# Deriving the Dirac equation & Dirac matrices

In pg. 20 of the Lectures on Quantum Field Theory book by Ashok Das, pg.20, the author starts with the energy momentum relation

$$p^\mu p_\mu=m^2$$ in matrix form and then defines the matrix square root of $$p_\mu p^\mu$$ as $$\overline{p}=\gamma^\mu p_\mu$$ where $$\overline{p}^2=p_\mu p^\mu$$. It was also stated that $$\gamma^\mu$$ are four linearly independent matrices.

He then wrote the following relationship for the $$\gamma^\mu$$ matrices in eqn. 1.78: $$\gamma^\mu\gamma^\nu={1\over 2}(\gamma^\mu \gamma^\nu+\gamma^\nu\gamma^\mu).$$

How did he arrive at this relationship? In general, linearly independent matrices don't commute, so $$\gamma^\mu \gamma^\nu \neq \gamma^\nu \gamma^\mu$$.

I'm think this relationship holds for Dirac matrices, but the author uses this relationship to later on derive the Dirac matrices. Isn't that a circular argument?

I assume that you've made this conclusion from the derivations in equation 1.78 in the cited book, which proceed as follows

$$\gamma^\mu \gamma^\nu p_\mu p_\nu = p^2 \mathbf{1} \Rightarrow \\ \Rightarrow \frac{1}{2}(\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu) p_\mu p_\nu = p^2 \mathbf{1}$$

This does not imply that $$\gamma^\mu \gamma^\nu = \frac{1}{2}(\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu)$$ (which, as you've already noticed, would in turn imply $$\gamma^\mu \gamma^\nu = \gamma^\nu \gamma^\mu$$, which is false).

Instead, what the author does is take the identity in the first row with indices swapped $$\nu \leftrightarrow \mu$$, which reads

$$\gamma^\nu \gamma^\mu p_\nu p_\mu = p^2 \mathbf{1}$$

and adds it to the original identity:

$$\gamma^\mu \gamma^\nu p_\mu p_\nu + \gamma^\nu \gamma^\mu p_\nu p_\mu = 2p^2 \mathbf{1}$$

Now, use the fact that $$p_\mu$$ and $$p_\nu$$ commute:

$$(\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu) p_\mu p_\nu = 2p^2 \mathbf{1}$$ $$\frac{1}{2}(\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu) p_\mu p_\nu = p^2 \mathbf{1}$$

• Thank a lot! I'll accept this answer when I can. Dec 16 '20 at 7:47
• @TaeNyFan Glad to be of help! Good luck on your QFT path. Dec 16 '20 at 8:04