Hermitian properties of the gamma matrices

[spaghetti3451]

The gamma matrices $\gamma^{\mu}$ are defined by

$$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}.$$

---

There exist representations of the gamma matrices such as the Dirac basis and the Weyl basis.

---

Is it possible to prove the relation

$$(\gamma^{\mu})^{\dagger}\gamma^{0}=\gamma^{0}\gamma^{\mu}$$

without alluding to a specific representation?

 

[vanhees71]

I don't think so, because given any set of matrices fulfilling the anti-commutation relations, any other set
$$\tilde{\gamma}^{\mu} = \hat{A} \gamma^{\mu} \hat{A}^{-1},$$
where $\hat{A}$ is an arbitrary $\mathbb{C}^{4 \times 4}$ matrix also fulfills them. It's of course more natural to use a simple set of matrices as suggested by the representation theory of the Lorentz group behind the bispinor representation, e.g., the chiral (or Weyl) representation. The pseudohermiticity relations, are only preserved with $\hat{A}$ unitary.