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The one-particle states in the Hilbert space of a quantized relativistic field theory are said to form irreducible representations of the Poincaré group. Why is that? I mean, popular texts in QFT do not explicitly construct any representation but simply state that one-particle states are representations. Is this so obvious? If not, how can one understand/ensure that they indeed form irreducible representation of the Poincaré group?

EDIT: Moreover, one-particle states are supposed to be the irreducible representations of Poincaré group. Does it mean that any representation which is labelled by unique values of Casimir invariants are irreducible?

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    $\begingroup$ The general idea is that any representation of the Poincare' group can be labelled with a pair $(m,s)$, with each of the two variables undergoing some particular conditions that lead us to read them as mass and spin. The complete argument is more complex, but that is somehow where one starts from. $\endgroup$ – gented Dec 18 '15 at 14:51
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/73593/2451 , physics.stackexchange.com/q/65839/2451 and links therein. $\endgroup$ – Qmechanic Dec 18 '15 at 17:02
  • $\begingroup$ @Gennaro Tedesco- do you mean that since the one-particle states are labeled by eigenvalues of the casimir operators, they belong to the representation of the Poincare group? $\endgroup$ – SRS Dec 18 '15 at 17:15
  • $\begingroup$ Yes, I do (the entire argument calls in the Casimir operators, as you pointed out). $\endgroup$ – gented Dec 18 '15 at 17:21
  • $\begingroup$ Note that in some sources fields which represents explicitly irreducible representations of the Poincare group are constucted explicitly. $\endgroup$ – Name YYY Dec 20 '15 at 8:36
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This is answered in depth in Weinberg's book on quantum field theory (Vol. I, Chapter 2).

Relativistic invariance means translation invariance and Lorentz invariance, hence - obviously - Poincare invariance, so that one has a representation of the Poincare group. Because of relativistic invariance and unitarity, the Hilbert space of a QFT carries a unitary representation of the Poincare group, and it splits (as any unitary representation) into a direct sum of irreducible ones. Being irreducible means being not further divisible, hence elementary. One can classify them, and finds that they describe single relativistic particles, hence elementary particles.

Irreducible representations have constant Casimirs, but the values of the constants do not always characterize the irrep. In particular, all massless irreducible representations of the Poincare group have the same values for the Casimirs but may differ in helicity.

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