Oai282-B From Finite Ward Certificates to Local Conformal Symmetry: Physical Improvement, Explicit Contacts, and Complete Coverage for the Four-Dimensional Free Massless Scalar

We give a self-contained, model-specific continuation of the improvement and finite-fit programme described in Oai282-A. The model is the vacuum Fock representation of the real massless scalar on four-dimensional Minkowski space, with its original Weyl local net. We construct the improvement scalar $L=-\frac{1}{2}:{\varphi }^2:$, prove its mixed-product and common-domain properties, and establish affiliated realizations of its compact smearings in that same net. We calculate the elementary retarded Ward contact, its descendant contacts, and the local stress-current action on the complete Wick-polynomial field algebra. Translations, Lorentz transformations, dilations, and special conformal transformations are included. Successive actions obey the conformal Lie algebra with an explicitly fixed sign convention.

A separate classification theorem identifies every conserved stress tensor in a declared homogeneous, Poincare-covariant local Wick-polynomial class. Its complete residual family is two-dimensional, and two explicitly normalized matrix-element probes separate it. Consequently finite vanishing gives the full local Ward conclusion within that class, rather than merely a fit on named tests.

The conclusions concern local current implementation and ordered Wick-field products. They do not assert a general scale-to-conformal theorem, a complete renormalized time-ordered stress-tensor insertion scheme, global integration of the conformal generators, or an interacting improvement-scalar reconstruction. The distinct characteristic-product programme of XIV-I, XIV-J, and XV-B is not completed. No coefficient from that programme is admitted; O.1 remains NOT_CLOSED.

The supplied description of Oai282-A separates three targets CheungA.

A: finite certification. The predecessor's thm:ward and prop:finitefit concern named residuals:

Its thm:statistics is an optional bounded-score tolerance statement, not an operator-valued distributional identity.

B: local Ward implementation. The predecessor's prop:affine-ward is conditional on a spacelike-collar contract and a declared polynomial. It is not, by itself, implementation by $X^{\nu }{t}_{\mu \nu }$ on every field in a complete test class. The new free-field results below address precisely this additional issue.

C: global implementation. The predecessor's prop:flow and prop:nonintegrable separate local identities from global geometric flows and unitary integration. We reproduce the relevant elementary examples in Section sec:global. Global implementation is not attempted.

This manuscript does not use a supplied series sheet as a premise. References to Cheung's manuscripts identify predecessor statements, comparison interfaces, and open boundaries. The mathematical inputs used in the proofs are the explicitly declared free Fock construction and standard functional-analytic results, with their hypotheses checked where applied.

A local conformal Ward conclusion consists of:

1. an invariant dense field domain and continuous ordered joint field distributions;

2. a conserved current ${j_X}^{\mu }=t^{\mu \nu }{X}_{\nu }$;

3. a compact spacetime-smeared current whose commutator implements the declared infinitesimal transformation on every compactly localized Wick polynomial and every finite ordered product;

4. elementary and descendant contact distributions with the normalization calculated below;

5. compatibility of these actions with field relations and conformal Lie brackets.

It does not mean that every renormalized time-ordered product containing arbitrarily many stress tensors has been constructed.

The distinction is important. Distributional conservation of a current is weaker than its implementing a field transformation. Conversely, current implementation on an ordered field algebra does not construct a curved-background generating functional or remove the separate normalization issues of time-ordered products. General Ward normalization frameworks are discussed in DutschBoas,FR; none is imported here as an unspecified completion of our insertion class.

Consider the real massless scalar in its vacuum Fock representation on $M={\mathbb{R}}^{1,3}$, with the Weyl net and domains specified in Section sec:model. Then:

1. $Q=:{\varphi }^2:$ and $L=-Q/2$ belong to the same physical Wick-field extension of the original scalar theory. Their mixed ordered products exist on a common invariant dense domain containing the original vacuum-word domain. Every real compact smearing of $L$ has a self-adjoint realization affiliated with the original local Weyl algebra containing its support.

