Landmark VII-B Cauchy harmonic insertions and second-order retarded products: a sharp-boundary obstruction in an off-shell polynomial domain

We investigate the first geometric coefficient of a relational scalar defined using harmonic coordinates with fixed zero Cauchy data. The quantum fields are a scalar and a gauge-fixed spin-two field, together with the ghost and auxiliary sectors of perturbative gravity. The proposed insertion space is the algebra of compactly supported microcausal polynomial kernels, with a fixed graded Hadamard contraction.

For the coordinate-response operator $L={\mathsf{E}}_{0}C$, we compute its distributional adjoint exactly. It contains a hypersurface delta distribution whose nonzero conormal covectors are timelike. We prove an exact criterion for a linearly smeared coordinate response to belong to the declared domain. More importantly, for every nonzero real test function supported strictly to the future of the initial surface, the first geometric insertion fails the microcausal condition. The proof exhibits a smooth scalar configuration at which the metric derivative has a nonvanishing hypersurface delta term. This excludes a coefficientwise, classically normalized quantization of that insertion in the declared off-shell algebra, and hence excludes the proposed termwise second-order construction in that algebra.

The obstruction is not a claim that the relevant distributions do not exist. Smooth-edge regularization is constructed and its sharp limit is analyzed. A smooth spacetime representative is also constructed for each linear coordinate observable on the free mass shell. Neither result supplies the missing nonlinear interacting products. The paper therefore closes the boundary-domain alternative, not the second-order existence alternative.

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

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

A classical relational construction does not, by itself, specify a quantum product domain. This distinction is especially important when the reference coordinates are obtained by solving a hyperbolic equation with data prescribed on a sharp initial hypersurface.

We consider the geometric functional

where the four functions ${{\mathsf{X}}_{\mathbf{g}}}^{\mu }$ are harmonic for $\mathbf{g}$, with their values and first time derivatives fixed on an initial surface. All classical ingredients used below are specified explicitly in Section sec:classical; no theorem about their diffeomorphism intertwining is assumed or reproved.

The intended second-order interacting coefficient is formally

Before this expression can be constructed term by term, its geometric insertions must be admitted by the chosen domain. That prerequisite fails for the domain studied here.

The main result is Theorem thm:main. For every

the first geometric coefficient

is not a microcausal functional on the full smooth configuration space. The obstruction is a proved wavefront component, rather than a failure to verify a sufficient criterion for multiplying distributions.

We use a finite polynomial-kernel domain. This avoids relying on closure assertions for arbitrary smooth microcausal functionals: Hawkins, Rejzner, and Visser identify a closure problem for that larger class, introduce equicausal functionals, and establish equicausality for Wick polynomials with admissible fixed kernels HRV. The present obstruction is more elementary: it violates the microcausal cone condition itself, and therefore also excludes an equicausal representative that is equal to the original functional off shell.

The gravitational interpretation is deliberately limited. The gauge-fixed field content belongs to perturbative gravity, rather than to a scalar theory with a prescribed external metric. However, admission to a gauge-fixed algebra is not the same as physicality. The latter requires a separate gauge-cohomological analysis, as in the perturbative framework of Brunetti, Fredenhagen, and Rejzner BFR.

is and is not proved. The following statements are established:

1. an exact sharp-Cauchy adjoint formula;

2. an exact domain criterion for linearly smeared coordinate responses;

3. exclusion of every nontrivially smeared first geometric insertion from the declared off-shell domain;

4. impossibility of an off-shell, classically normalized quantization in that domain;

5. a smooth-edge approximation and a linear on-shell repair.

No Ward identity, radiative response, classification of interaction counterterms, or second-order product-existence theorem is asserted. In particular, the result does not exclude a construction in an appropriately enlarged boundary algebra or an interacting on-shell formulation.

Let

Indices are raised and lowered with $\eta$, and

The metric perturbation $h_{\mu \nu }$ is a smooth symmetric covariant tensor. Write

The symbol $h$ without indices denotes the trace only where no confusion can arise.

Let $E_{\mathrm{R}}$ and $E_{\mathrm{A}}$ denote the retarded and advanced Green operators for $\square$, respectively:

The zero-Cauchy solution operator ${\mathsf{E}}_0$ is defined by

For smooth $s$, it can be realized for $t>0$ as

The right-hand side is understood as the retarded solution of the past-compact distributional source $\theta s$. Its restriction to $M_{+}$ has the prescribed zero Cauchy data.

