In ordinary quantum mechanics there are two related notions of an idealized elementary particle state of the system. The first one is the (often improper) energy eigenstates associated to the free particle Hamiltonian operator (Laplacian), and the second (due to Wigner) is the unitary irreducible representations of the Poincare group on the system's Hilbert space. These two definitions are equivalent thanks to the well known fact from classical harmonic analysis that the improper eigenfunctions of the Laplacian are generalized matrix elements of the Wigner representations. In curved spacetimes and with more general Hamiltonians the picture becomes much more intricate. In homogeneous cosmology one has an isometry group acting transitively on spatial hypersurfaces, which gives rise to the Wigner-type notion of an instantaneous symmetry elementary particle state. On the other hand, if the system admits a dynamical formulation than Hamiltonians at different times commute and possess a common system of energy eigenstates, i.e., dynamical elementary particle states. Now the relation between the two notions runs into an open problem in non-commutative harmonic analysis. The situation is even more perplexed for quantum fields in homogeneous cosmological spacetimes. In this talk we will discuss the mathematical problems arising on this way, some conjectures and recent advancements.