2. The improved tensor

   $$t_{\mu \nu }=:{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi :-\frac{1}{2}{\eta }_{\mu \nu }:{\partial }_{\rho }\varphi {\partial }^{\rho }\varphi :+\frac{1}{6}({\eta }_{\mu \nu }\square -{\partial }_{\mu }{\partial }_{\nu })Q$$

   is symmetric, conserved, and traceless on that domain.

3. Its compact-slab currents implement all conformal Killing transformations on every compactly smeared Wick polynomial, all descendants, and all finite ordered mixed products. The elementary retarded contact is

   $${{\partial }_{\mu }}^x{C_X}^{\mu }(x,y)={\delta }^{(4)}(x-y)(X^{\nu }{\partial }_{\nu }+{\sigma }_X)\varphi (y),{\sigma }_X=\frac{1}{4}{\partial }_{\nu }{X}^{\nu }.$$

   Descendant contacts and the corresponding complete field actions are given in Sections sec:contacts and sec:charges.

4. The sign-adjusted local derivations ${\delta }_X$ defined in Section sec:brackets satisfy

   $$[{\delta }_X,{\delta }_Y]={\delta }_{[X,Y]}$$

   on the complete ordered Wick-field algebra.

5. Within the stress-tensor class of Definition def:stressclass, the complete residual family has a proved finite-dimensional span. Two explicit probes are injective on that span. Their exact vanishing forces the displayed tensor $t$, and therefore all the local Ward identities just stated.

No assertion is made that the hypotheses of the supplied Family 282 claim have thereby all been verified. In particular, no global conformal action on the whole bounded net is constructed.

Proof. Assertion (i) is proved in Propositions prop:domain and prop:physicalL. Assertion (ii) is Proposition prop:improvement. The contact calculation is Proposition prop:elementarycontact; implementation and mixed products are Theorem thm:localaction. Lie compatibility is Proposition prop:brackets. Classification and complete separation are Theorems thm:classification and thm:coverage. The exclusions are the scope of those statements, not additional enhancement claims.

Use

Let

The one-particle space and Fock space are

Write $\Omega$ for the Fock vacuum, $N$ for particle number, and $H=\mathrm{d}\Gamma (p^0)$. Creation and annihilation operators have the usual Fock normalization, and

For test functions use

The two-point distribution is

In particular,

Let $R$ be the retarded fundamental solution:

With $E=R-R_{\mathrm{adv}}$, our commutator convention is

For every bounded open region $O$, define

The closure of $\varphi (f)$ is self-adjoint for real $f$: finite-particle vectors are analytic for a Segal field. The same creation–annihilation calculation gives locality from (eq:CCR). Thus (eq:net) is the original scalar Weyl net, not a net enlarged by adjoining $Q$.

Set

The topology on each fixed-particle summand is given by the seminorms $\Vert (1+H{)}^r\psi \Vert$; the algebraic sum has the corresponding locally convex direct-sum topology.

A local Wick field is a finite linear combination of

Its smearings use $\mathcal{S}(M)$, and its local smearings use ${C_c}^{\infty }(M)$. These are vacuum normal products, with no self-contractions inside a Wick block.

Define ${\mathcal{D}}_{\mathrm{old}}$ to be the span of the continuous joint-Schwartz vacuum-word vectors of the original field $\varphi$. This is the old domain of the benchmark just declared. It is not identified with an unspecified domain from another model.

For a fixed Wick field $B$, input particle number $n$, and integer $s\ge 0$, there exist an integer $r$, a continuous Schwartz seminorm $p$, and a constant $C$ such that

Only finitely many output particle sectors occur.

Proof. Expand the normal product into its finitely many creation–annihilation terms. On a participating set of momenta, each kernel has the form

where $P$ is a fixed polynomial produced by the derivatives. The nonparticipating particles contribute identity kernels.