The first coordinate response is

This operator is not switched.

For a compact set $K⋐\{t>0\}$, set

In Minkowski spacetime this is compact. Finite propagation speed implies that $w[h]{|}_K$ depends only on $h$ in ${\mathcal{D}}_K$, including its trace on the initial surface.

Introduce a single metric field by

Define

Their first coefficients are

The harmonic-coordinate equations are written in density form:

Fix

Writing

gives (eq:L-definition) and

Here $v$ denotes the coefficient of ${\kappa }^2$, without a factorial.

Formal inversion yields

Consequently,

These are geometric coefficients, expanded before any interacting field equations are solved.

For clarity, consider arbitrary smooth formal curves

Then direct substitution gives

Thus the second metric coefficient $k$ enters through the interacting expansion of one graviton field. It is not a second independent quantum field.

Equation (eq:classical-bookkeeping) is an algebraic identity on smooth configurations. It is not a theorem asserting the existence of quantum retarded products. In particular, ${\varphi }_1$ and ${\varphi }_2$ are not included in the bare coefficients (eq:B1)–(eq:B2).

The quantum field multiplet is

The fields $c,\bar{c}$ are odd; the remaining fields are even. The scalar has mass $m>0$.

Let the trace-reversal operator be

The free action, modulo an unswitched total divergence, is

This is the linear de Donder gauge with a decoupled, shifted Nakanishi–Lautrup field. An explicit nonlinear action representative and its switched vertices are specified in Appendix app:vertices.

With the metric used to identify dual tensor bundles, the free Hessian has diagonal even blocks

and odd block, in the order $(c,\bar{c})$,

The displayed odd signs correspond to writing the quadratic action as $-{\bar{c}}_{\mu }\square c^{\mu }$.

For $m\ge 0$, write

Fix the graded two-point matrix by

All remaining blocks vanish. In particular, the algebraic auxiliary field has zero two-point function.

No positivity assertion is made for this gauge-fixed matrix. Moreover, gauge fixing does not impose $C[h]=0$ as an identity on the off-shell configuration space.

Let ${\overline{V}}_{\pm }\subset T^{\ast }M$ be the future/past nonspacelike covector cones, including the zero covector in each fibre. For $n\ge 1$, define

Wavefront sets never contain the total zero covector.

The space ${\mathcal{P}}_{\mathrm{mc}}$ consists of finite polynomials in the field multiplet whose coefficient kernels are compactly supported distributions $u_n$ satisfying

The kernels have the graded symmetry appropriate to their field slots. In a purely even sector,

The quantum polynomial algebra is

with the graded Wick product determined by $\mathcal{H}$.

For fixed polynomial degree the Wick expansion has only finitely many contractions. The usual wavefront calculus gives the Wick product on this domain; the polynomial setting also falls within the fixed-kernel results of HRV. Background on these contractions and their microlocal formulation can be found in FR.

We shall use only a necessary property of this domain.

If $F\in {\mathcal{P}}_{\mathrm{mc}}$, then every functional derivative satisfies

at every smooth configuration $\Psi$. In particular, $F^{(1)}(\Psi )$ has no nonzero timelike covector in its wavefront set.

Proof. Differentiation produces finite sums of partial contractions of the compact kernels $u_n$ with smooth configurations. In the wavefront bound for a partial contraction, the covectors in the contracted slots must be zero. If the remaining covectors were all future nonspacelike, or all past nonspacelike, adjoining the zero covectors would give a forbidden element of $\mathrm{WF}(u_n)$. This proves the claim. The same argument applies componentwise in the graded theory.

The geometric coefficients are initially considered as functionals on the full space of smooth configurations. Quotienting by the free field equations is not part of Definition def:poly.

An admissible quantization of a classical insertion $F$ is an element

whose coefficient $Q_0$ equals $F$ on every smooth off-shell configuration.

This requirement distinguishes equality of the original functional from equality only after imposing field equations.

Let

be a test covector, and define

The field $b$ is smooth. It is supported in $J^{-}(\mathrm{supp}a)$, and it solves

In a full neighbourhood of ${\Sigma }_0$, it solves the homogeneous wave equation.

The formal adjoint of $C$, using the pairing $\int b_{\mu }{C}^{\mu }[h]$, is

Let $n_{\mu }={\partial }_{\mu }t$. The compactly supported distributional adjoint $q_a=L^{\ast }a$ is

Furthermore,

There is no ${\delta }^{\prime }(t)$ term in this adjoint.

