Windowed Quantum Field Theory: Domain-Restricted Actions, Standard Model Recovery, and the Vanishing of Delocalized Stress-Energy

1. Introduction

1.1. Motivation

Quantum field theory (QFT) and its crown achievement, the Standard Model (SM) of particle physics, are among the most precisely tested frameworks in science. Quantum Electrodynamics (QED) corrections to the electron anomalous magnetic moment agree with experimental data to better than one part in ${10}^{12}$ [3]. The electroweak theory predicted the existence and masses of the W and Z bosons before their discovery [3,4]. The Higgs mechanism produces fermion masses that match measured values across twelve orders of magnitude in $m_f$ [5,6]. These successes rest on a framework that presupposes a globally defined time parameter against which field operators evolve and with respect to which Noether conservation laws are formulated.

In practice, however, no physical interaction is instantiated over all of spacetime. Every realistic experiment involves a detector that is switched on and off, a laser pulse of finite duration, a scattering region of finite spatial extent, or a measurement apparatus that couples to the field for a bounded interval. This finite extent is routinely encoded by switching functions, pulse envelopes, and window functions that multiply the interaction Lagrangian or the coupling Hamiltonian. Such functions appear ubiquitously in quantum optics [7,8], open quantum systems [1], detector models in quantum field theory in curved spacetime [9,10,11], and Casimir and moving-mirror calculations [12,13]. In each context they are introduced as technical conveniences, and their precise profile is assumed to be removable in an appropriate infinite-time or adiabatic limit.

Two structural consequences of such windowing are, however, unavoidable regardless of the physical motivation for the window. First, the Noether currents of the windowed theory discussed in this work satisfy modified Ward identities: the globally conserved quantity is the windowed current $\diamond J^{\mu }$, not $J^{\mu }$ alone, and apparent non-conservation of $J^{\mu }$ is confined to the boundary layer where $\nabla \diamond \ne 0$. This structure is mathematically identical to the effective current non-conservation that arises when tracing over environmental degrees of freedom in open quantum systems [1], and has been noted in specific contexts such as Unruh–DeWitt detector models [11] and finite-time scattering theory [6]. Second, when the windowed matter stress-energy sources the Einstein field equations (EFE), the contracted Bianchi identity requires an additional boundary-supported stress-energy contribution to restore global covariant conservation. This requirement is not specific to any particular theory of time or localization: it follows from the product rule of covariant differentiation applied to any compact-support source, and it is already implicit in the Israel junction conditions [14] and dynamical horizon flux balance laws [15].

A companion paper [16] has argued, on foundational grounds, that the global time parameter of QFT is not primitive but arises operationally from the localization of quantum information under the Heisenberg uncertainty relation and the relativistic causal bound, within a pre-existing causally coherent substrate. Within that framework the window function $\diamond \left(x\right)$ is not a calculational device but a physical encoding of the domain over which field dynamics are operationally instantiated. The present paper is self-contained with respect to that motivation: its results follow from the window function structure alone and hold for any physical system in which a smooth domain-restricting scalar multiplies the matter action. The companion paper provides one physical interpretation; the present paper provides the field-theoretic machinery that any such interpretation requires.

1.2. Localization, Operational Time, and the Window Function

The physical picture motivating the windowed action prescription is briefly summarized here; full development is in the companion paper [16].

No physical interaction is instantiated over all of spacetime. Two universal constraints govern every physical interaction and jointly generate a directed ordering of events. The first is the Heisenberg uncertainty relation [17]:

$$\Delta x\Delta p\ge {\displaystyle \frac{\hslash }{2}}.$$

(1)

Localization is intrinsically finite and bounded; prior to localization there is no operationally defined temporal ordering, only a probabilistic description of potential outcomes. The second is the relativistic causal bound [18]:

$$\left|\Delta x\right|\le c\Delta t,$$

(2)

which enforces a minimum causal interval $\Delta t_{min}=\Delta x/c>0$ between any localization event and any subsequent observation of it. The ordering of localization events, possible only when causal propagation has connected them, constitutes operational time: a local, relational, physically instantiated distinction between before and after.

