Landmark VII-C A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions

We retain the sharp zero-Cauchy harmonic dressing and its exact off-shell classical functional, and examine a boundary enlargement of the free polynomial Wick domain. The outcome is a specified continuity obstruction, not an algebra construction. For the fixed Minkowski de Donder two-point matrix, the full tensor calculation gives $H_{ww}=0$, while $H_{wh}$ contains a Cauchy-reflected frequency branch. Linear products, the insertion self-product, and a mixed quadratic test admit compact distributional polynomial representatives. Nevertheless, the mandatory ordered word $B_f\star W_g\star B_{f^{\prime }}$, with $W_g=\int g{h_{00}}^2$, has a fully contracted coefficient whose regular-kernel approximations grow logarithmically. We compute the nonzero coefficient explicitly as a product of reflected light-ray transforms. The approximating insertion kernels converge in a specified normal topology, so this divergence rules out a sequentially continuous extension of that coefficient. Consequently, it rules out a hypocontinuous Wick algebra extension satisfying the stated admission and coefficient-continuity requirements. The logarithmic coefficient can be nonzero for three pairwise disjoint observation supports. No free equation is imposed on configurations, no additional coordinate field is introduced, and no retarded product is constructed.

Figure fig:split-VII-C records one view of the same separation. Figure fig:chain-VII-C records one view of the same separation. Figure fig:pair-VII-C records one view of the same separation. Figure fig:gate-VII-C records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-VII-C records that separation. The drawing adds no hypothesis.

The sharp dressing considered here is

The minus sign agrees with the first geometric coefficient ${{\mathcal{B}}_f}^{(1)}$. All classical configurations are smooth and unrestricted by their equations of motion.

The preceding obstruction is not interpreted as closing the interacting product problem. Here we ask a narrower question: does admitting the actual compact distributional kernel of (eq:B), with a specified normal topology, permit a continuous extension of the fixed free Wick product?

The answer proved below is negative for a named mandatory graph. The continuity assumptions are part of the theorem. An abstract algebra with a different topology, a smaller collection of bulk observables, or additional renormalized boundary operations is not excluded.

Let

Indices are raised with $\eta$, and

The zero-Cauchy inverse satisfies

We take $f\in {C_c}^{\infty }(M_{+})$. Thus its support has a positive distance in the time coordinate from ${\Sigma }_0$.

dictionary. Relative to the convention ${\eta }_{\mathrm{B}}=\mathrm{diag}(1,-1,-1,-1)$,

All raised-index expressions are recomputed using $\eta$. If one also changes $h=-h_{\mathrm{B}}$ so that the full metric changes sign, then $C[h]=-C_{\mathrm{B}}[h_{\mathrm{B}}]$ and ${\mathsf{E}}_{0}C[h]={{\mathsf{E}}_0}^{\mathrm{B}}{C}_{\mathrm{B}}[h_{\mathrm{B}}]$.

The active even blocks are

where $m\ge 0$ is fixed and

No positivity is assumed for the graviton block.

The remaining gauge-fixed multiplet components are spectators in every calculation below. An enlargement of the full multiplet algebra must, in particular, admit the displayed even-sector products. Therefore an obstruction in this sector is sufficient for the stated conclusion. No inference about physical observables in gauge cohomology is made; that is a separate problem in the perturbative gravitational framework BFR.

We use the Wick-symbol convention

For finite polynomials this sum is finite. A local expression such as $\int g{h_{00}}^2$ denotes its Wick symbol; no self-contraction is included inside that expression.

Let ${\mathcal{P}}_{\mathrm{mc}}$ denote finite polynomial functionals with compact distributional kernels satisfying the usual microcausal exclusion: for a kernel of degree $n$,

with the total zero covector omitted. Let ${\mathcal{P}}_{\mathrm{reg}}\subset {\mathcal{P}}_{\mathrm{mc}}$ denote the finite polynomials with smooth compact kernels.

Only finite polynomials are used. Results about unrestricted smooth microcausal functionals are not presumed. In particular, the distinction between polynomial and general functional domains identified by Hawkins, Rejzner, and Visser is respected HRV. Their generalized-cone framework motivates a cone test, but supplies no premise asserting that $B_f$ passes it.

