Can the definition of particles in QFT as states that transform under irreducible unitary representations of the Poincare group be extended to solids?

Question

I've been reading Matthew D. Schwartz's book on Quantum Field Theory and the Standard Model and on page 110 he states the following:

Particles transform under irreducible unitary representations of the Poincare group.

This statement can even be interpreted as the definition of what a particle is.

I am aware that in condensed matter physics quasiparticles can be loosely defined as diagonal matrix representations of the terms in a Hamiltonian, i.e. terms expressible as

$$\mathcal{H}\propto\sum_p\epsilon\left(p\right)a^\dagger_p a_p.$$

My question is, whether the aforementioned definition can be extended to the definition of a quasiparticle, i.e. if a statement s.a.

Quasiparticles transform under some representation of the space group of a solid.

is true or whether a somewhat similar statement exists.

Answer 1

(Dislcaimer: a view from non-QFT angle)
If we see the particles as excitations from the ground state, the symmetry group determines which excitations are possible and which are not... or perhaps more precisely, which excitations are (excited) eigenstates of the field. Then the Bloch waves, phonons, and some other quasiparticles in solid state can be seen as arising from the crystal symmetry group. The difference is that in QFT the ground state is called "vacuum" and contains no particles (in QFT sense, let's call the "fundamental particles"), whereas in a solid state the ground state is actually a state with all the particles (in QFT sense, i.e., the fundamental particles) still present in the ground state, and the excitations are called quasiparticles (although often, after applying effective mass approximation, it is forgotten that an electron in a band is not a real electron.) However, the particles constituting the ground state cannot be destroyed or created - at least not as long as the solid state is treated in non-relativistic limit.

Things however get more complicated when interactions are taken into account, and one introduces new quasiparticles as excitations of the interacting many-body system - plasmons, excitons, Landau quasiparticles. These also depend on the symmetries of the interactions, so they depart from the basic QFT picture; moreover (although I assume that QFT has and equivalent of those, although perhaps known under a different name.)

Something to ponder is that particles arising in interacting solid state typically have finite lifetimes, but not in the same sense as in QFT - rather than transforming into other particles, they "spread" and stop behaving as distinct particles (which perhaps can be traced to them not satisfying the crystal symmetries.)

It is also interesting to note that it is for the treatment of these latter excitations that solid state physics mostly has borrowed the QFT methods - these are then typically applied at the level when the crystal symmetries are already ignored (e.g., electrons and holes in effective mass approximation), and many of the QFT symmetries are restored (notably rotations and translations, as well as gauge transformations for potentials.) Yet, often these treatments are non-relativistic, although in graphene or strongly correlated systems they may actually aspire to the full symmetry of Poincaré group (or at least symmetry in 1+1 dimensions).

Related:
Is differentiating particle and quasiparticle meaningless?
Electrons and holes vs. Electrons and positrons
Particle/hole excitations have finite lifetime
If a non-interacting particle behaves like an undamped wave, can an interacting particle behave like a damped wave?