From scale symmetry to local conformal symmetry in four-dimensional QFT

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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.

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Theorem 1.1

In Scale and conformal symmetry in four-dimensional operational quantum field theory, 26 September 2026

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Suppose Assumption 2.1 holds, including the algebraic direct-sum decomposition with finite scaling support for every original field. Then the physical stress tensor admits a local symmetric conserved improvement with the same translation and Lorentz Ward charges and with zero trace. More precisely, let T^{(4)} and V^{(3)} be the Hermitian dimension-four and dimension-three components selected by the affine Ward identities. There is a Hermitian physical scalar field L of dimension two such that

\partial^\mu V^{(3)}_\mu=\Box L, \qquad t_{\mu\nu}=T^{(4)}_{\mu\nu} +\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L

is symmetric, conserved and traceless. The replacement T\mapsto T^{(4)} also preserves the affine charges. The currents X^\nu t_{\mu\nu} for conformal Killing vector fields X have unbroken local Ward identities on the physical field algebra, extending its Poincaré and dilation actions.

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Scale and conformal symmetry in four-dimensional operational quantum field theory

26 September 2026, current edition.

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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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Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
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