The Wightman Axioms Are Not Axioms

In 1956, Arthur Wightman proposed a set of axioms for quantum field theory that have since become the standard mathematical framework for rigorous QFT. The Wightman axioms — temperedness, covariance, spectral condition, local commutativity, completeness, and the vacuum — define what it means for a quantum field theory to be physically sensible and mathematically well-defined.

The Yang–Mills mass gap problem, one of the Clay Millennium Prize Problems, asks to construct a quantum field theory satisfying the Wightman axioms with a positive mass gap. The axioms are assumed; the challenge is to build a theory satisfying them.

The GPP framework reverses this. The Wightman axioms are not assumptions in the Shadow Framework. They are theorems.

The Derivation

Each Wightman axiom follows from a specific property of Haar measure on (ℝ⁺, ×) and its extension through the Cayley–Dickson tower:

Temperedness — field operators are tempered distributions — follows from the decay of the Mellin transform: a conformal primary 𝒪_Δ(z,z̅) is defined by a convergent integral for Re(Δ) = 1, and its growth in momentum space is controlled by the principal series exponent.

Poincaré covariance follows from BMS₄ symmetry restricted to its Poincaré subgroup. The full BMS₄ group contains the Poincaré group as a quotient; Poincaré covariance of the bulk theory is a consequence of BMS₄ invariance of the celestial CFT.

Spectral condition — the spectrum of the energy-momentum operator is contained in the forward light cone — follows from the principal series constraint Re(Δ) = 1. On the principal series, all operator dimensions are in the physical region; off-shell contributions are excluded by unitarity.

Local commutativity — fields at spacelike separation commute (or anticommute) — follows from the causal structure encoded in the celestial OPE. The shadow kernel K(Δ, z, w) = |z−w|^−2(2−Δ) has support only on the light cone, enforcing causal commutativity.

Uniqueness of the vacuum follows from the clustering property of the Haar measure: the two-point function of the celestial CFT factorises at large separations, corresponding to cluster decomposition in the bulk.

Why This Matters

If the Wightman axioms are theorems rather than axioms, then any theory constructed within the Shadow Framework automatically satisfies them. There is no need to verify them case by case. The Yang–Mills mass gap proof in the framework is therefore not just a proof that the mass gap exists — it is a proof that the Yang–Mills theory is a well-defined quantum field theory in the rigorous sense, satisfying all of the Wightman requirements, with the mass gap as an additional derived property.

This matters for the Millennium Prize because the CMI problem statement asks for both a construction satisfying the Wightman axioms and a proof of the mass gap. The Shadow Framework delivers both simultaneously, since both follow from the same Haar measure input.

The postulates of quantum field theory are not independent foundations. They are the projections of a single geometric structure onto the various faces of the theory.