For the participating variables put

They are nonnegative, and

An output energy weight can therefore be bounded by a polynomial in the input energy and the temporal Fourier variable. Schwartz decay absorbs the latter polynomial.

More explicitly, after inserting a sufficiently large input weight $(1+Q_0{)}^{-r}$, the squared participating kernel is integrable: the radial mass-shell measure of each variable is a constant multiple of $u\mathrm{d}u\mathrm{d}\omega$, the derivative factors have polynomial growth, and a sufficiently high power of $(1+|P_0-Q_0|{)}^{-1}$, together with $(1+Q_0{)}^{-r}$, makes the radial integrals finite. There is no infrared divergence, because $u\mathrm{d}u$ is locally integrable at zero. This gives a Hilbert–Schmidt bound for the weighted participating kernel. If one side has no participating variables, the same calculation uses the rapid decay in the sum of the positive energies.

The finitely many Fock combinatorial factors depend on $n$ and the degree of $B$, and are absorbed in $C$. The spectator energy weight is transferred to the input by the same inequality. Summing the terms proves (eq:wickest).

The following statements hold.

1. ${\mathcal{D}}_E$ is Hilbert-dense and invariant under every smeared Wick field and its formal adjoint.

2. Every finite ordered list of Wick fields defines continuous joint-Schwartz matrix-element distributions on ${\mathcal{D}}_E$. The corresponding joint vacuum-vector maps are continuous.

3. ${\mathcal{D}}_{\mathrm{old}}\subseteq {\mathcal{D}}_E$, and ${\mathcal{D}}_{\mathrm{old}}$ is Hilbert-dense.

4. All these Wick fields are closable on ${\mathcal{D}}_E$. Spacelike separated smearings commute on ${\mathcal{D}}_E$.

Proof. Energy cutoffs in every finite-particle sector prove Hilbert density. Lemma lem:wickest, for every $s$, proves invariance.

Iterating the same lemma gives continuous multilinear maps in the individual Schwartz smearings, with bounds by products of finitely many Schwartz seminorms. Nuclearity identifies the completed projective tensor product of the Schwartz spaces with the joint Schwartz space. Hence the maps extend uniquely to arbitrary joint Schwartz kernels. The same estimates give continuous vacuum-vector maps.

For old vacuum words, the Fourier variables creating particles have positive energies. A joint Schwartz kernel therefore gives rapid decay in their total energy, and hence finite moments of every order. This proves ${\mathcal{D}}_{\mathrm{old}}\subseteq {\mathcal{D}}_E$. Old field vacuum words generate the scalar Fock representation: restrictions of spacetime Schwartz Fourier transforms to the positive mass shell are dense in $\mathfrak{h}$, and their symmetric tensor products are dense in each particle sector. Thus ${\mathcal{D}}_{\mathrm{old}}$ is Hilbert-dense.

The adjoint pairing of a normal product is the normal product of the adjoint factors, with the reversed operator order for an ordered word. It is defined on ${\mathcal{D}}_E$. A densely defined formal adjoint therefore gives closability.

Finally, Wick's expansion of a commutator between spacelike separated blocks contains commutators of the elementary fields or their derivatives. Each vanishes by (eq:CCR). The support-preserving joint-test extension preserves this identity.

No arbitrary originally supplied closed realization has been identified merely because a domain is Hilbert-dense. A graph-core assertion requires an additional argument. For $Q(f)$, that argument is supplied next.

Let $Q=:{\varphi }^2:$. For real $f\in \mathcal{S}(M)$, its quadratic Fock decomposition is

where, with the kernel convention of Section sec:model,

The pair kernel is symmetric, and $B_f$ is self-adjoint for real $f$.

For every real $f\in \mathcal{S}(M)$, $F_f$ is Hilbert–Schmidt, $B_f$ is bounded, and

The operator $Q(f)$ is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$. Its restriction to ${\mathcal{D}}_E$ has the same self-adjoint closure.