Proof. For smooth $h$, the retarded representation and formal adjointness give

The pairings involve only the compact causal set $J^{-}(\mathrm{supp}a)\cap M_{+}$. Hence the calculation is valid without a global compact-support assumption on $h$.

Applying the product rule in distributions and using

gives (eq:q-adjoint). Since $C^{\ast }$ has differential order one, it differentiates the step function at most once. This proves the absence of a delta derivative. Differentiation does not enlarge support, giving (eq:q-support).

The map $v\mapsto \mathsf{B}(v)$ is injective. In the chosen coordinates,

Proof. The displayed components follow directly from (eq:boundary-map). The $00$ and $0i$ components recover all components of $v$.

Let

Every element of this conormal bundle is timelike.

For every $a$ as above,

If $b(0,{\mathbf{x}}_0)\ne 0$, then

In particular, the hypersurface delta term cannot be cancelled by the step-function term in (eq:q-adjoint).

Proof. Both coefficients in (eq:q-adjoint) are smooth. The standard wavefront bounds for a smooth multiple of a step or delta distribution therefore give (eq:q-wf-upper); see BDHwavefront.

For the lower bound, choose a tensor component for which $\mathsf{B}(b(0,{\mathbf{x}}_0))\ne 0$, possible by Lemma lem:B-injective. Locally that component is

with $r,v$ smooth.

After compact localization, Fourier transformation in the normal variable has the form

at fixed tangential frequency $\mathbf{\zeta }$, in either normal direction. For any localization nonzero at $(0,{\mathbf{x}}_0)$, the localized $v$ is not identically zero, so its tangential Fourier transform is nonzero at some fixed $\mathbf{\zeta }$. Thus rapid decay fails in every conic neighbourhood of each nonzero normal covector. The step term has one lower normal Fourier order and cannot cancel the nonzero leading term.

For $a\in {C_c}^{\infty }(\{t>0\};T^{\ast }M)$, the following are equivalent:

1. ${\ell }_a\in {\mathcal{P}}_{\mathrm{mc}}$;

2. $q_a=L^{\ast }a$ is smooth;

3. the advanced solution $b=E_{\mathrm{A}}a$ has vanishing full Cauchy data on ${\Sigma }_0$:

   $$b{|}_{{\Sigma }_0}=0,{\partial }_{t}b{|}_{{\Sigma }_0}=0.$$

When these conditions hold, $q_a$ is a smooth compactly supported test tensor.

Proof. For a linear functional, domain membership is exactly the condition $\mathrm{WF}(q_a)\subset {\Xi }_1$. By (eq:q-wf-upper), every possible singular covector of $q_a$ is timelike. Hence domain membership is equivalent to smoothness.

Suppose $q_a$ is smooth. Its delta coefficient must vanish: a distribution supported on the surface cannot cancel a locally bounded step term. Lemma lem:B-injective gives $b{|}_{{\Sigma }_0}=0$. Smoothness of the remaining distribution $\theta C^{\ast }b$ implies

Since the trace of $b$ vanishes identically on the surface, all of its tangential derivatives vanish there. Formula (eq:C-adjoint) then gives

Injectivity again yields ${\partial }_{t}b{|}_{{\Sigma }_0}=0$.

Conversely, choose $ϵ>0$ such that $a=0$ for $t<ϵ$. On that region $b$ solves the homogeneous wave equation. Vanishing Cauchy data imply, by uniqueness, that $b=0$ for $t<ϵ$. Thus $\theta b=b$ is smooth, and $q_a=C^{\ast }b$ is smooth. Its compact support follows from (eq:q-support).

The theorem does not say that a distribution with timelike wavefront set is intrinsically meaningless as a quantum smearing. It says that this smearing is excluded by the specific off-shell domain in Definition def:poly. Section sec:repairs constructs a linear on-shell representative.

The obstruction for the bilinear insertion is stronger than the existence of exceptional bad test covectors. Every nonzero real smearing $f$ produces a failure somewhere in the smooth configuration space.

