Scale and conformal symmetry: reconstructing the improvement field

Source and verification

This article explains family 282 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Scale symmetry rescales lengths uniformly. Conformal symmetry also supplies conserved currents for special conformal transformations. Family 282 studies the passage between them in four-dimensional quantum field theory. Its central task is to construct a dimension-two improvement scalar as a field of the prescribed physical theory, including its mixed products, operator domain, and localization. The manuscript’s conclusion is a local conformal Ward action under its operational hypotheses; a global conformal action on the entire bounded local net is a separate question. (OpenAI, 2026)

🏷️ Preliminaries: fields, local operations, and currents

The prerequisites are distributions, Hilbert-space operators, the spectral theorem, and basic complex analysis. The field-theoretic and modular notions needed for the proof outline are introduced below.

Work on Minkowski space with metric $\eta =\mathrm{diag}(1,-1,-1,-1)$ and wave operator $\square ={\partial }^{\mu }{\partial }_{\mu }$. A quantum field $A$ is an operator-valued distribution: its smearing $A(f)$ acts on a specified dense domain, whereas its value at a point is generally not an operator. Vacuum correlations of products are distributions in all insertion variables.

A bounded local net assigns an algebra $\mathcal{M}(O)$ of bounded operations to each spacetime region $O$. Inclusion of regions implies inclusion of algebras; spacelike separated operations commute, with the appropriate statistics convention for fermions. An unbounded closed operator is affiliated with $\mathcal{M}(O)$ when it commutes, including domain invariance, with every unitary in the commutant $\mathcal{M}(O{)}^{\prime }$. Weak commutation on a small field domain alone does not establish affiliation.

The vacuum wave of a smeared field is the vector $A(f)\Omega$. Knowing this vector for every $f$ is much less than knowing $A(f)$ on arbitrary states. A new vacuum wave therefore does not automatically define a new local field.

A symmetric conserved stress tensor $T_{\mu \nu }$ generates translations and Lorentz transformations through Ward identities. Here a Ward charge is implemented by a compactly smeared flux surrounding an insertion; one need not first construct an everywhere-defined global charge. A local dilatation current has the form

$$\begin{array}{c}{D}_{\mu }=x^{\nu }{T}_{\mu \nu }-V_{\mu },\\ {\partial }^{\mu }{D}_{\mu }=0,\Theta :=T^{\mu }{}_{\mu }={\partial }^{\mu }{V}_{\mu }.\end{array}$$

The vector $V$ is the virial field. Scale invariance permits a nonzero divergence of $V$, so it does not immediately give a traceless stress tensor.

🏷️ The theorem and its hypotheses

The manuscript assumes a unitary, positive-energy, Poincaré-covariant theory generated by original fields and their adjoints from a Poincaré- and dilation-invariant vacuum. It does not assume that the invariant vacuum vector is unique. The original fields and ordered products have a common invariant domain and satisfy locality and covariance.

Its scaling hypothesis is particularly concrete:

$$\begin{array}{c}\mathcal{F}=\underset{\Delta \in \Sigma }{⨁}{\mathcal{F}}_{\Delta },\\ A=\sum _{\Delta \in S_A}{A}_{\Delta },|S_A|<\infty .\end{array}$$

The real dimension spectrum $\Sigma$ is bounded below and locally finite. Each grade has finite multiplicity, counting Lorentz components, and consists of homogeneous fields. Differentiation raises the grade by one. Thus every original field has finite scaling support. This restricts field labels, without bounding word lengths or requiring joint test kernels to have finite tensor rank.

The hypotheses also require the fixed causally commuting bounded net, affiliated realizations of fields and adjoints, and physical local stress and dilatation currents with their compact-flux Ward identities. A reconstructed family is admitted only after all its joint products with the original fields, a compatible invariant domain, and affiliation in this same net have been established.

The manuscript's local enhancement theorem

Under these assumptions, the dimension-four stress component $T^{(4)}$ has the same translation and Lorentz Ward charges as the original stress tensor. There is a Hermitian physical scalar $L$ of dimension two such that

$$\begin{array}{c}\square L={\partial }^{\mu }{V}_{\mu }^{(3)},\\ t_{\mu \nu }=T_{\mu \nu }^{(4)}\\ +\frac{1}{3}({\partial }_{\mu }{\partial }_{\nu }-{\eta }_{\mu \nu }\square )L.\end{array}$$

The tensor $t$ is symmetric, conserved, and traceless, with unchanged translation and Lorentz charges. Its conformal currents give unbroken local Ward identities on the physical field algebra. (OpenAI, 2026)

Finite scaling support makes the weight selection legitimate: after testing homogeneous spectators, the Ward identities become finite sums of exponentials in logarithmic scale. Their independence selects weights four and three for the required charges. The discarded components have zero corresponding Ward charges; the argument does not assert that those fields vanish.