This paper does not claim to derive spacetime from non-spatiotemporal primitives. A causally coherent background substrate in which the constraints (1) and (2) are meaningful is assumed throughout. What the substrate provides is the possibility of physical interaction and the causal structure that determines which orderings are geometrically possible. What localization generates is operational temporal ordering: the directed, physically instantiated sequence of events that constitutes time as measured. The window function $\diamond \left(x\right)$ introduced in Section 2 encodes the domain over which this operational instantiation has occurred. It is a scalar on the substrate, not a modification of its geometry, not a dynamical field, and not a new degree of freedom. The mathematical scaffolding of covariant coordinates in the action encodes operational conditions under which physical quantities are defined, exactly as coordinate labels in general relativity (GR) encode geometric structure without implying a preferred frame [19].

This distinction resolves the apparent circularity in localization-based approaches to time: localization does not require a prior localization event, only a causally coherent background in which physical interactions can occur. Operational time-ordering is instantiated by localization within the substrate; it does not create the substrate itself. The position is consistent with relational approaches to time in which causal structure is presupposed while temporal ordering is derived [20,21].

1.3. Claims and Scope

The claims of this paper are deliberately conservative. No new particles, interactions, or modifications to the dynamical equations of QFT are introduced. The restriction is entirely one of domain, not of dynamics. The specific claims are as follows.

1.

The windowed action produces field equations for each SM sector that reduce identically to the standard curved-spacetime equations inside the localization window. All corrections are boundary terms confined to the decoherence layer and suppressed by the ratio $ϵ/L$ for any macroscopic scale L.

2.

Gauge invariance ($SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$) and local Lorentz invariance are preserved exactly within the window. Apparent violations are confined to the boundary layer and are structurally identical to open-system flux terms from decoherence theory [1].

3.

The non-local boundary term $T_{\mu \nu }^{\mathrm{nl}}$ appearing in the EFE of the companion paper decomposes as $T_{\mu \nu }^{\mathrm{nl}}=T_{\mu \nu }^{\mathrm{comp}}+T_{\mu \nu }^{\mathrm{Rem}}$. The compensator $T_{\mu \nu }^{\mathrm{comp}}$ is the boundary-layer term required locally by the contracted Bianchi identity. The remnant $T_{\mu \nu }^{\mathrm{Rem}}$ is the macroscopic, globally-conserved stress-energy that $T_{\mu \nu }^{\mathrm{comp}}$ becomes after coarse-graining in the high-localization-density regime. Neither component carries a window factor: $T_{\mu \nu }^{\mathrm{nl}}$ is unwindowed by necessity, because it must be nonzero precisely where and beyond where $\diamond \left(x\right)$ transitions to zero.

4.

For all regular quantum fields, $T_{\mu \nu }^{\mathrm{comp}}$ vanishes upon coarse-graining, so $T_{\mu \nu }^{\mathrm{Rem}}=0$ and $T_{\mu \nu }^{\mathrm{nl}}$ is macroscopically negligible. This is established as a formal Lemma. Standard QFT and GR are fully recovered in all accessible regimes.

1.4. Why $T_{\mu \nu }^{\mathrm{nl}}$ Carries No Window Factor

Before proceeding to the derivations, it is worth stating explicitly why the EFE shown below take the asymmetric form:

$$G_{\mu \nu }+g_{\mu \nu }\Lambda =8\pi G\left[\diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}+T_{\mu \nu }^{\mathrm{nl}}\right],$$

(3)

where $G_{\mu \nu }$ encodes spacetime curvature, $g_{\mu \nu }\Lambda$ is the cosmological constant term, $\diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}$ is the baryonic stress-energy modulated by the window, and $T_{\mu \nu }^{\mathrm{nl}}$ is an unwindowed boundary-support term. The asymmetry between the two right-hand-side terms is not an inconsistency. It is a structural necessity whose origin is the contracted Bianchi identity.

