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Is it correct for me to say that the symmetry of a quantum system with respect to the Poincare group leads to the concept of mass and spin?

The postulates of the special theory of relativity demand that the Poincare group must be a symmetry group of any closed physical system (one postulate says that inertial reference frames are related by Poincare transformations and the other says that the laws of physics are the same in any inertial reference frame).

If we are considering a single free particle then the physical states of the system are represented as state vectors in a Hilbert space (actually as rays). Wigners theorem says that a symmetry transformation of a system is (at least) represented by a unitary operator acting on the Hilbert space. Thus we are lead to classifying the (projective) unitary irreducible representations (UIR's) of the Poincare group (or equivalently, its universal cover).

It is well known that the UIR's of the Poincare group (in $D=4$) are labelled by two numbers; mass and spin (at least in the $m>0$ case). This is the group theoretic origin of the mass and spin of an elementary particle.

It seems to me, that all of this comes about from the special theory of relativity; by demanding Poincare invariance. So is it correct to say that Symmetry considerations lead to concept of mass and spin in a quantum system?

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    $\begingroup$ $\uparrow$ Yes. $\endgroup$ – AccidentalFourierTransform Oct 26 '17 at 11:25
  • $\begingroup$ Oh, well that's surprising... I was half expecting somebody to tell me that I am misinterpreting something... Thanks! $\endgroup$ – SigmaAlpha Oct 26 '17 at 11:31
  • $\begingroup$ Actually the numbers are three: there is also the sign of $P^0$ that is Poincaré invariant and one considers physically sensible those representations with $P^0>0$ among these with $m>0$. The classification of the case $m=0$ is more complicated... $\endgroup$ – Valter Moretti Oct 26 '17 at 11:50
  • $\begingroup$ You'd think after all of these questions I should be past this stuff by now... but the more deeper I try to understand it, the more questions it raises. I guess it just really fascinates me. $\endgroup$ – SigmaAlpha Oct 26 '17 at 12:03
  • $\begingroup$ Please mind that questions asking whether you are correct are not really a good fit for the SE model if you are indeed correct since "Yes" is too short to even submit as an answer. Please try to ask questions in a way that an answer can actually contain meaningful information. $\endgroup$ – ACuriousMind Oct 26 '17 at 12:18

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