Plateau-compatible relational scalars: a second-order classical intertwiner Landmark VII-A

We consider a minimally coupled scalar field on a formal metric $\mathbf{g}=\eta +\kappa h+{\kappa }^{2}k$ and specify a compactly switched scalar action together with unswitched harmonic coordinates having fixed Cauchy data. The admitted gauge transformations are supported strictly inside a region on which the interaction switch equals one and are the identity near the initial surface. For this restricted group we prove an exact first-order cancellation and a second-order classical intertwining identity. All switch derivatives are retained.

The proof uses the covariance of the switched, density-valued Euler operator under plateau-supported transformations and uniqueness of the coefficientwise Cauchy problems. We give the complete second-order relational coefficient, verify its transformation directly, and show that the construction restricts to a precisely defined plateau Einstein–scalar coupling jet. We also derive the first-order defect for larger gauge supports and explain why it cannot be repaired by adding a higher-order coefficient without changing the first-order construction.

A causal support bound and a classical comparison map between switches are proved. Neither equality for switches sharing only an observation plateau nor a quantum algebraic cutoff comparison is asserted. Renormalized products, quantum Ward identities, quantum responses, and counterterm separation are not constructed here. The result closes the classical gate of the finite-time programme, not its radiative gates.

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

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

This paper makes one choice throughout: harmonic coordinates are defined by a finite-time Cauchy problem, not by a Feynman inverse. The interaction switch is compactly supported, while the harmonic-coordinate equation is unswitched. Compatibility is obtained by restricting the admitted gauge transformations to a declared interaction plateau.

The effective-field-theory motivation is distinct from a claim of ultraviolet completion. Perturbative, locally covariant treatments of gravity provide an appropriate broader setting BFR. The present paper proves only a classical compatibility statement needed before attempting quantum products.

The principal result is the following, stated informally here and precisely in Theorem thm:second:

For the switched scalar action specified below, the scalar solution with fixed free Cauchy data and the unswitched harmonic-coordinate map transform by the same pullback under plateau-supported formal gauge transformations. Their relational composition is therefore invariant through second order.

The proof is off shell in the metric coefficients. Consequently it remains valid on any invariant restriction of those coefficients, including the Einstein–scalar jet defined in Section sec:einstein.

The motivating obstruction concerns a particular combination of unswitched coordinates, switched scalar response, fixed free input, and gauge parameters extending outside the interaction plateau. Restriction of the gauge domain does not refute that obstruction.

For self-containment, no numbered result from another Landmark manuscript is required here. Section sec:defect derives the relevant first-order defect and a concrete large-support witness directly. Thus the restriction adopted below is compatible with the large-group obstruction referred to in the production plan as Landmark VI, Theorem 12, without requiring its unavailable statement as a premise.

Several choices are mathematically essential.

1. A first-order response does not determine a second-order action. We use a single switch multiplying the entire difference between the curved and flat scalar Lagrangian densities.

2. The metric is a genuine second-order jet

   $$\mathbf{g}=\eta +\kappa h+{\kappa }^{2}k+\mathcal{O}({\kappa }^3).$$

   There is no factor of $1/2$ in the definition of $k$.

3. The scalar Euler operator is density valued. Its infinitesimal transformation consequently contains ${\partial }_{\mu }{\xi }^{\mu }$.

4. Compactly supported vector fields form a Lie algebra. The admitted group is generated by their flows; its second-order formal jets include both a first-order and a second-order gauge parameter.

5. Every inverse used below has prescribed Cauchy data. No identity of the form $\mathsf{E}P=\mathrm{id}$ is used on functions with unspecified initial data.

No global removal of the interaction switch is proved. No invariance under the full compactly supported diffeomorphism group is claimed for a fixed switch. We do not construct a quantum gravitational observable algebra, a quantum constraint algebra, a renormalized response, a counterterm catalogue, or an ultraviolet completion.

In particular, the classical result below is not itself a radiative prediction.

