In the script of my Professor it is stated that a general Poincaré transform $U(\Lambda,a)$ acting on a state $|\mathbf{p},s\rangle$ can be written as $$U(\Lambda,a)|\mathbf{p},s\rangle=e^{ia^\mu(\Lambda p_\mu)}\sum_{s'}{D^{(j)}_{ss'}(\omega)|\mathbf{\Lambda p},s\rangle},$$ where he denoted that $D^{(j)}_{ss'}(\omega)$ is called the Wigner function for $\omega$. He doesn't elaborate much further on it and thus I would like to read more on it in literature. Unfortunately, I found no mention of this function in neither Matthew D. Schwartz's QFT and the Standard Model nor M. Peskin & D. Schroeder's Introduction to QFT. Furthermore, there's some ambiguity with Wigner's quasiprobability distribution, making it increasingly difficult for me to find it on Google. If you know any textbook or online resource where this is explained well, I would be happy if you could share the name or link to this resource!
1 Answers
For some reason which I do not fully understand, many standard QFT textbooks such as Schwartz' or Peskin's do not give a detailed description of the unitary representations of the universal cover of the Poincaré group. They often discuss the non-unitary finite-dimensional representations of the spin group (the double cover of the Lorentz group), which are relevant for classifying fields, but not the infinite-dimensional unitary ones of the universal cover of the Poincaré group which are relevant for describing relativistic particle states.
That said, the best reference by far IMHO is Weinberg's The Quantum Theory of Fields, Volume 1, Chapter 2. In that chapter Weinberg derives the unitary irreducible representations of the Poincaré group in detail and then it becomes clear where these Wigner $D$ matrices come from, how they are defined, etc.
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