The spin–statistics theorem proves that the observed relationship between the intrinsic spin of a particle (angular momentum not due to the orbital motion) and the quantum particle statistics of collections of such particles is a consequence of the mathematics of quantum mechanics. In units of the reduced Planck constant ħ, all particles that move in 3 dimensions have either integer spin and obey Bose-Einstein statistics or half-integer spin and obey Fermi-Dirac statistics.^[1][2]

All known particles obey either Fermi-Dirac statistics or Bose-Einstein statistics. A particle's intrinsic spin always predicts the statistics of a collection of such particles and vice versa:^[3]

• Integral spin particles are bosons with Bose-Einstein statistics
• Half-integral spin particle are fermions with Fermi-Dirac statistics.

A spin-statistics theorem shows that the mathematical logic of quantum mechanics predicts or explains this physical result.^[4]

The statistics of indistinguishable particles is among the most fundamental of physical effects. The Pauli exclusion principle -- that every occupied quantum state contains at most one fermion -- controls the formation of matter. The basic building blocks of matter such as protons, neutrons, and electrons are all fermions. Conversely, particles such as the photon, which mediate forces between matter particles, are all bosons.^{[citation needed]} A spin-statistics theorem attempts explain the origin of this fundamental dichotomy.^[5]: 4

Naively, spin, an angular momentum property intrinsic to a particle, would be unrelated to fundamental properties of a collection of such particles. However, these are indistinguishable particles: any physical prediction relating multiple indistinguishable particles must not change when the particles are exchanged.

An elementary explanation for the spin-statistics theorem cannot be given despite the fact that the theorem is so simple to state. In the Feynman Lectures on Physics, Richard Feynman said that this probably means that we do not have a complete understanding of the fundamental principle involved.^[3]

Numerous notable proofs have been published, with different kinds of limitations and assumptions. They are all "negative proofs", meaning that they establish that integral spin fields cannot result in fermion statistics while half-integral spin fields cannot result in boson statistics.^[5]: 487

Proofs that avoid using any relativistic quantum field theory mechanism have defects. Many such proofs rely on a claim that

$${\displaystyle |\psi (\alpha _{1},\alpha _{2},\alpha _{3},...)|^{2}=|{\hat {P}}\psi (\alpha _{1},\alpha _{2},\alpha _{3},...)|^{2}}$$

where the operator ${\displaystyle {\hat {P}}}$ permutes the coordinates. However, the value on the left-hand-side represents the probability of particle 1 at ${\displaystyle r_{1}}$, particle 2 at ${\displaystyle r_{2}}$, and so on, and is thus quantum mechanically invalid for indistinguishable particles.^[6]: 567

The first proof was formulated^[7] in 1939 Markus Fierz, a student of Wolfgang Pauli, and was rederived in a more systematic way by Pauli the following year.^[8] In a later summary, Pauli listed three postulates within relativistic quantum field theory as required for these versions of the theorem:

1. Any state with particle occupation has higher energy than the vacuum state,
2. Spatially separated measurements do not disturb each other (they commute),
3. Physical probabilities are positive (the metric of the Hilbert space is postive definite).

Their analysis neglected particle interactions other than commutation/anti-commutation of the state.^{[9][5]: 374}

In 1949 Richard Feynman gave a completely different type of proof^[10] based on vacuum polarization which was later critiqued by Pauli.^{[9][5]: 368} Pauli showed that Feynman's proof explicitly relied on the first two postulates he used and implicitly used the third one by first allowing negative probabilities but then rejecting field theory results with probabilities greater than one.

A proof by Julian Schwinger in 1950 based on time-reversal invariance^[11] followed a proof by Frederik Belinfante in 1940 based on charge-conjugation invariance, leading to a connection to the CPT theorem more fully developed by Pauli in 1955.^[12] These proofs where notably difficult to follow.^[5]: 393

Work on the mathematical foundations of quantum mechanics by Arthur Wightman lead to a theorem that stated that the expectation value of the product of two fields, ${\displaystyle \phi (x)\phi (y)}$, could be analytically continued to all separations ${\displaystyle (x-y)}$.^[5]: 425 (The first two postulates of the Pauli era proofs involve the vacuum state and fields at separate locations.) The new result allowed more rigorous proofs of the spin-statistics theorems by Gerhart Luders and Bruno Zumino^[13] and by Burgoyne.^[5]: 393 In 1957 Res Jost derived the CPT theorem using the spin-statistics theorem and Burgoyne's proof the spin-statistics theorem in 1958 required no constraints on the interactions nor on the form of the field theories. These results are among the most rigorous practical theorems.^[14]: 529

In spite of these successes, Feynman, in his 1963 undergraduate lecture that discussed the spin-statistics connection, says: "We apologize for the fact that we cannot give you an elementary explanation."^[3] Neuenschwander echoed this in 1994, asking if there was any progress^[15] spurring additional proofs and books.^[5] Neuenschwander's 2013 popularization of the spin-statistics connection suggested that simple explanations remain elusive.^[16]

In 1987 Greenberg and Mohaparra proposed that the spin statistics theorem could have small violations.^[17][18] With the help of very precise calculations for states of the He atom that violate the Pauli exclusion principle,^[19] Deilamian, Gillaspy and Kelleher^[20] looked for the 1s2s ¹S₀ of He using an atomic beam spectrometer. The search was unsuccessful with an upper limit of 5×10⁻⁶.

The Lorentz group has no non-trivial unitary representations of finite dimension. Thus it seems impossible to construct a Hilbert space in which all states have finite, non-zero spin and positive, Lorentz-invariant norm. This problem is overcome in different ways depending on particle spin–statistics.

For a state of integer spin the negative norm states (known as "unphysical polarization") are set to zero, which makes the use of gauge symmetry necessary.

For a state of half-integer spin the argument can be circumvented by having fermionic statistics.^[21]

In 1982, physicist Frank Wilczek published a research paper on the possibilities of possible fractional-spin particles, which he termed anyons from their ability to take on "any" spin.^[22] He wrote that they were theoretically predicted to arise in low-dimensional systems where motion is restricted to fewer than three spatial dimensions. Wilczek described their spin statistics as "interpolating continuously between the usual boson and fermion cases".^[22] The effect has become the basis for understanding the fractional quantum hall effect.^[23][24]

• Parastatistics
• Anyonic statistics
• Braid statistics