Once $L$ exists, the elementary tensor calculation is immediate. Commuting derivatives makes the correction divergence-free, while its trace is $(1-4)\square L/3=-\square L$. Constructing $L$ with all the required operator properties is the difficult part.

🏷️ A free-field example

For the full massless real scalar theory, let $Q=:{\varphi }^2:$ denote the Wick square. The free equation gives

$$\begin{array}{c}\Theta =-:(\partial \varphi {)}^2:=-\frac{1}{2}\square Q,\\ V_{\mu }=-\frac{1}{2}{\partial }_{\mu }Q,L=-\frac{1}{2}Q.\end{array}$$

The improvement is therefore the familiar $({\eta }_{\mu \nu }\square -{\partial }_{\mu }{\partial }_{\nu })Q/6$. This example also explains why field membership matters. Restricting to shift-invariant Wick polynomials removes the displayed $Q$ and virial current. Such a restriction cannot simply inherit the theorem’s local-current hypothesis. Moreover, deleting a field name alone does not exclude a field reconstructible in the same bounded net: one must specify the operational theory.

🏷️ The trace wave and modular comparisons

After selecting homogeneous currents, write $M^2=p^2$ on the timelike translation spectrum. Positivity, scaling, and the finite Lorentz representations exclude nonzero light-cone and zero-momentum contributions to the stress and virial vacuum waves. The source defines

$$l=-\Theta /M^2,J_{\mu }=V_{\mu }-i{p}_{\mu }l$$

at the level of vacuum waves. The scalar $l$ has dimension two; its two-point measure has no new infrared divergence. Together with the conserved-vector and traceless-tensor wave channels, it carries an auxiliary conformal comparison representation. This representation describes two-point data, and has not yet established a conformal symmetry of the physical field theory.

The proof next uses modular theory. For a local algebra with cyclic and separating vacuum, the densely defined map $A\Omega \mapsto A^{\ast }\Omega$ has a polar decomposition whose positive part defines a one-parameter modular group. In the manuscript’s wedge construction this group is the Lorentz boost group. Appropriate inclusions of null-deformed regions provide positive generators.

A bounded transverse profile $g$ describes a null cut. A Ward calculation identifies the corresponding generator $H_g$ through stress-tensor matrix elements and proves linear dependence on $g$. Modular strip analyticity supplies a comparison of spectral subspaces, stronger than ordinary quadratic-form order. This stronger comparison is used to prove that the generators for different profiles commute.

In light-front variables, let $b>0$ be the longitudinal momentum and $Y$ the transverse position operator. The resulting representation is

$$\begin{array}{c}{H}_g=b{\int }_{{\mathbb{R}}^2}g(Y+z)\mu (dz),\\ \mu ({\mathbb{R}}^2)=I,\int z\mu (dz)=0,\\ \kappa :=\int |z{|}^2\mu (dz)\le h.\end{array}$$

Here $\mu$ is a positive operator-valued measure with commuting values, and $h$ is the internal parabolic comparison operator. The source does not identify $\mu$ with a projection-valued measure or claim $\kappa =h$.

The scalar radial comparison operators are

$$h_{\nu }=-{\partial }_M^2-M^{-1}{\partial }_M+{\nu }^2{M}^{-2}$$

on $L^2({\mathbb{R}}_{+},MdM)$, with their regular Hankel realization. Their Hankel transforms convert them into multiplication by the squared radial variable. The initial comparison for $l$ has order $\nu =2$, whereas the improvement requires order zero.

🏷️ Eliminating the critical spectral tail

The mechanism for that upgrade can be explained explicitly. Write ${\mathcal{H}}_{\nu }$ for the order-$\nu$ Hankel transform. The Bessel recurrence gives

$$\begin{array}{c}({\mathcal{H}}_2{\mathcal{H}}_{0}F)(v)\\ =\frac{2}{v^2}{\int }_0^{v}rF(r)dr-F(v).\end{array}$$

If $F$ is supported below $R$ and $\int vF(v)dv=0$, the right-hand side is also supported below $R$. The order-two comparison consequently excludes spectral leakage above $R$ for these zero-moment tests.

