Boson half-integer spin particles cannot be bosons and integer spin particles cannot be fermions. In large systems, the difference between bosonic and fermionic statistics is only apparent at large densities — when their wave functions overlap. At low densities, both types of statistics are well approximated by Maxwell–Boltzmann statistics, which is described by classical mechanics. All observed elementary particles are either fermions or bosons. The observed elementary bosons are all gauge bosons: photons, W and Z bosons, gluons, except the Higgs boson which is a scalar boson. Finally, many approaches to quantum gravity postulate a force carrier for gravity, the graviton,
Exchange operator Such motion is often not carried out in practice. Rather, the operation is treated as a "what if" similar to a parity inversion or time reversal operation. Consider two repeated operations of such a particle exchange: Therefore, formula_1 is not only unitary but also an operator square root of 1, which leaves the possibilities Both signs are realized in nature. Particles satisfying the case of +1 are called "bosons", and particles satisfying the case of −1 are called "fermions". The spin–statistics theorem dictates that all particles with integer spin are bosons whereas all particles with half-integer spin are fermions. The