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I've been studying Talagrand's What is a Quantum Field Theory? lately and I have some questions regarding the scheme he presents.

Essentially the state of affairs as of where I am in the book is that if one wants to model a certain type of particle, one finds a suitable representation of the Poincaré group (and all the double cover/ including parity versions). I, however, cannot help but to feel that this is all a bit misterious, about the fact that this works in itself, and about the choice of the representation as well. Talagrand's approach is just to state what the representation is, but I wonder how these were discovered and what the physical reasoning for them was.

So concretely I ask two things:

  1. Why should different representations encode information about different particles?

I get that representations are the way we associate group elements with observables (operators in some hilbert space), but am still a bit confused about what's going on.

  1. From what I can gather, this perspective was introduced in physics by mainly Weyl and Wigner. Is that so, and can someone tell me how these ideas were born?
Qmechanic
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Lourenco Entrudo
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    Can't you ask the very same question in non-relativistic QM and e.g. about the Galilean group? – Tobias Fünke Feb 19 '24 at 10:45
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    You may want to check Weinberg QFT Vol 1. – Oбжорoв Feb 19 '24 at 12:43
  • @TobiasFünke for the non-rel. QM that I have encountered, the general approach has been to quantize a Hamiltonian and see what we get, where things like spin are added as an ad hoc with historical justification. In some places I have found the standard group theoretic approach (Stone's theorem etc). But that seems to me less misterious than this. That is, the fact that quantum systems should carry a representation of their symmetry group is not surprising to me. What is surprising is that different representations yield different particle types, and how. – Lourenco Entrudo Feb 19 '24 at 13:25

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