Spin-statistics theorem

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:

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.

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.

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