The contracted Bianchi identity ${\nabla }^{\mu }{G}_{\mu \nu }=0$ is a geometric identity that holds everywhere on the spacetime manifold, $\mathcal{M}$, without exception. The right-hand side of Equation (3) must therefore be globally divergence-free. The windowed matter term $\diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}$ fails this requirement in the boundary layer ${\mathcal{B}}_{ϵ}$ where ${\nabla }_{\mu }\diamond \ne 0$, because even when ${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{SM}}=0$ holds inside the window, the Leibniz rule gives ${\nabla }^{\mu }(\diamond T_{\mu \nu }^{\mathrm{SM}})=({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}\ne 0$ in that layer. It also fails outside ${\Omega }_{\mathrm{loc}}$ where $\diamond =0$ but any geometry generated by past localization must still be supported. The term $T_{\mu \nu }^{\mathrm{nl}}$ exists to restore global conservation in both regions. Applying $\diamond \left(x\right)$ to $T_{\mu \nu }^{\mathrm{nl}}$ would suppress it precisely where it is required to be nonzero, causing the Bianchi identity to fail globally.

The asymmetry in Equation (3) therefore reflects a single physical statement: matter sources geometry only where it is localized, plus an unwindowed global bookkeeping correction that exists at and beyond the boundary. The baryonic source is windowed because it is active only during the localization epoch; the non-local term is unwindowed because it must maintain global covariant conservation after that epoch has ended. This is precisely the structure of the Israel junction conditions [14] in the sharp-window limit: the surface stress-energy is supported on the shell boundary where the bulk stress-energy transitions to zero, not where it is nonzero.

1.5. Organization

Section 1.2 summarizes the localization-based physical motivation for the window function, including the two universal constraints that generate operational temporal ordering and the precise sense in which the window encodes them. Section 2 develops the mathematical structure of the localization window, including sharp and smooth limits, variational consistency, and the decoherence-scale interpretation of the boundary thickness $ϵ$. Section 3 derives the windowed field equations for each SM sector and demonstrates standard-QFT recovery in the interior. Section 4 carries out the full Noether analysis and derives windowed Ward identities, with worked examples for electromagnetism and spinor fields. Section 5 develops the unifying bridge between the matter and gravitational sectors by showing that the windowed Ward and Ward–Takahashi identities and the Bianchi-forced compensator are parallel consequences of the same window-gradient term and therefore expressions of exact covariant closure of the full localized-plus-boundary system. Section 6 establishes the boundary conservation problem, introduces $T_{\mu \nu }^{\mathrm{comp}}$ precisely, and gives the Schwarzschild worked example. Section 7 states and proves the Vanishing Lemma, identifies its regime of failure, derives $T_{\mu \nu }^{\mathrm{Rem}}$ and its formal structure, and establishes the decomposition $T_{\mu \nu }^{\mathrm{nl}}=T_{\mu \nu }^{\mathrm{comp}}+T_{\mu \nu }^{\mathrm{Rem}}$. Section 8 collects the complete windowed field equations.

1.6. Notation

$(\mathcal{M},g_{\mu \nu })$ is a four-dimensional Lorentzian manifold with signature $(-,+,+,+)$. The metric determinant is $g=det{g}_{\mu \nu }$. Natural units $\hslash =c=k_B=1$ are used unless stated otherwise. The covariant d’Alembertian is ${\square }_g\equiv g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }$. The window function is written $\diamond \left(x\right)$ or ⋄ throughout.