Let $0\ne f\in {C_c}^{\infty }(\{t>0\};\mathbb{R})$. There exists ${\varphi }_f\in {C_c}^{\infty }(M;\mathbb{R})$ such that, for

the trace $b_0{|}_{{\Sigma }_0}$ is not identically zero. One explicit choice is

Proof. Let

On every bounded time interval, the spatial supports of $a_0$ and $b_0$ are contained in a common compact set. Integrating $\square b_0=a_0$ over space therefore yields

Since $b_0$ is advanced, $B=B^{\prime }=0$ for sufficiently large positive $t$. Consequently,

Set $u={\partial }_t(tf)$, so that ${\varphi }_f=-u$. Integration by parts gives

This number is strictly positive. Indeed, $u=0$ would imply that $tf$ is independent of $t$; compact time support would then force $f=0$. Hence $b_0{|}_{{\Sigma }_0}$ cannot vanish identically.

Fix the field multiplet and the off-shell domain of Section sec:domain. For every $0\ne f\in {C_c}^{\infty }(\{t>0\};\mathbb{R})$:

1. ${{\mathcal{B}}_f}^{(1)}$ is a continuous bilinear functional of smooth $h,\varphi$, with compact functional support contained in ${\mathcal{D}}_{\mathrm{supp}f}$.

2. It is not microcausal and, in particular,

   $${{\mathcal{B}}_f}^{(1)}\notin {\mathcal{P}}_{\mathrm{mc}}.$$

3. There is no admissible quantization of ${{\mathcal{B}}_f}^{(1)}$ satisfying Definition def:normalization.

4. There is no termwise retarded-product construction of (eq:four-pieces-intro) in ${\mathfrak{A}}_{\mathrm{poly}}$ that admits the original geometric insertion coefficientwise and preserves its off-shell classical normalization.

Proof. Continuity of $L$ on smooth fields, locally in the future half-space, follows from the continuous dependence of the inhomogeneous Cauchy problem on its source. Multiplication by $f$, differentiation of $\varphi$, and integration are continuous. This proves the first assertion, including polynomial smoothness. The support statement follows from finite propagation and (eq:DK).

For the second assertion, differentiate with respect to the metric field:

This derivative is independent of $h$.

Choose $\varphi ={\varphi }_f$ from Lemma lem:witness. Then $b=E_{\mathrm{A}}(f\mathrm{d}{\varphi }_f)$ has a nonzero trace at some point of ${\Sigma }_0$. Lemma lem:wavefront shows that $-L^{\ast }(f\mathrm{d}{\varphi }_f)$ has nonzero timelike covectors in its wavefront set. This contradicts the necessary microcausal condition of Lemma lem:derivative-condition.

If an admissible quantization existed, its coefficient of ${\hslash }^0$ would belong to ${\mathcal{P}}_{\mathrm{mc}}$ and would equal ${{\mathcal{B}}_f}^{(1)}$ on every smooth configuration. Differentiating this equality gives the same forbidden derivative, proving the third assertion.

Finally, a coefficientwise construction that admits the geometric insertion in ${\mathfrak{A}}_{\mathrm{poly}}[[\kappa ]]$ must admit its coefficient ${{\mathcal{B}}_f}^{(1)}$, before applying $R_1(V_1;\cdot )$. The third assertion excludes this prerequisite. This proves the fourth assertion under its stated termwise and off-shell requirements.

The compact bilinear coefficient kernel of ${{\mathcal{B}}_f}^{(1)}$ does not satisfy the cone condition ${\Xi }_2$.

Proof. If that kernel satisfied ${\Xi }_2$, contraction of its scalar slot with the smooth field ${\varphi }_f$ would produce a metric kernel satisfying ${\Xi }_1$, by the proof of Lemma lem:derivative-condition. This contradicts (eq:metric-derivative) and Lemma lem:witness.

Thus the obstruction is not an artefact of evaluating a nonlinear functional far from the zero configuration. The fixed degree-two kernel itself fails the required wavefront condition. The witness configuration can also be multiplied by any nonzero real constant, including arbitrarily small ones.

Let $Q_f$ be a smooth functional such that, for every smooth configuration, its metric derivative is supported in a fixed closed set disjoint from a neighbourhood of ${\Sigma }_0$. Then

is not microcausal.

In particular, adding a bilinear kernel supported on the ordinary diagonal, with its scalar projection contained in $\mathrm{supp}f$, cannot remove the obstruction.

Proof. At the witness configuration, the metric derivative of $Q_f$ vanishes in a neighbourhood of ${\Sigma }_0$. Therefore the derivative of the sum has exactly the same nonzero boundary wavefront component as (eq:metric-derivative).

For the final statement, a diagonal-supported kernel with scalar projection in $\mathrm{supp}f$ also has its metric projection in $\mathrm{supp}f$, which is a positive distance in time from ${\Sigma }_0$.

