Question about $\gamma^{0}$ matrix

Question

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?

Answer 1

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.