2. The Localization Window: Mathematical Structure

2.1. Definition

Let ${\Omega }_{\mathrm{loc}}\subset \mathcal{M}$ be the spacetime region in which a given quantum field description is physically instantiated: a localization event has occurred, quantum information has acquired bounded stress-energy, and causal propagation has established the before-after ordering discussed in Section 1.2 and developed in the companion paper [16]. The window function is the smooth scalar:

$$\diamond \left(x\right):\mathcal{M}\to [0,1],\diamond \left(x\right)\approx \left\{\begin{array}{cc}1,\hfill & x\in {\Omega }_{\mathrm{loc}},\hfill \\ 0,\hfill & x\notin {\Omega }_{\mathrm{loc}},\hfill \end{array}\right.$$

(4)

with smooth interpolation across $\partial {\Omega }_{\mathrm{loc}}$. The boundary layer ${\mathcal{B}}_{ϵ}$ is the set of points where ${\nabla }_{\mu }\diamond \ne 0$; its characteristic thickness $ϵ$ is set by the decoherence timescale ${\tau }_{\mathrm{dec}}$ of the system [2,22]:

$$ϵ\sim c{\tau }_{\mathrm{dec}}.$$

(5)

This scale is not Planck-scale, not a UV regulator, and not a new constant of nature. It is apparatus- and context-dependent: for a macroscopic detector, ${\tau }_{\mathrm{dec}}$ is extremely small, so $ϵ$ is correspondingly thin but remains finite. For measurement-driven localization [2],

$$ϵ\gtrsim max\left\{c\Delta t_{det},\Delta x_{det}\right\},$$

(6)

where $\Delta t_{det}$ and $\Delta x_{det}$ are the detector temporal and spatial resolutions. These conditions make $ϵ$ apparatus-dependent, not fundamental.

2.2. Essential Properties of $\diamond \left(x\right)$

Four properties of $\diamond \left(x\right)$ are essential for the field-theoretic results that follow.

1.

Diffeomorphism scalar.$\diamond \left(x\right)$ is invariant under coordinate transformations. This is what allows it to multiply any generally covariant Lagrangian density without breaking general covariance or introducing a preferred frame.

2.

Gauge singlet.$\diamond \left(x\right)$ carries no charge under $SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$. It multiplies gauge-invariant Lagrangians and therefore cannot break gauge symmetry.

3.

Non-dynamical.$\diamond \left(x\right)$ is not varied in the action. It encodes the localization geometry determined by the physical process, specifically by the Heisenberg and causal constraints applied to the system. It is not a new propagating degree of freedom.

4.

Matter-sector restriction only.$\diamond \left(x\right)$ gates the matter Lagrangian density. The Einstein tensor, the contracted Bianchi identity, and all Riemannian structure of GR are entirely unaffected in the bulk. Only the domain over which classical stress-energy sources the EFE is restricted.

2.3. Sharp Limit and Smooth Regularization

The conceptually natural choice is a step-function window,

$${\diamond }_{\mathrm{sharp}}\left(x\right)=\Theta \left(f\left(x\right)\right),$$

(7)

where $f\left(x\right)=0$ defines $\partial {\Omega }_{\mathrm{loc}}$ and $\Theta$ is the Heaviside function. Its gradient,

$${\nabla }_{\mu }{\diamond }_{\mathrm{sharp}}=\delta \left(f\left(x\right)\right){\nabla }_{\mu }f\left(x\right),$$

(8)

is a distribution supported on $\partial {\Omega }_{\mathrm{loc}}$. In this limit, the boundary terms in the action become distributional, and the windowed field equations reduce to standard junction conditions familiar from thin-shell constructions in GR [14]. Distributional sources require care: products of distributions are ill-defined without additional regularization, and the variational principle is ambiguous without specifying boundary conditions.

For these reasons smooth windows similar to Equation (9) are used as the working window objects throughout this work:

$${\diamond }_{ϵ}\left(x\right)={\displaystyle \frac{1}{2}}\left[1+tanh\left({\displaystyle \frac{f\left(x\right)}{ϵ}}\right)\right],$$

(9)

which satisfies ${\diamond }_{ϵ}\in C^{\infty }\left(\mathcal{M}\right)$, $|{\nabla }_{\mu }{\diamond }_{ϵ}|=O\left({ϵ}^{-1}\right)$ within ${\mathcal{B}}_{ϵ}$, and ${\nabla }_{\mu }{\diamond }_{ϵ}=0$ outside. The sharp limit is recovered distributionally: ${lim}_{ϵ\to 0}{\diamond }_{ϵ}={\diamond }_{\mathrm{sharp}}$.