The principal result is the following statement, whose topology and coefficient are defined precisely in Sections sec:topology and sec:triple.

For the fixed matrix (eq:Hhh)–(eq:Hscalar), there exist real nonnegative $f,f^{\prime },g\in {C_c}^{\infty }(M_{+})$, with pairwise disjoint supports, and regular polynomials $B_{f,\epsilon },B_{f^{\prime },\epsilon }$ such that:

1. their mixed kernels converge to the exact kernels of $B_f,B_{f^{\prime }}$ in the normal-topology stages specified in Section sec:topology;

2. with $W_g=\int g(y)h_{00}(y{)}^2{\mathrm{d}}^{4}y$,

   $${\mathrm{ev}}_0[{\hslash }^3]\left(B_{f,\epsilon }{\star }_H{W}_g{\star }_H{B}_{f^{\prime },\epsilon }\right)=\mathcal{C}(f,g,f^{\prime })\log ({\epsilon }^{-1})+O(1),$$

   where $\mathcal{C}(f,g,f^{\prime })>0$;

3. the corresponding fully contracted coefficient has no sequentially continuous extension at the pair of sharp kernels that agrees with the old Wick product on regular kernels.

In particular, no algebra extension satisfying Assumption ass:extension exists.

The evaluation ${\mathrm{ev}}_0$ sets all classical fields to zero and hence extracts the field-independent coefficient. It is not a replacement of the algebra-valued target by a matrix element: it is a necessary continuous coefficient map of the proposed algebra extension.

Write

All formulas act componentwise on $h$.

For every smooth $h$, on $M_{+}$,

Consequently,

is an exact off-shell identity.

For $K=\mathrm{supp}f$, set

Then ${\mathcal{D}}_K$ is compact, and $B_f$ depends on $h$ only in ${\mathcal{D}}_K$ and on $\varphi$ only in $K$.

Proof. Let $A=\sqrt{-{\Delta }_{\mathbf{x}}}$. In spatial Fourier notation,

and

The quotients at zero spatial frequency are interpreted by their smooth limits.

The difference between the two sides of (eq:reconstruction) solves the homogeneous wave equation with zero Cauchy data, and therefore vanishes. Applying ${\mathsf{E}}_{0}C$ gives (eq:wsplit), and substituting into (eq:B) gives the claimed split.

Finite propagation speed implies that $w[h](x)$, for $x\in K$, depends only on the restriction of $h$ to the causal segment between ${\Sigma }_0$ and $x$. The same holds for its derivatives. The union of these segments is contained in ${\mathcal{D}}_K$. Compactness follows directly from compactness of $K$ and the Minkowski causal bound.

The triple $({\gamma }_{0}h,{\gamma }_{1}h,j)$ is a presentation of the same off-shell field, not a new collection of propagating fields. In particular, $j$ is not set to zero.

For linear source smearings, the bisolution identities imply

Thus such source variables are central for the free Wick product. This does not make them zero classical functionals, and it does not by itself define products involving arbitrary singular source-sector kernels.

We order the mixed kernel variables as $(x,z)$, with $x$ the scalar slot and $z$ the graviton slot. There is a compact tensor-valued distribution $b_f(x,z)$ such that

The compact-kernel assertion follows either from the explicit construction below or from continuity of the bilinear functional (eq:B) and Proposition prop:reconstruction.

Let $E_{\mathrm{ret}}$ denote the retarded fundamental solution of $\square$. For $x^0>0$,

The identity is local in $x$, so harmless compact cutoffs can be inserted in the integration variables.

For covectors in the fixed Cartesian coordinates, define

The case $\xi =0$ is included in ${\Lambda }_{\partial }$.

The kernel $b_f$ satisfies

More precisely, its possible singular branches arise from:

1. the bulk diagonal $x=z$;

2. direct null propagation from $z$ to $x$;

3. $z^0=0$ together with null propagation, with covectors

   $$(\xi ,-\xi +\tau \mathrm{d}t);$$

4. pure boundary conormals

   $$(0,\tau \mathrm{d}t).$$

This is an upper bound, not an assertion that every listed covector occurs.

