Identical particles fermions, which do not share quantum states as described by the Pauli exclusion principle. Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei. The fact that particles can be identical has important consequences in statistical mechanics. Calculations in statistical mechanics rely on probabilistic arguments, which are sensitive to whether or not the objects being studied are identical. As a result, identical particles exhibit markedly different statistical behaviour from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibbs' mixing
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