Spin–statistics assumptions and consequences

A standard Minkowski field-theoretic setting uses a positive-definite Hilbert space, a unitary representation of the connected Poincaré symmetry or its cover, and an invariant vacuum. The joint energy–momentum spectrum lies in the closed forward cone. Fields are suitably covariant operator-valued distributions on a common dense domain, with regularity and vacuum-cyclicity assumptions in the usual axiomatic formulation. Local commutation or graded commutation supplies causal compatibility rather than following from positivity alone.

Under the appropriate full hypotheses, a nontrivial physical local field cannot freely choose the wrong relation between spin and statistics. A free Dirac calculation displays a failure of positive energy or positive norm under commuting alternatives; a scalar with an inappropriate exchange algebra similarly conflicts with causal requirements. These examples diagnose the tension but are not a proof for arbitrary interacting models. An interacting theory must actually satisfy the assumptions, not merely have a classical Lagrangian that looks relativistic.

Once bosonic or fermionic ladder relations are fixed, exchange of two different creators produces the corresponding plus or minus sign. For fermions, duplicate creation of the same complete mode vanishes. Bosons permit arbitrary mode occupation, with square-root factors needed to normalize repeated creation. The mode label includes spin and every other one-particle quantum number, so two opposite-spin fermions in one spatial orbital do not violate exclusion.

The minus sign from a single exchange is a global phase if it changes only the naming convention for one state. It becomes physically significant when amplitudes for different exchange histories interfere, generating familiar differences in correlations and occupation counting. A spinor's minus sign under a full spatial rotation is consistent with the theorem, but that geometric observation by itself is not its relativistic derivation. The connection requires relating transformations, locality, and positivity in the full field theory.

Gauge-fixed calculations can introduce anticommuting scalar ghosts and an auxiliary space that lacks the required physical positivity. Such variables help implement constraints and cancel unphysical contributions; they are not observable scalar fermions violating the physical theorem. A physical state space or cohomology must be identified before applying the spin–statistics conclusion. Gauge-dependent potentials also need care because a component count is not a count of physical particle polarizations.

In two spatial dimensions, exchange paths can have braid topology and permit anyonic sectors under an appropriate framework. Nonrelativistic models also need not derive statistics from relativistic causality in the same manner. These observations mark the theorem's scope rather than undoing the ordinary three-spatial-dimensional result. Likewise, constructing finite-mode Fock spaces is elementary compared with proving existence of an interacting continuum model, its local observables, positive spectrum, and the required symmetry representation.