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In Weinberg I Ch2. he proceeds to classify irreducible unitary reps of the Poincare group. He then goes on to say that these do not hold for interacting particles. So what is the representation of the Poincare group on interacting particles and how did it escape his original classification?

Interacting particles are subject to external interactions that break the Poincare symmetry. Only the complete system (including all interacting particles) is Poincare invariant. This is described by a highly reducible representation.

What do the irreducible components of this interacting rep look like? So they fall under the classification in the book or is there something I missing?

Weinberg classified the physically relevant irreducible representations, a subset of all (determined by Wigner). But the reducible reps of QFT consist of uncountably many irreducibles, so his classification is quite useless since it tells nothing how these are combined.

The point is that in interactive QFT, one can start perturbatively with Fock spaces made from the irreducibles, and gets after renormalization a sensible perturbation theory. But nonperturbatively one does not have a Fock structure (Haag's theorem). This is the basic reason for the unsolved foundational problems in QFT.

He then goes on to say that these do not hold for interacting particles.

Yes, it only holds for interacting "particles". Those irreducible unitary reps of the Poincare group are interacting particles - they are in fact quasi-particles of compound systems (like normal modes in a solid body, for example). They look as "independent", "free", "non interacting", but they belong to an environement ("sources" and "sinks") to be observable with those quantum numbers. The quasi-particle interaction is small by definition (by their choice), like nearly separated variables of an equation system.

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