Smooth windows of this type appear in several established contexts: thin-shell stress-energy constructions [14], the Gibbons–Hawking–York boundary terms for gravitational actions on manifolds with boundary [23], and effective field theory constructions with compact-support sources [24]. The present construction is a direct field-theoretic generalization of these.

The smoothness conditions required for all boundary terms in this paper to be finite and covariant are [24]:

$$\begin{array}{cc}\hfill \diamond & \in C^1\left(\mathcal{M}\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \exists K\subset \mathcal{M}\mathrm{compact}& :\diamond {|}_{\mathcal{M}\setminus K}=0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\int }_{\mathcal{M}}|{\nabla }_{\mu }\diamond |\sqrt{-g}{d}^{4}x& <\infty .\hfill \end{array}$$

(12)

2.4. The Windowed Action Prescription and Its Variational Structure

For any generally covariant Lagrangian density $\mathcal{L}\left(x\right)$, the windowed action is:

$$S_{\mathrm{loc}}=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\mathcal{L}\left(x\right).$$

(13)

Variation of Equation (13) with respect to a generic field $\Phi$ yields:

$$\delta S_{\mathrm{loc}}={\int }_{\mathcal{M}}\diamond \mathcal{E}\left(\Phi \right)\delta \Phi \sqrt{-g}{d}^{4}x+{\int }_{\mathcal{M}}({\nabla }_{\mu }\diamond ){\Pi }^{\mu }\left(\Phi \right)\delta \Phi \sqrt{-g}{d}^{4}x,$$

(14)

where $\mathcal{E}\left(\Phi \right)=0$ are the bulk Euler–Lagrange equations and ${\Pi }^{\mu }$ is the canonical momentum density. The second integral is supported only within ${\mathcal{B}}_{ϵ}$. This structure is directly analogous to the role of Gibbons–Hawking–York terms [23]: the window introduces an effective internal boundary even when $\mathcal{M}$ itself has none, and smoothness ensures finite, covariant boundary contributions.

3. Windowed Field Equations and Standard QFT Recovery

3.1. Operational Interpretation of the QFT Time Parameter

QFT, including the SM, is formulated with respect to a global time parameter that orders field operators. The Schrödinger equation [3],

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\Psi =\widehat{H}\Psi ,$$

(15)

and its relativistic generalization, the flat-spacetime Dirac equation [3],

$$(i\hslash {\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0,$$

(16)

presuppose a meaningful global time coordinate. Within the localization-based picture of Section 1.2, this global time is an effective bookkeeping parameter that becomes physically meaningful only within finite spacetime regions where quantum information has been localized. Field evolution in QFT should be understood as evolution within a localization interval, not as a statement about global dynamics. Outside such intervals, the formal time dependence of field operators persists mathematically but does not produce localized stress-energy or define causal structure.

This distinction is made explicit by the windowed action prescription (13). Within the window, standard QFT applies without modification. Outside the window, the field contributes nothing to the right-hand side of the EFE and defines no physical time evolution. No field equation is algebraically altered; the restriction is solely one of domain.

It is important to note that localization here does not correspond to sharp particle position eigenstates. In standard QFT, strict particle localization is known to be problematic due to relativistic and field-theoretic effects [3]. The present definition is regional and operational: a field is localized if and only if it contributes to stress-energy within a finite spacetime region and can causally influence other localized regions. This aligns with the algebraic notion of local observables [24].