Let

and write

All coefficient fields are smooth up to ${\Sigma }_0$. Indices are raised and lowered with $\eta$ unless stated otherwise.

For $m\ge 0$ set

The operator ${\mathsf{E}}_m$ denotes the zero-Cauchy-data solution operator:

For sources supported away from ${\Sigma }_0$, it agrees on $M_{+}$ with the ordinary retarded Green operator applied to their extension by zero.

We use ${\mathsf{E}}_m$ on smooth sources. Sources need not have globally compact spatial support: on every compact observation region, finite propagation reduces the Cauchy problem to a compact portion of the source and initial data. The relevant existence, uniqueness, and finite-propagation statements are standard Cauchy theory for normally hyperbolic operators; see BGP (Chapter 3).

If $\psi \in C^{\infty }(M_{+})$ satisfies

then

More generally,

where $H_m[\psi ]$ is the homogeneous solution with the same Cauchy data as $\psi$.

Proof. The difference $\psi -{\mathsf{E}}_m{P}_m\psi$ solves the homogeneous equation and has the same Cauchy data as $\psi$. Uniqueness gives (eq:generalleft); the zero-data case gives (eq:leftinverse).

Fix a smooth free scalar

Its Cauchy data will remain fixed under all comparisons and admitted gauge transformations below.

Fix

Choose a nonempty relatively compact open set $U$ such that

In particular, $\overline{U}$ is separated from ${\Sigma }_0$.

Let

It is closed under the ordinary Lie bracket.

The group ${\mathcal{G}}_U$ is the subgroup of orientation-preserving diffeomorphisms of $M$ generated by flows of vector fields in ${\mathfrak{g}}_U$. Its elements are the identity outside a compact subset of $U$ and near ${\Sigma }_0$.

At second perturbative order we use pullbacks of the form

The Lie algebra of formal transformations (eq:formalF) is closed modulo ${\kappa }^3$ by the Baker–Campbell–Hausdorff formula. In particular, omitting $\zeta$ would not give a convenient composition-stable parametrization.

For scalars define

For scalar densities, represented relative to ${\mathrm{d}}^{4}x$, define

Thus

Every $F\in {\mathcal{G}}_U$ preserves $g$:

Indeed, $g=1$ on $U$, $F(U)=U$, and $F$ is the identity elsewhere.

Work coefficientwise in

No convergence of the formal series or global invertibility of a perturbed coordinate system is assumed.

Let

where $h$ and $k$ are smooth symmetric tensors. Put

and define

Writing

one obtains

Equations (eq:rhoexp)–(eq:Adensityexp) follow from (eq:metricjet).

Proof. The inverse metric is

Expanding $\exp (\frac{1}{2}\mathrm{tr}\log (\mathrm{id}+\kappa {\eta }^{-1}h+{\kappa }^2{\eta }^{-1}k))$ gives (eq:rhoexp). Multiplication gives (eq:Adensityexp).

Define

The switched scalar action is

The action is understood variationally, using compactly supported variations or variations respecting the prescribed boundary data; its numerical value need not be finite on every smooth configuration.

This prescription is equivalent to multiplying the difference between the curved and flat scalar Lagrangian densities by $g$. In particular, the quadratic metric vertex carries one factor of $g$, not a factor of $g^2$. Two first-order insertions will nevertheless produce two switch factors in the iterated response.

The corresponding Euler operator is

Its output represents a scalar density relative to ${\mathrm{d}}^{4}x$.

Define

Then

where

In expanded form,

These identities display all derivatives of the switch in the coefficient operators. In particular, ${\mathsf{V}}_g[h]\ne g{{\mathsf{Q}}_{\mathsf{h}}}^{(m)}$ in general.

The prescription (eq:switchedaction) is part of the theorem, not a consequence of the first-order repair. A different assignment of switch factors to higher vertices must be declared separately. The proof below applies to the action actually written here.

