Lorentz transformation of Dirac spinor

Question

I'm wondering again what I'm missing in my understanding. In Peskin and Schroeder, as well as in other sources, the spinor representation of Lorentz transformation is given by $$\Lambda_\frac{1}{2}=exp(-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu})$$. Where $\omega_{\mu\nu}$ is antisymmetric tensor representing the Lorentz transformation and $S^{\mu\nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu]$. In the same sources they write that $$\Lambda^\dagger_\frac{1}{2}=exp(\frac{i}{2}\omega_{\mu\nu}(S^{\mu\nu})^\dagger)$$. The question, using the matrix identity $(AB)^T=B^TA^T$ shouldn't we have $$\Lambda^\dagger_\frac{1}{2}=exp(\frac{i}{2}(S^{\mu\nu})^\dagger\omega_{\mu\nu})$$?

Answer 1

$\omega_{\mu\nu}$ are simply real numbers, while $S^{\mu\nu}$ are matrices that act on the Dirac spinors. The components of the matrix $S^{\mu\nu}$ would be $(S^{\mu\nu})_{\alpha\beta}$ where $\alpha$ and $\beta$ are the "Dirac indices". I.e. the components of the spinor $S^{\mu\nu} \psi(x)$ are $(S^{\mu\nu} \psi(x))_\alpha = (S^{\mu\nu})_{\alpha\beta} \psi_\beta(x)$.

In general the components $(S^{\mu\nu})_{\alpha\beta}$, $(\gamma^\mu)_{\alpha\beta}$, $\eta_{\mu\nu}$ etc. are all just real numbers and therefore commute with each other. The exception is the components $\psi_\alpha$, which anticommute because they are fermion fields: $\psi_\alpha \psi_\beta = -\psi_\beta \psi_\alpha$ and $\{\psi_\alpha(\vec{x}), \psi_\beta^\dagger(\vec{y})\} = \delta_{\alpha\beta} \delta^{(3)}(\vec{x} - \vec{y})$. Even in the classical limit, they are anticommutative Grassmann numbers: $\psi_\alpha \psi_\beta = -\psi_\beta \psi_\alpha$ and $\psi_\alpha \psi_\beta^\dagger = -\psi_\beta^\dagger \psi_\alpha$

Answer 2

Expanding the exponential:

\begin{align} Λ = e^{−\frac{i}{2}ω_{μν}S^{μν}} = e^{−\frac{i}{2}(ω_{01}S^{01} + ω_{02}S^{02}...+ ω_{32}S^{32})} \end{align}

$ω_{0i}$ and $ω_{i0}$ are infinitesimal angles for boost (so that $cosh(ω_{0i}) = \gamma$, and $sinh(ω_{0i}) = v_i\gamma$ for finite boosts) and $ω_{ij}$ are infinitesimal angles for rotations. Just real numbers.