Proof. The retarded wave kernel has diagonal conormals and, away from the diagonal, the null propagation relation $(x,\xi ;z,-\xi )$. Multiplication by $\theta (z^0)$ is permitted: a wavefront covector of that factor has zero $x$-component, whereas a wavefront covector of the retarded kernel has nonzero $x$-component. Hence opposite covectors cannot sum to zero.

The product theorem yields the direct branches, the boundary sums $(\xi ,-\xi +\tau \mathrm{d}t)$, and the pure boundary conormals. At $z^0=0$, the bulk diagonal does not meet the support of $f$, because $f$ is supported strictly in $M_{+}$.

Integrating the first-order operator $C$ by parts in $z$ gives the kernel of $L$. This can produce a $\delta (z^0)$, but not a ${\delta }^{\prime }(z^0)$. Finally, moving ${\partial }_{\mu }$ from $\varphi$ onto the $x$-kernel and multiplying by $f(x)$ do not enlarge the wavefront set. This proves the assertion.

The distinctions between multiplication, pullback, and pushforward used here are the standard distributional ones; membership in a wavefront-restricted vector space is not itself an operation theorem BDH.

Let ${\mathcal{N}}_{\pm }$ be the two null cones in the cotangent bundle and put ${\tilde{\mathcal{N}}}_{\pm }={\mathcal{N}}_{\pm }\cup \{0\}$. The proposed characteristic-cone condition is

For every nonzero $f\in {C_c}^{\infty }(M_{+})$,

In particular, its full mixed kernel cannot be admitted merely by replacing the causal exclusion cones by the null exclusion cones.

Proof. Fix $\mathbf{n}\in S^2$, let $r>0$, and consider the smooth complexified configuration

A direct use of (eq:C) gives

Solving the zero-Cauchy equation yields

Choose $a\in {C_c}^{\infty }(M_{+})$ with $\int fa{\mathrm{d}}^{4}x\ne 0$, and set

Substitution into (eq:B) gives

Indeed, the nonoscillating term is obtained from

all terms containing $e^{-2irt}$ decrease rapidly after the $t$-integration, including the term with the prefactor $rt$. The term containing ${\partial }_{t}a$ is $O(r^{-1})$.

This is a Fourier test of the compact kernel $a(x)b_f(x,z)$, contracted with the constant tensor $P_{\alpha \beta 00}$, along a ray for which both slot covectors are null and have the same temporal orientation. If the characteristic-cone condition held, compactness and the definition of the wavefront set would imply rapid decrease along that ray. The nonzero limit contradicts it.

If desired, the plane wave in the graviton slot can be multiplied by a compact cutoff equal to one on a neighborhood of ${\mathcal{D}}_{\mathrm{supp}f}$, without changing the calculation.

Define

These pullbacks exist because the conormals of the spacelike Cauchy surface are timelike, whereas the nonzero Hadamard covectors are null. The same argument permits the bulk–boundary blocks. This is an application of the pullback criterion, not a multiplication of boundary delta distributions BDH.

For the scalar massless factor, spatial Fourier transformation gives

multiplied by $P_{\alpha \beta \rho \sigma }$ in the graviton sector. The entries are locally integrable at $r=0$ in three spatial dimensions. The full matrix, including the off-diagonal phases, is retained.

For the matrix (eq:Hhh),

The induced free Cauchy contractions satisfy

Proof. In four dimensions,

The trace contribution in ${C_x}^{\mu }$ cancels the trace term in the first divergence of $P$, leaving (eq:CH). Applying ${C_y}^{\nu }$, using ${\partial }^y=-{\partial }^x$ on ${\Delta }_0^{+}(x-y)$, cancels the two mixed second-derivative terms and gives (eq:CHC).

To justify the Cauchy operations without multiplying singular unsmeared kernels, use (eq:positive). On each massless momentum mode, the twice-contracted tensor numerator in (eq:CHC) vanishes before either zero-Cauchy solution operator is applied. The remaining mode factors have polynomial growth and locally integrable infrared behavior. Thus the smeared contraction is zero. Equivalently, the Cauchy covariance construction based on (eq:Cauchymatrix) gives the zero distribution.