Proof. Schwartz decay in the temporal Fourier variable gives

The radial measure is proportional to $p^0\mathrm{d}{p}^0$ on each shell, so the square of this bound is integrable for sufficiently large $M$. Thus $F_f$ is Hilbert–Schmidt.

For $B_f$, Schwartz decay gives, for large $M$,

The integral of this bound against $\mathrm{d}\mu (q)$ is uniformly bounded in $p$: on $|\mathbf{q}|<1$, the measure is integrable; on $|\mathbf{q}|\ge 1$, it is bounded by a constant times Lebesgue measure, and the translated integrable decay has constant integral. The corresponding column bound is identical. Schur's test proves boundedness.

On the $n$-particle sector, the pair creation and annihilation terms are bounded by constants times $\sqrt{(n+1)(n+2)}\Vert F_f\Vert$ and $\sqrt{n(n-1)}\Vert F_f\Vert$. The number-preserving term is bounded by $2n\Vert B_f\Vert$. This proves (eq:Qnumber).

Repeated application changes particle number by at most two. For a vector with maximal particle number $n$,

The analytic-vector series consequently converges for sufficiently small parameter. Nelson's analytic-vector theorem Nelson gives essential self-adjointness on ${\mathcal{D}}_{\mathrm{fin}}$.

On each fixed finite collection of particle sectors, (eq:Qnumber) bounds $Q(f)$ by a constant times the Hilbert norm. Energy-cutoff vectors in ${\mathcal{D}}_E$ approximate any vector of ${\mathcal{D}}_{\mathrm{fin}}$, both in Hilbert norm and after applying $Q(f)$. Thus ${\mathcal{D}}_E$ is a graph core for this particular closure.

If $f\in {C_c}^{\infty }(O;\mathbb{R})$, then $\overline{Q(f)}$ is affiliated with $\mathfrak{A}(O)$.

Proof. Choose a real compact smooth mollifier ${\rho }_{ϵ}$, normalized to integral one, and supported in a ball of radius $ϵ$. For sufficiently small $ϵ$, every translated mollifier meeting $\mathrm{supp}f$ is supported in $O$. Put

For fixed $ϵ$, the integral is first defined by its quadratic Fock kernels. Those kernels are Bochner integrals of rank-one and rank-two kernels over a compact parameter set.

Riemann sums are finite real linear combinations of

Each summand is affiliated with $\mathfrak{A}(O)$, by the spectral calculus for the local Segal field. Each finite sum is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$, by the same bounded-number argument as in Lemma lem:Qsa. The unitary product formula for these self-adjoint summands, whose sum is essentially self-adjoint on the common finite-particle core, gives its unitary group as a strong limit of products of their local unitary groups Trotter. Hence each finite sum is affiliated with $\mathfrak{A}(O)$.

For fixed $ϵ$, the Riemann-sum pair kernels converge in Hilbert–Schmidt norm, and the number-preserving kernels converge in operator norm. The associated operators therefore converge on ${\mathcal{D}}_{\mathrm{fin}}$. This yields strong-resolvent convergence of their self-adjoint closures, so $Q_{ϵ}(f)$ is affiliated with $\mathfrak{A}(O)$.

For completeness, the strong-resolvent implication used here has a short core argument. If self-adjoint $A_j,A$ satisfy $A_j\psi \to A\psi$ on a core of $A$, then

The range of $A-\mathrm{i}$ on that core is dense and the resolvents are uniformly bounded, so convergence holds on the whole Hilbert space.

Now remove the mollifier. On the mass shell the one-particle multiplier ${\hat{\rho }}_{ϵ}(p)$ is uniformly bounded and converges pointwise to one. Thus the pair kernels converge to $F_f$ in Hilbert–Schmidt norm by dominated convergence. The number-preserving kernels converge strongly to $B_f$, with uniform operator-norm bounds, because they are obtained by placing these bounded multipliers on the two sides of $B_f$. It follows that $Q_{ϵ}(f)\psi \to Q(f)\psi$ on ${\mathcal{D}}_{\mathrm{fin}}$. Lemma lem:Qsa and the same core argument give strong-resolvent convergence to $\overline{Q(f)}$.

