# Particle as an irreducible representation

I want to get a better understanding on particle being an irreducible representation. Does that mean one particular type of particles (say particle $$A$$) is a subspace of the "total" Hilbert space $$H$$ (which contains all the types?), and the restriction (to the vector space $$V_A$$ corresponds to particle $$A$$) of the representation of the Lorentz transformation group (which govern all the transformation of all types of particles) which has no proper subrepresentation.

Or is the space always stay as $$H$$ no matter what type of the particle we are talking about, then it is just a matter of choosing different representation on $$H$$, and different representations mean different particles? In another word, I am confused what vector space are we talking about if we say particle $$A$$ being an irreducible representation. is it the entire $$H$$? or some subspace $$V_A$$? Because we say spin $$\frac{1}{2}$$ corresponds to the two dimensional representation, I am unsure what is two dimensional.

• Whether the irreducible representation is "the entire $H$" depends on your definition of "entire"! What is the context here? Are we doing non-relativistic QM, Fock spaces in QFT, something else? – ACuriousMind Nov 1 '20 at 12:03
• QFT. I guess the question is "are we sticking with one vector space (which is the so-called "the entire $H$") and choosing different representation (on the same vector space) or we implicitly change the vector space when talking about another type of particles". – Tea_de Nov 1 '20 at 12:12
• What is the difference between "changing the representation" and "changing the vector space", given that all the infinite-dimensional Hilbert spaces featuring in QFT as unitary representations of the Lorentz group are isomorphic as vector spaces anyway? This recent question seems relevant. – ACuriousMind Nov 1 '20 at 12:22

Wigner built on this intuition and classified all the unitary, positive energy, discrete mass representations of the Poincare group. For massive particles of spin $$j$$ in 4d, you can build these representations by taking the direct sum of a whole bunch of copies of the spin $$j$$ irrep of $$SU(2)$$ (thought of as the double cover of the rotation group). The direct sum is over all the momentum states with a given invariant mass. For spin $$1/2$$, the $$SU(2)$$ irrep is $$2$$-dimensional, but the Poincare irrep is a sum of infinitely many copies of this, one for each momentum vector you can reach by Poincare actions. So the Poincare irrep is infinite-dimensional.