My question is that is the fact, that all particles are either boson or fermions, only an empirical fact, or can it be argued theoretically too. The reason why I'm asking this is that I have not encountered anyone stating honestly, that it is an empirical fact only. But on the other hand, I have not encountered a theoretical proof either. An most importantly, I have encountered one incorrect theoretical proof repeatedly. The incorrect proof goes like this. If the Hamiltonian commutes with the particle swapping operator [itex]\psi(x_1,x_2)\mapsto \psi(x_2,x_1)[/itex], then the eigenstates of the Hamiltonian must be eigenstates of the swapping operator. But possible eigenvalues of the swap are only [itex]\pm 1[/itex]. The truth is that the eigenstates of the Hamiltonian don't need to be eigenstates of the swapping operator. The Hamiltonian will be degenerate, and an eigenstate of the Hamiltonian can be written as a linear combination of other eigenstates, which are common eigenstates of the Hamiltonian and the swapping operator.