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I'm studying QFT with the book "Quantum Field Theory" by Dr. David Tong, and I have difficulties with understanding some used transformations.

On the page 88 the author calculates Hermitian conjugate of representation of Lorentz group using chiral representation of gamma matrices: $S[\Lambda]^\dagger = \exp(\frac{1}{2}\Omega_{\rho \sigma}(S^{\rho \sigma})^\dagger)=\exp(-\frac{1}{2}\Omega_{\rho \sigma}\gamma^0S^{\rho \sigma}\gamma^0)=\gamma^0\exp(-\frac{1}{2}\Omega_{\rho \sigma}S^{\rho \sigma})\gamma^0=\gamma^0S[\Lambda]^{-1}\gamma^0$ as far as $(S^{\mu \nu})^\dagger = - \gamma^0S^{\mu \nu}\gamma^0$,

where $S^{\mu\nu} = \frac{1}{4}[\gamma^\mu,\gamma^\nu]$.

I'm struggling with the equality $\exp(-\frac{1}{2}\Omega_{\rho \sigma}\gamma^0S^{\rho \sigma}\gamma^0)=\gamma^0\exp(-\frac{1}{2}\Omega_{\rho \sigma}S^{\rho \sigma})\gamma^0$, I have no idea how to perform such a transition

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For a matrix, $$ \exp A=1+A+\frac12 A^2+... $$ and the $\Omega_{\rho\sigma}$ are just scalar numbers, so it is nothing but $$ \exp \gamma_0 S\gamma_0 = 1+\gamma_0 S \gamma_0 + \frac12\gamma_0 S \gamma_0 \gamma_0 S \gamma_0 +... $$ where I've omitted the suffix $\sigma\rho$ on $S$, then use $\gamma_0^2=1$.

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