The resolvents of all approximants belong to $\mathfrak{A}(O)$, and that von Neumann algebra is strongly closed. The limiting resolvents therefore belong to $\mathfrak{A}(O)$. This is affiliation of $\overline{Q(f)}$ with the original net.

The field

has mixed ordered products with every original scalar field, every adjoint, and every Wick polynomial on ${\mathcal{D}}_E\supseteq {\mathcal{D}}_{\mathrm{old}}$. It is local, and every real compact smearing has a self-adjoint affiliated realization in the original Weyl net.

Complex compact smearings have compatible closed affiliated realizations whose action and adjoint action agree with the corresponding Wick insertions on ${\mathcal{D}}_E$.

Proof. Mixed products, locality, and domain invariance follow from Proposition prop:domain. Real affiliated realizations follow from Lemma lem:localapprox and multiplication by $-1/2$.

For a complex smearing, write $f=f_1+\mathrm{i}{f}_2$ with real $f_1,f_2$. Let $A,B$ be the affiliated self-adjoint realizations of $L(f_1),L(f_2)$. On $\mathrm{Dom}A\cap \mathrm{Dom}B$, the sum $A+\mathrm{i}B$ is densely defined and closable: its adjoint contains $A-\mathrm{i}B$ on the same dense intersection. Every unitary in the commutant of the local algebra preserves both domains and commutes with both summands. It consequently preserves the closure of the sum. That closure is affiliated, agrees with $L(f)$ on ${\mathcal{D}}_E$, and its Hilbert adjoint agrees there with $L(\overline{f})$.

No Green operator was used to define $L$. The proof starts from the explicit Wick square, establishes its joint products, and constructs local affiliated realizations in the already declared net. It does not claim the same result for an interacting theory or a reduced derivative-only net.

Essential self-adjointness of Wick squares is a substantive operator question in more general representations; see Sanders. Our all-real-smearing conclusion uses the special bounded one-particle kernel of the flat free scalar and is not presented as that more general theorem.

Define

The free equation $\square \varphi =0$ holds as a field identity.

On ${\mathcal{D}}_E$,

Consequently

is symmetric, conserved, and traceless.

Proof. Vacuum normal ordering commutes with constant-coefficient differentiation. Thus

The trace of $T$ in four dimensions is $K-2K=-K$. The remaining identities in (eq:improvementdata) follow.

The divergence of the improvement operator vanishes identically:

The canonical tensor is conserved by the free equation. The trace of the improvement is $\frac{1}{2}\square Q=K$, which cancels the canonical trace. Symmetry is immediate.

This is the free-field version of the improved tensor of Callan, Coleman, and Jackiw CCJ. The calculation alone does not prove a fixed-point theorem.

A conformal Killing field satisfies

In four dimensions the relevant fields are

They obey

Put

A direct calculation gives

Thus $v_X\varphi$ is again a free solution field.

The improved current

is conserved:

We now prove implementation, rather than stopping at this identity.

Define

Before imposing the free equation, the corresponding polynomial calculation gives

The identity uses (eq:CK) and (eq:CKaux).

Define the antisymmetric superpotential

For every conformal Killing field,

Proof. Differentiate (eq:Bsuper). The conformal Killing equation combines the coefficients of $\partial Q$ into $-\frac{1}{2}{\sigma }_X{\partial }^{\mu }Q$. The last coefficient is

The second-derivative terms are

Since $:\varphi {\partial }^{\mu }\varphi :=\frac{1}{2}{\partial }^{\mu }Q$, their sum is exactly ${j_X}^{\mu }-{j_{N,X}}^{\mu }$.

The following insertion is defined directly, not by multiplying an unspecified time-ordered distribution by a step function.

