Exchange operator

In quantum mechanics, the exchange operator is a quantum mechanical operator that acts on states in Fock space. The exchange operator acts by switching the labels on any two identical particles described by the joint position quantum state . Since the particles are identical, the notion of exchange symmetry requires that the exchange operator be unitary.

Construction

In three or higher dimensions, the exchange operator can represent a literal exchange of the positions of the pair of particles by motion of the particles in an adiabatic process, with all other particles held fixed. 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, 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.