Consider a system of N identical particles with hamiltonian
Indistinguishability of the particles means that and all measurable quantities are invariant under permutation of the particle labels. The eigenstates of then must be representations of the permutation group. According to the spin-statistics theorem in quantum field theory, particles with integral spins belong to the symmetric representation and are called bosons. Particles with half-integral spins belong to the anti-symmetric representation and are called fermions.