Browse › Mathematical physics › 282 From scale symmetry to local conformal symmetry in four-dimensional QFT
Scale and conformal symmetry in four-dimensional operational quantum field theory
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Abstract
We prove that scale symmetry implies local conformal symmetry for a class of four-dimensional unitary, positive-energy quantum field theories with a Poincaré- and scale-invariant vacuum. The class has a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, and each original field has finite scaling support. The original fields and their adjoints have compatible affiliated realizations in a causally commuting bounded local net, and a physical local dilatation current generates scale transformations through its local Ward identities. The operational framework admits sharply localized reconstructed fields after their joint products and compatible affiliated realizations in the same net have been established. Under these hypotheses, the stress tensor admits a symmetric, conserved, traceless improvement with unchanged translation and Lorentz charges. The corresponding conformal currents satisfy unbroken local Ward identities on the physical field algebra. The conclusion is local; integration to a global conformal action on the whole bounded net remains a separate question.
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Possibly relevant literature (found by the agent, not checked)
- D. Dorigoni and S. Rychkov, Scale Invariance + Unitarity => Conformal Invariance? (arXiv:0910.1087, 2009). Explains why the Zamolodchikov–Polchinski two-dimensional argument does not extend to higher dimensions, and discusses "fake counterexamples" without a stress tensor; this bears on the paper's history section and its stress-tensor/virial-current hypotheses.
- K. Farnsworth, M. A. Luty and V. Prelipina, Scale Invariance plus Unitarity Implies Conformal Invariance in Four Dimensions (arXiv:1309.4095, 2013; withdrawn 2014). An earlier non-perturbative claim of the same implication, withdrawn because its momentum-space OPE step fails; this is relevant to the paper's discussion of nonperturbative arguments and of DFKLP2016.
- F. Ceyhan and T. Faulkner, Recovering the QNEC from the ANEC (arXiv:1812.04683, 2018). Uses half-sided modular inclusions for null-cut algebras; this is related to the paper's step 1, which builds positive null-cut generators from half-sided modular inclusions.
- V. Morinelli, Y. Tanimoto and B. Wegener, Modular operator for null plane algebras in free fields (arXiv:2107.00039, 2021). Decomposes the modular operator of null-cut algebras into lightlike fibres for the free scalar; this is related to the paper's bounded-cut modular Ward formula and its free-scalar example.
- H. Bostelmann, Phase space properties and the short distance structure in quantum field theory (arXiv:math-ph/0409070; J. Math. Phys. 46 (2005) 052301). Recovers pointlike fields from a net of local algebras under a phase-space condition; this is related to the paper's reconstruction of the improvement scalar L as a field affiliated with the bounded net, which explicitly avoids phase-space assumptions.
Digests and expositions
- AI Proved What?, Scale implies conformal, proved at last? (episode 282, October 2026). This AI-assisted video and transcript explain the paper's hypotheses, dimension-two improvement scalar and modular-theory strategy, distinguishing the claimed local Ward identities from a global conformal action. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 282
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026,
author = {{OpenAI}},
title = {{Scale and conformal symmetry in four-dimensional operational quantum field theory}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf}{OAI:Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026}},
year = {2026}
}Family 282: From scale symmetry to local conformal symmetry in four-dimensional QFT
Under the stated bounded-local-net and field-reconstruction hypotheses, proves that scale symmetry implies local conformal symmetry for four-dimensional unitary positive-energy theories with a discrete bounded-below scaling spectrum of finite multiplicity, finite scaling support and a physical local scale current. The stress tensor has a traceless improvement with unchanged spacetime charges. The conclusion concerns local Ward identities, not a global conformal action on the whole net.
This is the only paper in this family.