Analyticity, CPT, and Spin–Statistics

CPT and spin–statistics share an analytic backbone—positive-energy tube analyticity, complex Lorentz covariance, and locality at Jost points—but they are different theorems. CPT produces an antiunitary spacetime-reflection symmetry and a reversed-field correlator identity. Spin–statistics constrains the spacelike exchange law of a nontrivial field from its Lorentz representation and Hilbert-space positivity. Neither conclusion follows from analyticity alone.

Required background. Positivity, spectrum, covariance, and locality separates the common hypotheses; tube analyticity supplies holomorphic continuation; and Jost points, edge-of-the-wedge, and locality supplies the ordering identities.

Helpful background. CPT hypotheses, content, and limits and the spin–statistics connection give the physical interpretations.

The shared analytic mechanism

For Wightman fields in four-dimensional Minkowski spacetime, the chain is:

$$\text{spectrum support} \Longrightarrow \text{tube holomorphy} \Longrightarrow \text{complex Lorentz continuation} \Longrightarrow \text{identities at Jost points}.$$

The complex Lorentz group contains a transformation connected in the complex group to the identity whose action on real vectors is total inversion $x\mapsto-x$. This is not an element of the real proper orthochronous Lorentz group, but covariance can reach it through the extended tube. Local commutativity then reverses a string of fields at a Jost configuration. Edge-of-the-wedge and uniqueness of analytic continuation extend the identity, and distributional boundary values return it to real spacetime. This is the core of the Wightman CPT proof in Streater and Wightman 2016, §§ 4-1–4-3, pp. 134–145.

For spin–statistics, one compares two-point boundary values related by spacelike exchange and complex Lorentz transformations. The Lorentz representation supplies the sign acquired by the relevant continuation; positivity fixes which exchange sign is compatible with a nonzero norm. The detailed Wightman argument and its hypotheses are in Streater and Wightman 2016, § 4-4, pp. 146–160.

Different conclusions

For a neutral scalar, CPT gives an antiunitary operator $\Theta$ with

$$\Theta\Omega=\Omega, \qquad \Theta U(a,\Lambda)\Theta^{-1}=U(-a,\Lambda), \qquad \Theta\phi(x)\Theta^{-1}=\eta\,\phi(-x),$$

where the intrinsic phase is constrained by the field and $\Theta$ is antilinear. Charged fields are mapped to conjugate fields, and spinorial/tensor fields carry the appropriate finite-dimensional matrices and phases. At the level of vacuum functions, antiunitarity appears as complex conjugation together with inversion and reversal of the field order.

The spin–statistics theorem instead says, schematically, that local fields transforming with integer spin use bosonic spacelike commutation, whereas half-integer-spin fields use fermionic spacelike anticommutation, assuming positive Hilbert metric, the spectrum condition, the relevant Poincaré covariance, locality, and nontriviality. A wrong assignment forces the corresponding field to vanish in the theorem’s setting. It does not say that every observable carrying integer angular momentum is an elementary boson, nor does it classify braid statistics in two spatial dimensions.

The two conclusions are therefore not converses. CPT symmetry alone does not determine exchange statistics, and a correct spin–statistics assignment alone does not prove total-inversion symmetry.

Free Dirac field as a diagnostic

The physical statement and its limits are organized on CPT: hypotheses, content, and limits; the calculation here traces the exact Wightman proof ingredients for the free Dirac two-point function.

For the free Dirac field, the positive-frequency two-point matrix distribution has Fourier support

$$\widehat S^+(p)=2\pi\,(p\!\!\!/+m)\theta(p^0)\delta(p^2-m^2), \qquad p\!\!\!/\equiv\gamma^\mu p_\mu.$$

Its support creates the same tube damping as for the scalar, while the spin representation continues on the double cover of the complex Lorentz group. The spacelike anticommutator vanishes. Under CPT, particle and antiparticle fields are interchanged, spacetime is inverted, and the reversed correlator identity includes the spinor matrices fixed by the chosen gamma-matrix convention. This checks why a scalar phase-only formula must not be copied to spinors.

Replace the positive Hilbert inner product by an indefinite form, or remove local anticommutativity. Formal Lorentz-covariant two-point distributions may still be written, but the positivity step that excludes the wrong sign, or the boundary identity that connects the orderings, is gone. The Wightman theorem no longer applies. That is a failure of hypotheses, not a counterexample to its conclusion.

An independent check comes from applying the free equation: $(p\!\!\!/-m)(p\!\!\!/+m)=p^2-m^2$, which vanishes on the mass shell. The matrix numerator is therefore consistent with the Dirac equation, while $\theta(p^0)$ independently enforces spectral support.

Theorem-status boundaries

The Wightman proofs are theorems for finite-component fields on Minkowski space under their stated positivity, covariance, spectrum, cyclicity, temperedness, and locality assumptions. Curved spacetime CPT results, algebraic spin–statistics theorems, gauge theories with indefinite-metric potentials, thermal states, and braid-group statistics are distinct variants with distinct hypotheses. They may preserve analogous conclusions, but not by automatic inheritance from the Wightman proof.

In particular, “Lorentz invariant and local” omits positive energy and positivity; “unitary and relativistic” omits the local field structure; and “analytic correlator” omits the boundary-value and ordering hypotheses. None is a valid shorthand for the full theorems.

Exercises

Why is total inversion available to the analytic proof even though it is not in the real proper orthochronous Lorentz group?

Solution

The Wightman functions first extend from real spacetime to complex tubes. Within the connected complex Lorentz group, one can continue the covariance identity to a transformation whose restriction to real vectors is $x\mapsto-x$. The conclusion is then brought back to real boundary values. This does not assert that total inversion was a connected real Lorentz transformation.

References

• Pauli, Wolfgang. 1940. “The Connection Between Spin and Statistics.” Physical Review 58: 716–722. DOI.
• Streater, Raymond F., and Arthur S. Wightman. 2016. PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics. Princeton University Press. DOI.