Well, one of course has to make a distinction between the physical thing and the mathematical representation of said thing. The set of states which model a type

Well, one of course has to make a distinction between the physical thing and the mathematical representation of said thing. The set of states which model a type of elementary particle furnish an irreducible representation of the appropriate group. Folks usually (somewhat imprecisely, but physicists aren't generally known for their pedantry) say that this is 'defining a particle'. The idea gets across, but for pedagogical purposes, being more precise is better, I'd reckon; thank you for bringing this to our attention.