- #1

- 104

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$$\phi(x)\longmapsto \exp\left(\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\right)\phi(x),$$

where ##\omega## is anti-symmetric and ##S^{\mu\nu}## satisfy the commutation relations of the Lorentz group, namely

$$\left[S_{\mu \nu}, S_{\rho \sigma}\right]=-\eta_{\mu \rho} S_{\nu \sigma}+\eta_{\mu \sigma} S_{\nu \rho}-\eta_{\nu \sigma} S_{\mu \rho}+\eta_{\nu \rho} S_{\mu \sigma}.$$

These matrices ##S^{\mu\nu}## are sometimes (in my lecture notes for example) called "Spin matrices" but I'm having a hard time associating the existence and their specific form (f.e. for a scalar field ##S^{\mu\nu}=0## and for a bispinor ##S^{\mu\nu}=\frac{1}{4}[\gamma^\mu,\gamma^\nu]##, where the ##\gamma##-matrices satisfy the Clifford algebra, etc.) with the actual spin of the particle. I've already been pointed out to [1], but I honestly don't really see how this is related to the question (since there we construct a projector that gives us a spinor pointing into a specific direction, instead of explaing what this has to do with the transformation matrices at all).

I've already taken a look into Perskin & Schroeders and Bjorken & Drell's book, but I'm having a hard time finding the passage that explains this connection... I obviously don't expect anyone to give me an incredibly long and complicated explanation if this should be necessary, I'd already be happy if someone could point me to a specific chapter/subsection of a book that discusses this.

[1] https://en.wikipedia.org/wiki/Dirac...spinor_with_a_given_spin_direction_and_charge