I'm learning about how the Dirac equation came about.
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?