The preceding result does not say that $w[h]$ is the zero off-shell functional. It does not say that $H_{wh}=0$. It depends on the fixed matrix (eq:Hhh); no claim is made for an arbitrary gauge choice or an arbitrary smooth change of covariance.

The component needed for the obstruction is

By (eq:CH),

For $r>0$,

Consequently,

The terms $r^{-1}$ and $r^{-2}$ are locally integrable at $\mathbf{p}=0$. At large momentum this is an oscillatory distribution with symbol amplitudes.

The last term in (eq:K) is the Cauchy-reflected branch. Its first time frequency is opposite to that of the direct Hadamard branch, while its second time frequency is unchanged. This is the branch responsible for the obstruction.

Reality of ${\mathsf{E}}_{0}C$ and Hermiticity of the fixed covariance imply

The results in this section are existence statements for specified compact polynomial kernels. They are not a closure theorem.

Independently smeared Cauchy fields are admitted using (eq:Cauchymatrix). Their pairings with ordinary spacetime smearings are well defined. This does not define coincident products of boundary fields.

For a compact smooth graviton test tensor $a$, write

Since $H_{hh}a$ is smooth,

Likewise, for $u\in {C_c}^{\infty }(M)$,

The reverse products use the transposed covariance. All displayed coefficients are compact distributional polynomial functionals with the causal support inherited from Proposition prop:reconstruction.

For $f,f^{\prime }\in {C_c}^{\infty }(M_{+})$, the following compact distributional polynomial is well defined:

It is the Wick contraction formula for the proposed $B_f{\star }_H{B}_{f^{\prime }}$. The potential ${\hslash }^2$ coefficient vanishes.

Proof. For fixed smooth $h$, the factor multiplying the differentiated Hadamard distribution in (eq:self) is a smooth compact test function. Hence the pairing exists.

The map from $h$ to $w[h]$, restricted to any fixed compact observation region, is continuous in the smooth topology and depends only on a fixed compact causal region. Polarization of (eq:self) therefore gives a continuous bilinear form on the corresponding smooth graviton configurations. The Schwartz kernel theorem gives a compact distributional quadratic kernel.

There are no $h$–$\varphi$ contractions. Every term involving a $w$–$w$ contraction vanishes by Proposition prop:tensor. This leaves exactly (eq:self).

The conclusion is stronger than a merely formal inventory of self-contractions, but weaker than an algebra theorem. The mixed triple below shows why the distinction is necessary.

Let

The finite contraction formulas for $B_f{\star }_H{Q}_q$ and $Q_q{\star }_H{B}_f$ define compact distributional polynomial kernels. In particular, the double contraction

is a well-defined distribution.

Proof. Each component of $H_{wh}$ has a direct and a reflected Fourier representation of the form illustrated in (eq:K), with polynomial tensor numerators. In either branch, every nonzero wavefront covector in the $y$-slot has the same temporal orientation as the second slot of ${\Delta }_m^{+}$; it is nonzero and null. This follows directly from the phase $irs-i\mathbf{p}\cdot \mathbf{y}$. The direct branch has the usual null relation and the reflected branch has the relation obtained by reflecting the first time coordinate.

The two factors in the displayed product therefore cannot have opposite total covectors: their $y$-components already have the same orientation in a pointed cone. The multiplication criterion applies.

A single $h$-contraction leaves a distribution multiplied by the smooth classical scalar factors. A single scalar contraction leaves a differentiated Hadamard distribution multiplied by the smooth factors $w[h](x)h(y)$. As in Proposition prop:self, continuity in the smooth configurations and the kernel theorem give the remaining compact polynomial kernels. The reverse product has the reversed, but again common, orientation.

The product with

also exists at the level of $B_f{\star }_H{W}_g$: at most one graviton leg of $B_f$ can contract with $W_g$, and the resulting mixed covariance is a distribution paired with smooth remaining configuration factors. What fails is the mandatory subsequent multiplication by a second insertion in the topology specified next.

For a compact $K\subset M\times M$, write

componentwise in the finite-dimensional tensor indices. It carries the normal topology.

Explicitly, this topology is generated by strong distribution seminorms

where $\mathcal{B}$ is a bounded set of test functions, and by

where $V$ is a closed cone and

The normal topology and the distinction between continuity and hypocontinuity are important here DB,BDH.

