Adjoint Spinor

Because the combination $\psi^\dagger\gamma_0$ occurs so often, we define

$$\overline{\psi}(x) \equiv \psi^\dagger\gamma_0,$$ (5.162)

where $\overline{\psi}(x)$ is the adjoint spinor.

Its Lorentz transformation properties are

$$\psi^\prime(x^\prime) = S\psi(x)$$

$$\psi^{\prime^\dagger}(x^\prime) = \psi^\dagger(x)S^\dagger = \psi^\dagger(x)\gamma_0S^{-1}\gamma_0$$

$$\psi^{\prime^\dagger}(x^\prime)\gamma_0 = \overline{\psi}^\prime(x^\prime) = \overline{\psi}(x) S^{-1}.$$ (5.163)