A particle's intrinsic spin always predicts the statistics of a collection of such particles and conversely:[3] A spin–statistics theorem shows that the mathematical logic of quantum mechanics predicts or explains this physical result.
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.
[citation needed] A spin–statistics theorem attempts to explain the origin of this fundamental dichotomy.
A pair of distinct state vectors are physically equivalent if they differ only by an overall phase factor, ignoring other interactions.
Additionally, the assumption (known as microcausality) that spacelike-separated fields either commute or anticommute can be made only for relativistic theories with a time direction.
However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained.
A boost transfers to a frame of reference with a different velocity and is mathematically like a rotation into time.
By analytic continuation of the correlation functions of a quantum field theory, the time coordinate may become imaginary, and then boosts become rotations.
This is the Pauli exclusion principle: two identical fermions cannot occupy the same state.
In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law.
a numerical function with complex values) creates a two-particle state with wavefunction
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
[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.
[8] In a later summary, Pauli listed three postulates within relativistic quantum field theory as required for these versions of the theorem: 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.
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.
[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 Peter Burgoyne.
[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.
[5] Neuenschwander's 2013 popularization of the spin–statistics connection suggested that simple explanations remain elusive.
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.
For a state of half-integer spin the argument can be circumvented by having fermionic statistics.
[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".