The translation-invariant angular cone ${\Lambda }_B$ in (eq:GammaB) is deliberately an upper-bound cone. The theorem does not presume that it is the smallest possible cone.

Choose a real, nonnegative, even, Euclidean-radial $\rho \in {C_c}^{\infty }({\mathbb{R}}^4)$ with $\int \rho =1$, and put

Define

Its mixed kernel is

In particular, $B_{f,\epsilon }\in {\mathcal{P}}_{\mathrm{reg}}$.

These are approximation devices for the exact sharp kernel. They are not proposed as replacement observables.

For every fixed $f$, choose a compact $K$ containing a neighborhood of $\mathrm{supp}{b}_f$. Then, for all sufficiently small $\epsilon >0$,

with its normal topology.

Proof. Smoothness and the fixed compact support follow from convolution with a compact approximate identity.

Strong distribution convergence follows from the finite order of $b_f$ and the convergence

in all smooth seminorms, uniformly for $\psi$ in a bounded set of test functions.

For completeness, the microlocal estimates can be checked directly because ${\Gamma }_B$ has a translation-invariant angular cone. Compactness and $\mathrm{WF}(b_f)\subset (M\times M)\times {\Lambda }_B$ imply rapid decrease of ${\hat{b}}_f$ in every closed cone separated from ${\Lambda }_B$. Moreover,

The multipliers are uniformly bounded and tend to one on compact frequency sets.

On a closed cone separated from ${\Lambda }_B$, split frequency space into a fixed ball and its complement. On the ball, the multiplier converges uniformly to one. On the complement, arbitrarily high rapid-decay bounds for ${\hat{b}}_f$ give a uniformly small tail. Thus the weighted Fourier seminorms converge to zero.

For a localized seminorm, use

A slightly larger cone still disjoint from ${\Lambda }_B$ controls the nearby-frequency part. On the complementary part, the angular separation forces the argument of $\hat{\chi }$ to be large; its arbitrary rapid decrease dominates the uniform polynomial growth of ${\hat{b}}_f$. This proves convergence of every normal-topology seminorm.

The approximants have support in arbitrarily small neighborhoods of the exact compact support; the limit retains the exact causal support of Proposition prop:reconstruction. The theorem uses an ordinary fixed-support distribution stage containing those neighborhoods. It does not impose an equation of motion and does not replace $\theta$ by a new smooth-edge observable.

Let $f,f^{\prime },g$ be real compact smooth functions, with $f,f^{\prime }$ supported in $M_{+}$, and set

Define

This coefficient is unambiguously defined in the old polynomial domain.

There is exactly one fully contracted graph type: the scalar legs of the two insertions are joined, and the two graviton legs are joined to the two legs of $W_g$. The two assignments of the $W_g$ legs give a factor of $2$. No $ww$ contraction is present in this graph.

Let

Euclidean radiality makes this independent of $\mathbf{n}$. It is real, smooth, rapidly decreasing, and $\mu (0)=1$. For the scalar field define

Here and below ${\omega }_m(r)=\sqrt{r^2+m^2}$.

Let $K_{\epsilon }$ be the right-hand side of (eq:K) with the additional multiplier $\mu (\epsilon |\mathbf{p}|)$ in the momentum integral. This is exactly the covariance between $w[S_{\epsilon }h](x)$ and $h_{00}(y)$.

For $q\in {\mathbb{R}}^3$, put

For every $\epsilon >0$,

For $f^{\prime }=f$ and $g\ge 0$, this expression is nonnegative.

Proof. By (eq:CH), a contraction with $h_{00}$ forces the insertion vector index to be $0$. Thus the scalar contraction is

The factor $2$ from the two $W_g$-leg assignments cancels the factor $1/2$ in this spectral density. The two minus signs in the definitions of the insertions cancel as well. Hermiticity gives the complex conjugate factor in (eq:spectral).

For fixed $\epsilon >0$, the mollifier multipliers and the compact smooth smearings make the resulting integrals convergent. The positivity assertion uses only the positive scalar spectral measure, not positivity of the graviton covariance.