Define

by

and by requiring all positive-order scalar coefficients to have zero Cauchy data. Coefficient comparison gives

Each source to which ${\mathsf{E}}_m$ is applied in these formulas is smooth and compactly supported, because the coefficients of ${\mathsf{V}}_g$ and ${\mathsf{W}}_g$ are compactly supported. Thus these formulas define smooth classical coefficients without any product of singular distributions.

Equations (eq:phi1)–(eq:phi2) give the unique scalar jet satisfying the switched equation and the specified Cauchy data.

Proof. At order $\kappa$ the equation is

At order ${\kappa }^2$ it is

Existence follows from the Cauchy solution operator, and uniqueness follows successively at the two orders.

The four functions ${{\mathsf{X}}_{\mathbf{g}}}^{\mu }$ are defined coefficientwise by

with Cauchy data

There is no switch in (eq:harmonic).

Write

Define

Then

Equations (eq:w)–(eq:v) uniquely solve (eq:harmonic)–(eq:harmonicdata) through second order.

Proof. At first order,

At second order,

Both coefficients have zero Cauchy data. Existence and uniqueness follow as in Lemma lem:scalarunique.

The sources in (eq:w)–(eq:v) need not be compactly supported. Their use is legitimate as smooth Cauchy sources; the causal support bound proved later makes the dependence on a compact observation region explicit.

A map with leading term $\mathrm{id}$ has a unique formal inverse. In the present notation,

This is a formal statement. It does not assert that a nonperturbative harmonic-coordinate map is a global diffeomorphism.

Proof. Substitution into ${{\mathsf{X}}_{\mathbf{g}}}^{\mu }(x)=x^{\mu }+\kappa w^{\mu }(x)+{\kappa }^2{v}^{\mu }(x)$ cancels the first-order terms. At second order, Taylor expansion contributes $-w^{\nu }{\partial }_{\nu }{w}^{\mu }$, which is cancelled by the displayed inverse coefficient.

For a fixed label-space test density represented by $f\in {C_c}^{\infty }(U)$, define

The smearing is performed in coordinate-label space. Therefore no spacetime volume density is to be inserted into (eq:Of). An equivalent spacetime expression would contain the Jacobian of ${\mathsf{X}}_{\mathbf{g}}$.

Expanding,

where

The notation ${A_1}^{(g)}$ emphasizes that the first-order expression itself is unchanged; its admitted gauge domain is changed.

Define

For every smooth vector field $\xi$,

If $\xi$ vanishes near ${\Sigma }_0$, then

Proof. The trace of $K\xi$ is $2{\partial }_{\rho }{\xi }^{\rho }$, so

The second term is cancelled by half the derivative of the trace. Equation (eq:wK) then follows from Lemma lem:leftinverse, componentwise.

For every smooth scalar $\psi$,

In particular, if $P_m{\varphi }_0=0$, then

Proof. The unswitched density-valued operator

satisfies the change-of-variables identity

Differentiate at $\mathbf{g}=\eta$ along the flow of $\xi$. The variation of the operator is ${{\mathsf{Q}}_{\mathsf{K}\xi }}^{(m)}$, the variation of the scalar is $X_{\xi }\psi$, and the variation of the output density is $D_{\xi }{P}_m\psi$. Rearrangement gives (eq:QK).

Let the data satisfy Section sec:data. For every smooth metric coefficient $h$ and every $\xi \in {\mathfrak{g}}_U$,

More explicitly,

Consequently, the first-order coefficient of ${\tilde{\mathcal{O}}}_f[g]$ is invariant under the infinitesimal transformations admitted by ${\mathcal{G}}_U$.

Proof. All derivatives of $\xi$ are supported in $U$, where $g=1$. Hence $gK\xi =K\xi$. By linearity and Lemma lem:QK,

The last equality is valid because $\xi$ vanishes near ${\Sigma }_0$. Lemma lem:CK gives the coordinate variation. Substitution into (eq:A1) proves (eq:firstinvariance).