Changing a prescription only in positive powers of $\hslash$ cannot change the contradiction in the classical coefficient. Allowing a new order-${\hslash }^0$ boundary term changes the off-shell insertion unless that term vanishes as a functional. Theorem thm:main does not prohibit changing the observable; it prohibits calling such a change an off-shell quantization of the unchanged observable under Definition def:normalization.

Choose a smooth function $\chi :\mathbb{R}\to [0,1]$ such that

and set

Define

For each fixed $a$:

1. $q_{a,ϵ}$ is a smooth compactly supported test tensor, and ${\ell }_{a,ϵ}[h]=⟨q_{a,ϵ},h⟩$ is a regular linear element of ${\mathcal{P}}_{\mathrm{mc}}$;

2. the supports of $q_{a,ϵ}$ are contained in the fixed compact set $J^{-}(\mathrm{supp}a)\cap M_{+}$;

3. $q_{a,ϵ}\to q_a$ in the strong topology of compactly supported distributions;

4. if $b{|}_{{\Sigma }_0}\ne 0$, this sequence cannot converge to an element of any fixed prescribed-wavefront space ${{\mathcal{E}}^{\prime }}_{\Gamma }$ with $\Gamma \subset {\Xi }_1$.

Proof. Smoothness and support follow directly from (eq:smooth-edge). For a smooth test tensor $h$,

The integrand is supported in a slab of time thickness $ϵ$, inside a common spatial compact set. It follows that

The estimate is uniform on bounded sets of smooth test tensors, which proves strong distributional convergence.

If convergence held in ${{\mathcal{E}}^{\prime }}_{\Gamma }$ to an element of that space, continuity of its inclusion into distributions would identify the limit with $q_a$. But Lemma lem:wavefront gives a timelike wavefront component of $q_a$, incompatible with $\Gamma \subset {\Xi }_1$.

The prescribed-wavefront topology in the last assertion can be taken to be the normal topology used for distributional operations in BDHtopology. The assertion only requires that its inclusion into distributions be continuous.

At fixed $ϵ$, the source equation is

rather than the unswitched equation on the whole future half-space. Thus a fixed nonzero edge width changes the coordinate prescription. Proposition prop:smoothing establishes a distributional sharp limit for linear smearings, not a sharp limit inside the declared insertion domain and not a theorem for the bilinear composite or its retarded products.

The failure of off-shell membership does not preclude an on-shell representation by smooth test tensors.

Let

be the causal propagator of $\square$, acting componentwise.

If $q$ is a compactly supported tensor distribution with

then $u=Eq$ is a smooth, spacelike-compact solution of $\square u=0$.

Proof. The causal propagator extends to compactly supported distributions. Its distribution kernel has only paired characteristic covectors: both nonzero covectors in each wavefront pair are null. In the kernel-composition wavefront bound, a singularity in $Eq$ would require a null input covector from $\mathrm{WF}(q)$, or a zero input covector in the wavefront set of the propagator kernel. Neither occurs: $\mathrm{WF}(q)$ is timelike, and the causal propagator has no such zero-input wavefront component. Hence $Eq$ is smooth. This is an application of the standard kernel wavefront calculus reviewed in BDHwavefront,FR.

The bisolution identity gives $\square Eq=0$, and the support bound

gives spacelike compactness.

Let $q=q_a=L^{\ast }a$. There exist a smooth compactly supported tensor $q_{\mathrm{sm}}$ and a compactly supported tensor distribution $r$ such that

Consequently,

for every smooth solution of $\square h=0$.

A construction is obtained by setting $u=Eq$, choosing a smooth time function $\rho (t)$ equal to zero sufficiently far in the past and one sufficiently far in the future, and defining

Proof. Lemma lem:causal-smoothing gives a smooth, spacelike-compact homogeneous solution $u$. The derivatives of $\rho$ have compact time support. Therefore $q_{\mathrm{sm}}$ is smooth and compactly supported.

The identities

follow from the wave equation and uniqueness with zero data in the appropriate time direction. Thus

Set $v=q-q_{\mathrm{sm}}$. Then $v$ is compactly supported and $Ev=0$, so

Its support is contained in

a compact set. Moreover, $\square r=v$, proving (eq:on-shell-relation). Pairing that identity with a smooth solution $h$ and using the formal self-adjointness of $\square$ proves (eq:on-shell-equality).

The construction gives compact support, but it does not assert that $q_{\mathrm{sm}}$ has support in the original ${\mathcal{D}}_{\mathrm{supp}a}$. More importantly, (eq:on-shell-equality) is an equality on the free mass shell, not on the full configuration space.

