quantum mechanics - Showing Dirac equation's Lorentz invariance and use of unitary matrix $U$

Dirac equation is $i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0$

To show its Lorentz invariance, we convert spacetime into $x'$ and $t'$ from $x$ and $t$ and then

$( iU^\dagger \gamma^\mu U\partial_\mu^\prime - m)\psi(x^\prime,t^\prime) = 0$

The question is, how does one show from the above equation the following equation follows?:

$U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi(x^\prime,t^\prime) = 0$

where $U$ is some unitary matrix for lorentz transformation for $\psi$.

Answer

Actually, "unitary representation" is meant with respect to the spinors, which do not form a finite-dimensional space and therefore allow a unitary representation of the proper Lorentz group. The action is defined by $D(\Lambda)\psi(x)=U(\Lambda)\psi(\Lambda^{-1}x)$, and you can simply calculate that this is unitary on your spinor space. However, this does not(!) mean that the matrix $U$ is actually unitary. Therefore I also assume, that you mean $U^{-1}$ instead of $U^\dagger$.

To your problem: Just notice that $U$ is actually a matrix constant with respect to $x'$, therefore it commutes with $\partial'_\mu$, and therefore you have $(iU^{-1}\gamma^\mu U\partial'_\mu-m)=(iU^{-1}\gamma^\mu \partial'_\mu U-U^{-1} mU)=U^{-1}(i\gamma^\mu \partial'_\mu-m)U$.