I'm currently having fun with the Dirac matrices $\gamma^\mu$ which appear in the relativistic Fermion field equation (aka the Dirac equation). They need to satisfy the anticommutation relation $$ \{\gamma^\mu,\gamma^\nu\}=\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu} $$ in order for the Dirac equation ($i\gamma^\mu\partial_\mu-m)\psi=0$ to provide the correct relativistic energy-momentum expression.
Famously, the $\gamma^\mu$ must be at least 4-dimensional matrices, in order to satisfy the required anticommutation relation. However, they are not unique. Indeed, if $\gamma^\mu$ is one possible 4x4 set, and V is a 4x4 invertible matrix, then $\gamma'^\mu = V\gamma^\mu V^{-1}$ is also a possible set (i.e. it too will satisfy the anticommutation relation). This is trivial to prove, since $V^{-1}V =1$.
Now I'm wondering if the converse is true: suppose I have two distinct sets of 4x4 matrices, $\gamma^\mu$ and $\gamma'^\mu$, which both satisfy the anticommutation requirement. Is it then true, that there exists an invertible 4x4 Matrix V, such that $\gamma'^\mu=V\gamma^\mu V^{-1}$? Is there a way to obtain an explicit expression for V in terms of $\gamma'^\mu$ and $\gamma^\mu$?