The theorem is stronger than a statement restricted to $h=0$ or $h=K{\xi }^{\prime }$: no field equation for $h$ is required.

of the first-order statement. Theorem thm:first alone proves neither second-order covariance, independence of $(g,U)$, independence of the dressing prescription, a full gravitational gauge quotient, nor a quantum Ward identity. Second-order covariance is established separately below; the other claims are not made.

Let $\xi \in {C_c}^{\infty }(M_{+}^{\circ };TM)$, without requiring $\mathrm{supp}\xi \subset U$. Then

Proof. The coordinate variation is still $\xi$, since the initial-data condition is unchanged. Thus the left-hand side is

By (eq:QKon) and Lemma lem:leftinverse,

Linearity of $\mathsf{Q}$ in its tensor argument gives (eq:defect).

There are compactly supported gauge parameters after ${\Sigma }_0$ for which (eq:defect) is nonzero, including parameters that are constant on a neighbourhood of $\mathrm{supp}g$.

Proof. Choose $\chi \in {C_c}^{\infty }(M_{+}^{\circ })$ with $\chi =1$ on a neighbourhood of $\mathrm{supp}g$, and let

for a nonzero constant vector $c$. Then $K\xi =0$ on a neighbourhood of $\mathrm{supp}g$, hence

The switched scalar correction has zero variation, while the coordinate correction changes by $\xi$. Therefore

Choose free Cauchy data such that $c^{\mu }{\partial }_{\mu }{\varphi }_0$ is nonzero at some point of $U$. Such data exist, for example by taking an appropriate real Klein–Gordon plane wave and restricting its smooth Cauchy data. A test function supported sufficiently close to that point gives a nonzero smeared variation.

This $\xi$ is not supported in $U$: its nontrivial cutoff to zero takes place outside the region on which it is constant, and that region contains $\mathrm{supp}g$.

With fixed free input and the same first-order coefficient ${A_1}^{(g)}$, adding a regular coefficient ${\kappa }^2{B}_2$ cannot make the observable invariant under every gauge parameter of Proposition prop:witness.

Proof. In a formal series with nonnegative powers of $\kappa$, invariance requires the coefficient at each order to vanish. A nonzero coefficient at order $\kappa$ cannot be cancelled by a coefficient at order ${\kappa }^2$.

Let $F_{\kappa }$ be as in (eq:formalF). The transformed metric jet is

with

The second equation is indispensable. Keeping $k$ fixed while changing $h$ by $K\xi$ does not describe the second-order pullback of a metric.

For $F\in {\mathcal{G}}_U$ and every smooth scalar $\psi$,

The identity also holds coefficientwise for the formal transformations (eq:formalF).

Proof. Regard ${\mathsf{A}}_{\mathbf{g}}$ as a contravariant tensor density and ${\rho }_{\mathbf{g}}$ as a scalar density. Their unswitched definitions are natural under pullback.

By (eq:Ag),

Since $F^{\ast }g=g$,

The difference between this expression and ${\mathsf{A}}_g[F^{\ast }\mathbf{g}]$ is

It vanishes: on $U$ the factor $1-g$ is zero, while outside $U$ the diffeomorphism and its derivative are the identity. Hence

The same argument applies to ${\rho }_g[\mathbf{g}]$.

The divergence of a vector density and the pairing with a scalar derivative are natural. Applying these operations proves (eq:plateaucov). Its formal coefficient identities follow by expansion.

Lemma lem:plateaucov is an identity for the complete divergence-form operator. It does not replace ${\partial }_{\mu }(g{A}^{\mu \nu }{\partial }_{\nu })$ by $g{\partial }_{\mu }(A^{\mu \nu }{\partial }_{\nu })$. The terms in (eq:Vgexpanded) and (eq:Wgexpanded) remain part of the operator.

For the transformed jet (eq:hprime)–(eq:kprime),

The free input ${\varphi }_0$ is the same on both sides.