Any remaining leakage depends on the single moment $m(F)=\int vF(v)dv$. In a spectral representation $\mathcal{U}$ of $\lambda =\sqrt{\kappa }$, homogeneity fixes its shape:

$$\begin{array}{c}\mathcal{U}(l{\mathcal{H}}_{0}F)(\lambda )=m(F){\lambda }^{-2}{d}_l,\\ \lambda >R.\end{array}$$

A stress/virial dispersion identity compares the two null horizons. The vector and tensor tail coefficients satisfy $d_{j_z}=-d_l$ and $d_{s_z}=-d_l/3$, where $j_z,s_z$ are their longitudinal polarization labels. A radial test extending over a logarithmically long range produces the leading pairing

$$\frac{|c_{\ast }{|}^2}{3}\log \Lambda \Vert d_l{\Vert }^2+o(\log \Lambda ).$$

The same identity expresses it through terms of size $o(\log \Lambda )$; $c_{\ast }\ne 0$. Dividing by $\log \Lambda$ forces $d_l=0$. Thus the critical tail vanishes and the order-zero spectral comparison follows. The source controls this argument with differences of dilated tests whose tails cancel; it never assigns a finite weighted norm to an individually divergent tail. (OpenAI, 2026)

This is a spectral comparison for the moment $\kappa$, still short of constructing a scalar operator.

🏷️ From physical modular motion to an accessible scalar

The improved comparison first gives forms with one scalar insertion and identifies quadratic charges with genuine self-adjoint null moments on specified domains. Future and past cone comparisons then remove a residual dilation charge. A vacuum thermal analyticity identity identifies the actual cone modular groups with dilations; half-sided inclusions supply actual transformations of diamonds along a time axis.

These actual unitaries act on bounded operations in the original net. Applying them to the virial wave extracts a dimension-two scalar wave $S$, independent of the chosen timelike axis. Its localization is quantitative: there are bounded local approximants $A_R(f)$ with

$$\begin{array}{c}\Vert A_R(f)\Vert \le C{R}^q,\\ \Vert A_R(f)\Omega -S(f)\Vert \\ +\Vert A_R(f{)}^{\ast }\Omega -S(\bar{f})\Vert \\ \le C_N{R}^{-N}\text{for every }N.\end{array}$$

The polynomial exponent $q$ is fixed; constants and finitely many test seminorms may depend on the desired accuracy. Such control allows products of finitely many approximants to converge in complex spacetime tubes.

There remains a possible residual $\ell =l-S$. Products through total length four supply inner products of two-insertion vectors and a conformal representation on their span. Positivity of its radial energy controls the possible singular terms in a harmonic residual correlation. Normalized current fluxes eliminate those terms. Finally, a centered dilation flux yields $0=2c\Vert \ell {\Vert }_{\mathrm{rad}}^2$ with $c\ne 0$, hence $l=S$.

The three representations have different roles: the auxiliary action compares one-field waves; actual diamond modular motions transport bounded physical operations; the action on pairs supplies four-point positivity. None can silently replace the missing reconstruction on arbitrary field products.

🏷️ Mixed products and the operator domain

Even after $l=S$, analytic products need real-axis distributional boundaries in every ordering. The source proves this by induction on the number of scalar insertions. Orderings with all $L$ entries on the right follow from pure scalar vectors and Hilbert-space pairings with original words.

For an adjacent interchange of $L(x)$ and an old field $A(y)$, applying ${\square }_x$ replaces $L$ by the original trace $\Theta$. This lowers the induction index. A causal solution handles the difference of the two known sources. The remaining homogeneous wave ambiguity is a finite sum of derivatives of the massless commutator, with coefficients depending on the merged variables.

Those coefficients are initially holomorphic functions without an established distributional boundary. Gaussian tests on the characteristic branch excluded by the target ordering provide polynomial bounds. Scaling then controls growth at infinity and establishes the required spectral support. This closes the adjacent-interchange argument. The estimates may depend on the fixed field list and length.

The reconstructed functions obey the cross-word Gram identity

$$W(P^{\ast }Q)=⟨\Psi (P),\Psi (Q)⟩.$$

Their vacuum words therefore define an invariant dense domain $\mathcal{D}$ in the original Hilbert space. Positivity ensures that insertion is well-defined: if $\Psi (P)=0$, Cauchy–Schwarz applied to $P$ and $B^{\ast }BP$ gives $\Vert \Psi (BP){\Vert }^2=0$. Null polynomials form a left ideal.

The domain contains the original vacuum-word domain, and original fields retain their actions there. On $\mathcal{D}$, $\square L=\Theta$ is now an operator insertion identity. Exchange with every bounded element of the original local commutant supplies compatible closed affiliated extensions. For real smearings the constructed extension is symmetric; essential self-adjointness is not asserted.

🏷️ Local conformal Ward identities

For a conformal Killing vector $X$, the conformal Killing equation and tracelessness give

$${\partial }^{\mu }(t_{\mu \nu }{X}^{\nu })=0.$$

For affine Poincaré Killing fields, the improvement changes the current by an antisymmetric superpotential divergence. Its compact commutator flux vanishes, preserving translation and Lorentz charges. The residual-charge argument identifies the dilation action as well.