For the gauge-fixed graviton in (eq:free-action), the free equation is $\mathsf{T}\square h=0$, equivalent to $\square h=0$. Thus Theorem thm:onshell applies to the linear graviton observable. It does not construct an off-shell polynomial representative of the bilinear insertion, and it does not control contact terms in interacting retarded products.

For each fixed pair of coordinate tests $a,a^{\prime }$, the preceding construction also gives well-defined free on-shell pairings

where $t$ is a smooth compactly supported graviton test tensor. These expressions are independent of the time cutoff used to construct the smooth representatives.

Indeed, two such representatives have a smooth compactly supported difference $d$ with $Ed=0$. Then

and $E_{\mathrm{R}}d=E_{\mathrm{A}}d$ is smooth and compactly supported. The bisolution property of ${\mathcal{H}}_{hh}$ annihilates this difference in either argument.

Equations (eq:linear-wh)–(eq:linear-ww) are statements about smooth-test pairings of free linear observables. They are not a licence to multiply the corresponding unsmeared kernels with scalar propagators, or to pull them back to the coincidences required by a composite insertion.

If $q_a$ is singular, there is no smooth compactly supported tensor $s$ such that

for every smooth $h$.

Proof. The equality would in particular hold for all compactly supported smooth $h$, and hence would imply $q_a=s$ as distributions, contradicting singularity.

Thus the on-shell repair and the off-shell obstruction are compatible statements, with different equality requirements.

Suppose

and a retarded map is normalized formally by

Substituting the geometric expansion produces

This is a formal coefficient identity conditional on the existence of its terms.

The present result stops before the first mixed product:

Its proposed geometric argument is not an element of the declared off-shell domain. No graph inventory or extension prescription for this product is therefore claimed here.

The obstruction is not a failure of a diagonal extension theorem. The distribution $L^{\ast }a$ already exists, with compact support and finite transverse singular order. It fails the selected wavefront condition. Distribution-extension methods such as those of Brunetti and Fredenhagen BF, and local covariant time-ordered products of local Wick polynomials as constructed by Hollands and Wald HW, do not by themselves place this nonlocal sharp-Cauchy insertion in the declared domain.

Nor is the result an off-diagonal multiplication obstruction: no claim is made that a characteristic product of a coordinate kernel and a quantum two-point function is intrinsically undefined. Such a claim would require an additional analysis of the actual tensor-valued graph distributions.

The precise conclusion is therefore

A continuation must change at least one of these three requirements and prove that the new formulation preserves the intended observable in its stated sense.

No connected two-point response is declared in this paper. Accordingly, ${{\mathcal{B}}_f}^{(1)}\star {{\mathcal{B}}_{f^{\prime }}}^{(1)}$ is not constructed. The boundary result neither computes a quantum response nor settles a later Ward or renormalization problem.

This appendix fixes a perturbative Einstein–scalar model with linear de Donder gauge fixing. Its purpose is to identify the quantum continuation unambiguously. None of the vertices is needed for the boundary obstruction, which is already present in the geometric insertion.

For $\mathbf{g}=\eta +\kappa h$, let

Choose the first-derivative Einstein density representative

The curvature-sign convention is the one for which this is the Einstein density modulo a total divergence. The representative (eq:grav-density), rather than its equivalence class modulo divergences, is used before switching.

Because $\Gamma =O(\kappa )$, the density has a regular formal expansion

Its quadratic part, modulo an unswitched divergence, is

For an odd vector field $c^{\mu }$, define

The notation ${\mathsf{D}}_{\kappa }(h)$ here denotes the infinitesimal metric gauge generator, not an Euler operator.

With an unshifted auxiliary field $B_{\mu }$, choose

The identity

gives

Adding $C_{\mu }{C}^{\mu }$ to (eq:FP) gives the graviton quadratic density in (eq:free-action), modulo an unswitched divergence. The shifted auxiliary field has no interaction vertices in this parametrization.

The chosen action is sufficient to define the gauge-fixed perturbative field content. No switched BRST or Ward identity is inferred from these formulas.

Let

Choose real compactly supported switches $g,g_{\mathrm{grav}},g_{\mathrm{gh}}$. The scalar interaction is

Thus one switch multiplies the complete curved-minus-flat density. The quadratic metric vertex is multiplied by $g$, not by $g^2$.

The full switched interaction is