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I am studying my first QFT course, and there seems to be a lot that I was not taught in previous courses. In my first assignment, I have to prove several properties about the Poincare group, but I have barely any knowledge about them. I looked at the book "Group Theory in a Nutshell for physicists", by A. Zee, but I did not find anything on Poincare transformations. Two examples of the kind of questions I wish to be able to answer are:

  1. Determine the inverse and unity element of the Poincare group, as well as the multiplication rule $(\Lambda_1,a_1)(\Lambda_2,a_2)$

  2. Show that the traslation generators commute, $[P^{\mu},P^{\nu}]=0$

What book/paper/resource would be a good place for me to start studying this subject?

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    $\begingroup$ Try Wu-Ki Tung's Group Theory in Physics. It's basically designed precisely to give this kind of background. $\endgroup$ – knzhou Aug 31 at 0:45
  • $\begingroup$ The first few chapters of Weinberg's QFT should do the trick. $\endgroup$ – AccidentalFourierTransform Aug 31 at 0:56
  • $\begingroup$ Poincaré transformations are just Lorentz transformations (Lorentz boosts combined with spatial rotations) combined with spacetime translations. From the Poincaré transformation $x’=\Lambda x+a$ you can figure out the identity element, the inverse, and the composition rule. No book necessary! $\endgroup$ – G. Smith Aug 31 at 4:19
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Given the suggestions, the two books I found most useful were:

1- Wu-Ki Tung's Group Theory in Physics

2-Weinberg's Quantum Field Theory

I Highly recommend both of them if you encounter the same problems as me.

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