Proof. By Lemma lem:plateaucov, the right-hand side of (eq:scalarintertwiner) solves the transformed switched equation. Since $F_{\kappa }$ is the identity near ${\Sigma }_0$, its Cauchy data are exactly those of ${\varphi }_0$. Uniqueness in Lemma lem:scalarunique proves the first identity.

For the second identity, each component of ${\mathsf{X}}_{\mathbf{g}}$ is a scalar solution of the unswitched, massless density equation. Ordinary naturality of that equation implies that ${\mathsf{X}}_{\mathbf{g}}\circ F_{\kappa }$ solves the harmonic equation for $F_{\kappa }^{\ast }\mathbf{g}$. Its Cauchy data agree with (eq:harmonicdata), again because $F_{\kappa }$ is the identity near ${\Sigma }_0$. Lemma lem:coordinateunique gives the result.

Coefficient extraction yields

For the switched action (eq:switchedaction), the harmonic Cauchy prescription (eq:harmonic)–(eq:harmonicdata), and the formal gauge transformations admitted by ${\mathcal{G}}_U$,

Consequently,

The coefficients are (eq:A1) and (eq:A2), with (eq:phi1), (eq:phi2), (eq:w), and (eq:v).

Proof. Formal inversion of (eq:coordinateintertwiner) gives

Together with (eq:scalarintertwiner),

This proves (eq:Phiinvariant). Smearing proves (eq:Ofinvariant). Taylor expansion gives the stated coefficients.

The pointwise identity is valid wherever the formal coefficients are evaluated. The restriction $\mathrm{supp}f\subset U$ is the observation contract used in this paper; it is not needed for the algebraic cancellation itself.

It is useful to verify that the second-order assertion contains the correct density terms. Write

Set

Expanding Lemma lem:plateaucov gives

These are identities of differential operators, not only identities on free solutions. In particular, they retain the terms proportional to $P_m$.

To verify the second scalar transformation directly, put

Since $P_{m}a=-\mathsf{V}{\varphi }_0$ and $P_{m}b=-\mathsf{V}a-\mathsf{W}{\varphi }_0$, equations (eq:Videntity)–(eq:Widentity) imply

The candidate has zero Cauchy data. Uniqueness proves (eq:phi2transform) independently of any informal cancellation of switch terms.

Substituting the constructed scalar coefficients into (eq:A2) gives the explicit expression

There are five displayed contributions: iterated scalar propagation, the second-order scalar vertex, the mixed first-order term, the second-order inverse-coordinate term, and the quadratic Taylor term.

None may be omitted merely because another term is described as a dressing correction.

For definiteness, normalize the unswitched Einstein–scalar action as

Its metric equation is

This fixes the normalization of $\kappa$ for this section.

The expansion is a coupling expansion. A nonzero free field ${\varphi }_0$ is allowed at order ${\kappa }^0$ because its gravitational source enters (eq:Einstein) at order ${\kappa }^2$. It is not being identified with a nonzero background expectation value in a linearization about a different Einstein–scalar background.

Define the Einstein-tensor coefficients by

An explicit coefficient definition is given in Appendix app:curvature.

The leading stress tensor is

The space ${{\mathcal{J}}_2}^g$ consists of tuples

such that:

1. $P_m{\varphi }_0=0$;

2. the scalar coefficients are the zero-data solutions (eq:phi1)–(eq:phi2);

3. the metric coefficients satisfy

   $$\begin{array}{rl}{G}^{(1)}[h]& =0,\\ G^{(1)}[k]+G^{(2)}[h,h]& =T^{(0)}[{\varphi }_0].\end{array}$$

No zero-Cauchy-data condition is imposed on $h$ or $k$.

The metric equations in this definition are unswitched at the retained order; the scalar equation is switched. On $P_g$ the scalar operator equals the unswitched curved operator, so the tuple satisfies the Einstein–scalar equations there through the stated orders.