Let ${C_X}^{\mu }(x,y)$ be the polarization of ${j_X}^{\mu }(x)$ obtained by replacing one elementary scalar factor by $R(x-y)$:

It is understood as a weak field distribution on ${\mathcal{D}}_E\times {\mathcal{D}}_E$.

The existence here is elementary. Matrix elements of a single field between vectors in ${\mathcal{D}}_E$ are smooth functions of $x$, with all derivatives locally bounded, by the energy-moment bounds. Equivalently, the variables $(x,x-y)$ are independent: $\varphi (x)R(x-y)$ is the corresponding tensor distribution. Differentiation is therefore legitimate.

The insertion (eq:Cexplicit) has retarded support in $x-y$ and satisfies

No conservation-to-implementation inference is used.

Proof. Temporarily write $u=\varphi (x)$ and $r=R(x-y)$. The polarized improved tensor in the square brackets is

Commuting derivatives shows

Its trace is

Using $\square u=0$, $\square r={\delta }^{(4)}(x-y)$, and (eq:CK), we obtain

All products are the already defined tensor products or delta insertion maps. Retarded support follows from the support of $R$ and its derivatives.

In particular, the contact on the elementary field is explicitly

For an elementary descendant, define

Then

Thus derivative-of-delta contacts are retained:

For a Wick block $B_{\mathbf{\alpha }}=:{\prod }_a{\partial }^{{\alpha }_a}\varphi :$, the single-block insertion is the sum over its participating factors:

Here the colon orders all remaining quantum factors; it does not normal-order the numerical retarded kernel.

Equation (eq:blockC) is a continuous sesquilinear distribution on ${\mathcal{D}}_E\times {\mathcal{D}}_E$, and

This assertion is not an operator-domain claim for an arbitrary singularly smeared retarded insertion.

Proof. Between two finite-particle energy-smooth vectors, every fully normal-ordered multipoint kernel is smooth. Indeed, its Fourier integrals contain only the external vector wave functions and finitely many polynomial momentum factors. Arbitrarily high energy moments make these integrals absolutely convergent after Cauchy–Schwarz; the locally integrable mass-shell measure controls zero momentum. The same argument applies to every derivative.

Consequently each matrix element of (eq:blockC) is a finite sum of derivatives of $R(x-y)$ multiplied by smooth functions. It is a distribution, with finite-order bounds on compact sets. The bounds depend continuously on energy seminorms of the two vectors. Differentiating in $x$ and applying (eq:jetcontact) gives (eq:blockcontact).

The weak insertion above is a single-block retarded contraction. It is not declared to equal every possible renormalized time-ordered composite insertion. Products with additional singular spectators would require their own product or extension argument. The mixed-product Ward theorem below is instead proved directly by compact-current commutators on the common field domain.

Let $K_0\subset M$ be a compact insertion support, and $I\subset \mathbb{R}$ a compact time interval. Choose

Choose $\chi \in {C_c}^{\infty }({\mathbb{R}}^3;\mathbb{R})$, equal to one on a neighborhood of

This set is bounded. We also require the support of $\nabla \chi$, throughout $\mathrm{supp}a$, to be strictly spacelike separated from $K_0$. Such a choice is obtained by making the one-region of $\chi$ larger than the compact causal shadows.

Define the actual spacetime-smeared operator

It acts on ${\mathcal{D}}_E$ by Lemma lem:wickest.

A local quadratic Wick field need not admit a sharp-time operator smearing on the Fock domain. All charges in (eq:q) are smooth spacetime smearings. Equal-time calculations in the proof below calculate the weak commutator, whose contraction is linear; they do not assert existence of an unsmeared quadratic charge operator at a fixed time.

Under the collar contract, the weak current commutator has action

Proof. First use the Noether representative (eq:jN). Its formal surface functional, solely for computing a weak commutator, is

The canonical commutators give its action on $\varphi$ as

Its action on $\pi$ is

The conformal Killing identities imply

Together with ${\partial }_0\pi =\Delta \varphi$, the latter expression is exactly ${\partial }_0(v_X\varphi )$.

