I'm trying to understand better the idea of the standard model, where particle states are described within vector spaces corresponding to irreducible representations of the group of symmetry of physics (e.g. the Poincaré group, or some covering of it).
My question is simple:
Why do we only consider linear representations of the underlying symmetry group?
I know that the maths are much simpler when studying linear reps. because we know a lot about linear algebra, but why would the physical "reality" be described in a vector space?
Are we implicitly studying only a first order approximation of that reality?