Compact fluxes of the improved currents define local derivations ${\delta }_X$ and annihilate vacuum correlations after summing over insertions. With the source’s active-action convention, coordinates multiplying fields are held fixed, giving $[{\delta }_X,{\delta }_Y]=-{\delta }_{[X,Y]}$. The same currents extend to further finite families only after their joint reconstruction with the original fields has been established.

The conclusion concerns flat-space local currents and their Ward action. It does not establish a global conformal action on every bounded-net operation or a curved-background Weyl identity.

🏷️ Reading and verification scope

The theorem, framework, weight selection, critical-tail argument, reconstruction interfaces, and final improvement/Ward calculation were checked against selected source sections. The complete modular cutoff estimates, dispersion continuation and subtraction proof, scalar extraction, radial classification, and boundary estimates were not independently verified. They remain substantive inputs to this exposition.

🏷️ Key steps and proof outlines

The operator extension is constructed by commuting bounded operations

The final localization step has a short functional-analytic proof worth separating from the long field reconstruction. Let $A,A^{♯}$ act on the reconstructed dense domain $\mathcal{D}$. For a compact localization set $K$, the required input is the full exchange identity

$$⟨\varphi ,CA\psi ⟩=⟨A^{♯}\varphi ,C\psi ⟩,C\in \mathcal{M}(O{)}^{\prime },\varphi ,\psi \in \mathcal{D},$$

for every neighborhood $O\supset K$, together with its counterpart with $A,A^{♯}$ interchanged. The reconstruction establishes this identity with every bounded commutant element. It is stronger than exchanging $A$ with a list of spacelike fields.

Enlarge the domain to finite sums of vectors obtained by applying such bounded commutant operations:

$${\mathcal{D}}_K=\mathrm{span}\{C_1\cdots C_m\psi :\psi \in \mathcal{D}\}.$$

Define the extension by moving $A$ through the coefficients:

$$\tilde{A}\left(\sum _i{C}_i{\psi }_i\right)=\sum _i{C}_{i}A{\psi }_i.$$

Here each $C_i$ may itself be a product. This formula needs a well-definedness proof, since a vector can have different representations. If ${\sum }_i{C}_i{\psi }_i=0$, exchange and density give

$$⟨\varphi ,\sum _i{C}_{i}A{\psi }_i⟩=⟨A^{♯}\varphi ,\sum _i{C}_i{\psi }_i⟩=0,$$

so the proposed image is zero.

For finitely many neighborhoods, choose a common smaller neighborhood of $K$. Isotony places all the coefficients in its commutant, allowing products such as $D_j^{\ast }{C}_i$ in the same exchange identity. This proves

$${\tilde{A}}^{♯}\subset {\tilde{A}}^{\ast },\tilde{A}\subset ({\tilde{A}}^{♯}{)}^{\ast }.$$

The dense adjoint domains imply closability. Every allowed commutant unitary and its inverse preserve the enlarged domain and commute with the extension. Taking the closure preserves this graph invariance, which is exactly affiliation. If $A=A^{♯}$, the resulting closed extension is symmetric; the argument does not promote symmetry to self-adjointness. (OpenAI, 2026)

Why positivity must include cross terms between scaling components

Finite scaling support lets each old field be expanded into finitely many homogeneous labels, but it does not allow one to reconstruct each component independently and discard mixed terms. A zero vector can result from cancellation between words of different component labels. The identity

$$W(P^{\ast }Q)=⟨\Psi (P),\Psi (Q)⟩$$

must therefore hold for whole finite polynomials $P,Q$, with all cross terms.

Its proof first establishes splitting for homogeneous tuples, then expands the labels under the same joint test kernel and sums the resulting cross-tuple identities. No finite tensor-rank assumption on that kernel is introduced. Positivity then makes zero polynomials a left ideal, as in the preceding domain construction, and ensures that every insertion respects their cancellation. This is the step that places the improvement field and the old fields on one common invariant domain in the original Hilbert space. The separate commutant extension above subsequently gives localization in the original bounded net. (OpenAI, 2026)

See Also

• BFSS normalizable eigenstates gives another quantum-mechanical setting where specifying the closed operator and its domain is essential to passing from a formal expression to a physical conclusion.
• OpenAI collection places the family among the mathematical physics articles and records the graduate prerequisites.

🏷️ Manuscripts and verification scope

• Scale and conformal symmetry in four-dimensional operational quantum field theory — pinned manuscript and companion source.

No family scope page was present at lean/docs/282.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Scale and conformal symmetry in four-dimensional operational quantum field theory.