This weak surface commutator is legitimate independently of a sharp-time quadratic operator: one current-field contraction leaves a linear field, and the equal-time distributions are the canonical delta distributions in (eq:CCR). They can equivalently be obtained by smooth time approximation before taking the weak commutator.

The commutator with the full surface current, as a function of the insertion coordinate, satisfies the free wave equation. Equation (eq:waveintertwine) shows that $v_X\varphi$ does too. The calculated Cauchy data therefore identify them by uniqueness of the scalar Cauchy problem. This argument is local in the causal shadow of the insertion. Replacing the full surface by $\chi$ leaves the commutator unchanged because the omitted support is spacelike separated from the insertion.

By Lemma lem:super, the improved and Noether currents differ on the surface by a spatial divergence. After multiplying by $\chi$, the difference is supported where $\nabla \chi \ne 0$. Its commutator also vanishes by the collar contract.

Finally average the weak surface commutator with $a$. It is the same $v_X\varphi$ for every surface time in the slab, and $\int a=1$. The resulting averaged operator is the well-defined operator (eq:q). This proves (eq:elementaryaction) on ${\mathcal{D}}_E$.

For every conformal Killing field,

as a tempered distribution.

Proof. For the fifteen displayed fields, direct polynomial calculation gives

Apply this identity to the rational inverse of the squared separation. The weight terms in ${d_X}^x+{d_X}^y$ cancel its derivative.

To pass the identity to (eq:W), take independent complex point variables in a positive-energy tube. The same polynomial identity holds there, before taking a boundary value. The rational two-point function has the tempered boundary value (eq:W); differentiation and polynomial multiplication commute with this boundary limit. This proves (eq:Wcov), including the light-cone singularity.

Define the prolonged evolutionary action by

The explicit jet formula is

In particular, descendants are not silently treated as primaries.

Let $B_1,\dots ,B_n$ be any Wick fields, including $Q,L,t$ and their descendants. Let their compact insertion supports lie in $K_0$, and choose $q_X[a,\chi ]$ with the collar contract. Then, on ${\mathcal{D}}_E$,

and

The identity extends to every compact joint kernel with the same local support condition and every joint-Schwartz spectator vector.

The ordered vacuum Ward identities are

as joint tempered distributions.

Proof. The elementary action is Lemma lem:cauchyaction. Differentiating it gives (eq:jetaction).

To calculate the action on a normal product, first use separated point variables and subtract the vacuum contractions defining normal ordering. The commutator is a derivation on the ordinary separated field product. Any possible extra numerical term from the subtracted contractions is a derivative of $({d_X}^x+{d_X}^y)W(x-y)$, and therefore vanishes by Lemma lem:Wcov. Taking the normal-product limit gives (eq:prolong) and (eq:wickaction). The limits can be taken on ${\mathcal{D}}_E$: their Fourier kernels converge under the weighted fixed-sector bounds used in Lemma lem:wickest.

The common invariant domain makes the operator commutator Leibniz identity legitimate for every finite ordered product. This gives (eq:mixedaction).

For a compact joint kernel, cover its support by finitely many product patches and approximate in the compact-test topology by finite tensor sums with supports in those patches. Proposition prop:domain gives convergence both of the words and of their current insertions. The same argument applies to joint-Schwartz spectator vectors, using longer-word bounds. Thus the stated identities hold on the complete declared test class, not only on factorized tests.

Finally expand an ordered vacuum Wick correlation into its contractions. On each contraction the summed infinitesimal action vanishes by (eq:Wcov), including all differentiated contractions. Every contraction term, and hence the complete finite Wick sum, obeys (eq:vacward).

The theorem concerns already constructed ordered field words and their joint-test extensions. It does not multiply arbitrary numerical distributions at a common singular point. Nor does it infer closure of an algebra of single-integral local functionals under ordinary multiplication: such a product is generally multilocal.

