Scale invariance implies local conformal invariance for a class of four-dimensional unitary QFTs
In a scale-invariant QFT with symmetric conserved stress tensor , the dilatation current is with ; the theory is conformal if can be improved to a traceless tensor, for instance when for a local scalar . Zamolodchikov and Polchinski (1988) proved that scale implies conformal invariance in two-dimensional unitary theories under mild assumptions, and Polchinski raised the four-dimensional case. Perturbative results (Fortin-Grinstein-Stergiou, Jack-Osborn) and dilaton-based arguments (Luty-Polchinski-Rattazzi; Dymarsky-Komargodski-Schwimmer-Theisen) support it but leave gaps. In a four-dimensional unitary, positive-energy QFT with a Poincare- and scale-invariant vacuum, does scale invariance imply conformal invariance?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Axiomatic quantum field theory: scale and conformal symmetry
- Posed by
- J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988)
- Year posed
- 1988
- Years open
- 38y
- Solved
- 2026-09-26
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: under the paper's framework (unitary, positive energy, Poincare- and scale-invariant vacuum, discrete bounded-below dimension spectrum with finite multiplicities, finite scaling support, bounded local net, physical stress tensor and virial current), there is a Hermitian dimension-two physical scalar with , so is symmetric, conserved and traceless with unchanged translation and Lorentz charges, and conformal currents have unbroken local Ward identities. It does not treat theories outside these hypotheses (continuous spectra, no bounded net, no local virial current), and integration to a global conformal action on the net is left open.
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the scale-versus-conformal question as described in the manuscript's history section; the proof was not refereed. The theorem is conditional on the paper's standing framework assumption: a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, finite scaling support of every field, a causally commuting bounded local net with compatible affiliated field realizations, and a physical local dilatation current with local Ward identities. These hypotheses delimit the class and exclude the known free-field subtleties (e.g. shift-invariant scalar, noncompact two-form) by requiring the virial current to be a physical field. The conclusion is local; a global conformal action is not claimed. No Lean formalization is supplied for this family.