For $y=(s,\mathbf{y})$ and $\mathbf{n}\in S^2$, define

Only a compact interval contributes. Geometrically, this samples a future observation ray issuing from the spatial point $\mathbf{y}+s\mathbf{n}$ on ${\Sigma }_0$.

Let $Y\subset M$ be compact. Uniformly for $y\in Y$, $\mathbf{n}\in S^2$, $r\ge 1$, and $0<\epsilon \le 1$,

where

For $|q|\le 1$, $F_{f,\epsilon }(y,q)$ is uniformly bounded on $Y$, independently of $0<\epsilon \le 1$.

Proof. Separate (eq:K) into its direct and reflected terms.

term. After integration against $f$, the direct term contains the Fourier transform of $f$, or of $tf$, at temporal frequency $|p|+{\omega }_m(r)$ and spatial frequency $p+r\mathbf{n}$. Since $f$ is smooth and compact,

and the same holds with $tf$. Integrating against $|p{|}^{-1}+|p{|}^{-2}$ proves arbitrary rapid decrease in $r$, uniformly in $\epsilon$ and in $y\in Y$.

term. Write

On $|a|\le r/2$,

The constants in the last estimate may depend on the fixed mass. The phase is

uniformly for $t$ in the support of $f$ and $y\in Y$.

The spatial Fourier transform of $f(t,\cdot )$ decreases rapidly in $a$, uniformly in $t$, with all required derivatives. The errors above therefore contribute $O(r^{-1}+\epsilon )$ after extracting $r^{-2}$.

The leading $a$-integral is Fourier inversion:

Its $t$-integral is $A_f$.

On $|a|>r/2$, arbitrary rapid decrease in $a$ dominates the locally integrable factor $|p{|}^{-2}$. After extracting $r^{-2}$, that tail is $O(r^{-N})$ for any prescribed $N$, by taking sufficiently many Fourier-decay powers. The direct term can be absorbed into the same error. This proves (eq:Fasymptotic)–(eq:Ebound).

For bounded $q$, integrate the spatial Fourier-decay bounds against $|p{|}^{-1}+|p{|}^{-2}$. Both are locally integrable, and the large-$|p|$ tails are rapidly decreasing. This proves the last assertion.

For real $f,f^{\prime },g$ as above,

where

The $O(1)$ bound is for the fixed three test functions and fixed mass. The coefficient of the logarithm is independent of the chosen normalized radial mollifier.

Proof. Insert Lemma lem:asymptotic into (eq:spectral) and use polar coordinates $q=r\mathbf{n}$. For $r\ge 1$,

The leading contribution is therefore

We first check that the remainder is uniformly bounded. By (eq:Ebound), the error in the product of the two square brackets in (eq:Fasymptotic) is bounded by a constant times

Also, for every $N$,

Thus the error integral is controlled by

This is uniformly bounded: terms containing $r^{-1}$ are integrable without the cutoff, and

The correction from ${\omega }_m(r)/r-1=O(r^{-2})$ is integrable as well. The region $0\le r\le 1$ is uniformly bounded by the final assertion of Lemma lem:asymptotic.

It remains to evaluate $I_{\epsilon }$. For $r\ge 1$,

so boundedness of ${\mu }^{\prime }$ gives

Consequently,

Changing variables $u=\epsilon r$,

because $\mu (0)=1$, $\mu (u{)}^4-1=O(u)$ near zero, and $\mu$ decreases rapidly at infinity. This proves (eq:log) with the coefficient (eq:residue).

There exist nonnegative $f,f^{\prime },g\in {C_c}^{\infty }(M_{+})$ with pairwise disjoint supports and

Their supports may also be chosen disjoint from ${\Sigma }_0$.

Proof. Choose $S,T,T^{\prime }>0$ pairwise distinct and ${\mathbf{n}}_0\in S^2$. Consider

These three spacetime points are distinct. Choose nonnegative smooth bumps $g,f,f^{\prime }$, respectively positive near these points, with sufficiently small pairwise disjoint supports contained in $M_{+}$.

For $y$ near $y_0$ and $\mathbf{n}$ near ${\mathbf{n}}_0$, the rays in (eq:ray) meet the regions where $f$ and $f^{\prime }$ are positive. Hence both ray transforms are positive on an open set on which $g$ is positive. Formula (eq:residue) then gives the result.