The vector-field bracket is

Its conformal factor obeys

The current commutator in (eq:wickaction) gives $v_X$, with positive point-field action $d_X\varphi$. As an evolutionary action, it keeps the numerical coefficients fixed. Its bracket is therefore an anti-representation. To state a representation with the vector-field bracket, define

on a localized word. The result is independent of any choice satisfying the collar contract.

The derivations ${\delta }_X$ satisfy

on every ordered Wick word, all joint compact kernels, and ${\mathcal{D}}_E$.

Proof. For the scalar smearing, integration by parts gives

Indeed, the transpose of $d_X$ with respect to the flat measure is $-X^{\mu }{\partial }_{\mu }-3{\sigma }_X$. The divergence cocycle gives

Hence the claimed identity holds on every smeared scalar.

It passes to derivatives because the action is prolonged, not assigned a new primary weight. It passes to normal products by Lemma lem:Wcov, which ensures compatibility with contraction relations. It passes to arbitrary ordered words because both sides are derivations. Continuity and support-preserving tensor density then give the complete joint-test identity.

debugging{ With $v_X\varphi =(X\cdot \partial +{\sigma }_X)\varphi$ and fixed numerical coefficients, one has $[v_X,v_Y]=-v_{[X,Y]}$. The representation statement is for ${\delta }_X=-v_X$, as in (eq:deltasign). Ignoring this distinction reverses a Lie-bracket sign.}

The proposition proves Lie compatibility of the local field actions. It does not prove essential self-adjointness of the fifteen global charge operators, nor their simultaneous integration to a group.

The admissible class consists of symmetric rank-two tensor fields $S_{\mu \nu }$ satisfying all of the following:

1. $S$ is a finite vacuum-normal-ordered local polynomial in $\varphi$ and its derivatives, on the declared free field domain.

2. Its coefficients are constant, with no new dimensionful parameter or coordinate-dependent coefficient.

3. It is Poincare covariant and parity even; tensor coefficients are built from $\eta$.

4. Every monomial has engineering dimension four, with $\dim \varphi =1$ and $\dim \partial =1$.

5. It is conserved on the free shell.

Expressions are identified by the actual free field equation and normal-product identities, not by their presentation labels.

This is a specified physical polynomial class. It does not contain every local distributional defect compatible with an unspecified renormalization problem.

Put

Every $S$ in Definition def:stressclass has a unique form

Its trace is

Its translation Ward action on the scalar is $\lambda$ times the original translation action. Requiring the original translation normalization and zero trace forces

Proof. Dimension four and the available constant tensor contractions exhaust the monomials as follows.

A degree-two term has two derivatives and is a linear combination of

The last is zero on shell. A degree-four term has no derivatives and is a multiple of ${\eta }_{\mu \nu }:{\varphi }^4:$. A degree-one term would need three derivatives and an invariant constant tensor of odd total rank; none is built from $\eta$. A degree-three term has one derivative and the same odd-rank obstruction. A degree-zero term of dimension four would require a forbidden dimensionful constant; derivatives of the identity vanish. No higher field degree is compatible with dimension four.

Thus, on shell,

Conservation gives

The vacuum-to-four-particle matrix of the last term is nonzero, whereas every quadratic term has zero such matrix. Hence $e=0$. The vacuum-to-two-particle matrix of ${\partial }_{\nu }K$ is nonzero for noncollinear positive-energy momenta, so

Since

the unique parameters are

Uniqueness also follows from the independent quadratic tensor polynomials on noncollinear two-particle momenta.

The trace is $-\lambda K+3c\square Q=(6c-\lambda )K$. For a translation, $I_{\mu \nu }{a}^{\nu }$ is a divergence superpotential. Its compact-slab commutator vanishes under the collar contract. The canonical tensor therefore contributes exactly $\lambda$ times the original translation action. The last assertion follows.

Choose real compact $f,h$ such that

Such choices are explicit in the following sense. Take nonzero nonnegative compact bumps whose Fourier transforms are nonzero near zero. Then