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In quantum mechanics, the spin–statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum (that is, the contribution to the total angular momentum that is not due to the orbital motion of the particle). All particles have either integer spin or half-integer spin (in units of the reduced Planck constant ''ħ''). The theorem states that: * The wave function of a system of identical integer-spin particles has the same value when the positions of any two particles are swapped. Particles with wave functions symmetric under exchange are called ''bosons''. * The wave function of a system of identical half-integer spin particles changes sign when two particles are swapped. Particles with wave functions antisymmetric under exchange are called ''fermions''. In other words, the spin–statistics theorem states that integer-spin particles are bosons, while half-integer–spin particles are fermions. The spin–statistics relation was first formulated in 1939 by Markus Fierz and was rederived in a more systematic way by Wolfgang Pauli. Fierz and Pauli argued their result by enumerating all free field theories subject to the requirement that there be quadratic forms for locally commuting observables including a positive-definite energy density. A more conceptual argument was provided by Julian Schwinger in 1950. Richard Feynman gave a demonstration by demanding unitarity for scattering as an external potential is varied, which when translated to field language is a condition on the quadratic operator that couples to the potential. == General discussion == In a given system, two indistinguishable particles, occupying two separate points, have only one state, not two. This means that if we exchange the positions of the particles, we do not get a new state, but rather the same physical state. In fact, one cannot tell which particle is in which position. A physical state is described by a wavefunction, or – more generally – by a vector, which is also called a "state"; if interactions with other particles are ignored, then two different wavefunctions are physically equivalent if their absolute value is equal. So, while the physical state does not change under the exchange of the particles' positions, the wavefunction may get a minus sign. Bosons are particles whose wavefunction is symmetric under such an exchange, so if we swap the particles the wavefunction does not change. Fermions are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the Pauli exclusion principle: two identical fermions cannot occupy the same state. This rule does not hold for bosons. In quantum field theory, a state or a wavefunction is described by field operators operating on some basic state called the ''vacuum''. In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law. The operator : (with an operator and a numerical function) creates a two-particle state with wavefunction , and depending on the commutation properties of the fields, either only the antisymmetric parts or the symmetric parts matter. Let us assume that and the two operators take place at the same time; more generally, they may have spacelike separation, as is explained hereafter. If the fields commute, meaning that the following holds: :, then only the symmetric part of contributes, so that , and the field will create bosonic particles. On the other hand, if the fields anti-commute, meaning that has the property that : then only the antisymmetric part of contributes, so that , and the particles will be fermionic. Naively, neither has anything to do with the spin, which determines the rotation properties of the particles, not the exchange properties. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Spin–statistics theorem」の詳細全文を読む スポンサード リンク
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