4 by 4 matrix? $\endgroup$ – mathematics poi Apr 11 '18 at 12:43
  • 1
    $\begingroup$ It is as I suspected, here 1 means the 2 by 2 identity matrix. $\endgroup$ – JamalS Apr 11 '18 at 13:18
0
$\begingroup$

This is just the very common block matrix notation. When you see

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

you should interpret this as

$$\gamma^0 = \begin{pmatrix} I_{2\times 2} & 0_{2\times 2} \\ 0_{2\times 2} & I_{2\times 2} \end{pmatrix}, $$

where $I_{2\times 2}$ and $0_{2\times 2}$ are the $2\times 2$ identity and zero matrices. Written out in full the matrix would be

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

There are two things going on here, really. One is that physicists like to write the identity matrix as $1$ instead of $I$, and the other is the block matrix notation, where it is assumed that you know from context that $\gamma^0$ should be a $4 \times 4$ matrix. Many times we don't even write the zero matrices, so that you could even see

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

being used! You just have to know from context that this is supposed to be a $4 \times 4$ matrix, and the $1$'s are $2 \times 2$ identities.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.