Spin–Statistics Theorem

In the CPT framework, the spin-statistics connection is a theorem derived from the FR phase. Half-integer spin ↔ fermionic statistics, integer spin ↔ bosonic statistics, follows from the parity of the self-linking number.

The standard statement

Particles with half-integer spin are fermions (Fermi–Dirac), integer spin are bosons (Bose–Einstein).

In standard QFT

Proved via the requirement of relativistic causality and positive energy — complicated proof.

CPT derivation: one page

. odd → FR phase −1 → half-integer spin → fermionic. even → FR phase +1 → integer spin → bosonic.

The two cases

odd: at least one of p,q is odd and at least one is odd — includes T(1,1) (pq=1), T(1,3) (pq=3), etc.
even: includes T(2,3) (pq=6), T(2,5) (pq=10), etc.

Why this is non-trivial

The connection between spin and statistics is emergent from a single topological number — no appeal to relativity or causality required.

Limitations

The 8D extension (for quark generations) modifies the classification for higher winding states — see /nuclear/octonion-flavour.

Parity of the self-linking number

Source Documents

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