3.2. Dirac Equation in Curved Spacetime

3.2.1. Spinorial Geometry

Introduce a vierbein ${e_{\mu }}^a$ satisfying $g_{\mu \nu }={e_{\mu }}^a{e_{\nu }}^b{\eta }_{ab}$, with curved-space gamma matrices ${\gamma }^{\mu }\left(x\right)={e^{\mu }}_a{\gamma }^a$ obeying $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$. The spinor covariant derivative is [12]:

$${\nabla }_{\mu }\psi =\left({\partial }_{\mu }+{\displaystyle \frac{1}{4}}{\omega }_{\mu ab}{\gamma }^{ab}\right)\psi ,$$

(17)

where ${\omega }_{\mu ab}$ is the torsion-free spin connection with Latin vierbein indices $a,b$. The standard curved-spacetime Dirac equation, with Higgs-mechanism generated mass $m\left(H\right)=y(v+h)/\sqrt{2}$ [5,25,26], is:

$$\begin{array}{cc}\hfill (i{\gamma }^{\mu }{D}_{\mu }-m\left(H\right))\psi & =0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill D_{\mu }\psi & ={\partial }_{\mu }\psi +\frac{1}{2}{\left({\omega }_{ab}\right)}_{\mu }{\Sigma }^{ab}\psi ,\hfill \end{array}$$

(19)

where ${\Sigma }^{ab}=\frac{1}{4}[{\gamma }^a,{\gamma }^b]$ are the spinor representation generators of the Lorentz group. Note that ${\Sigma }^{ab}$ and the stress-energy tensor $T_{\mu \nu }^{\mathrm{SM}}$ are distinct objects; the former carries only Lorentz-algebra indices and acts on spinors, while the latter is a symmetric rank-2 spacetime tensor, hence the use of Latin indices here.

3.2.2. Windowed Dirac Action

The windowed Dirac action is:

$$S_D=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\left[{\displaystyle \frac{i}{2}}\left(\overline{\psi }{\gamma }^{\mu }{\nabla }_{\mu }\psi -\left({\nabla }_{\mu }\overline{\psi }\right){\gamma }^{\mu }\psi \right)-m\overline{\psi }\psi \right].$$

(20)

The symmetrized kinetic term ensures Hermiticity before localization. Varying with respect to $\overline{\psi }$:

$$\diamond \left(x\right)(i{\gamma }^{\mu }{\nabla }_{\mu }-m)\psi +{\displaystyle \frac{i}{2}}({\nabla }_{\mu }\diamond \left(x\right)){\gamma }^{\mu }\psi =0.$$

(21)

Recovery inside the window. Where $\diamond =1$ and ${\nabla }_{\mu }\diamond =0$, Equation (21) reduces identically to the standard curved-spacetime Dirac equation. The second term is nonzero only in ${\mathcal{B}}_{ϵ}$.

3.2.3. Worked Example: Spinor Rescaling Across the Decoherence Boundary

The physical content of the boundary term in Equation (21) can be made explicit. Define a locally rescaled spinor:

$$\chi \equiv \diamond {\left(x\right)}^{1/2}\psi ,\diamond \left(x\right)>0.$$

(22)

Substituting $\psi ={\diamond }^{-1/2}\chi$ into Equation (21), noting that ${\nabla }_{\mu }\left({\diamond }^{-1/2}\chi \right)={\diamond }^{-1/2}{\nabla }_{\mu }\chi -\frac{1}{2}{\diamond }^{-3/2}({\nabla }_{\mu }\diamond )\chi$, and multiplying through by ${\diamond }^{1/2}$:

$$\begin{array}{cc}\hfill & {\diamond }^{1/2}·\diamond ·i{\gamma }^{\mu }{\nabla }_{\mu }\left({\diamond }^{-1/2}\chi \right)-{\diamond }^{1/2}·\diamond ·m{\diamond }^{-1/2}\chi +\frac{i}{2}({\nabla }_{\mu }\diamond ){\gamma }^{\mu }{\diamond }^{-1/2}\chi =0,\hfill \\ \hfill & {\diamond }^{1/2}\left[i{\gamma }^{\mu }{\nabla }_{\mu }\chi -m\chi -\frac{i}{2}{\diamond }^{-1}({\nabla }_{\mu }\diamond ){\gamma }^{\mu }\chi +\frac{i}{2}{\diamond }^{-1}({\nabla }_{\mu }\diamond ){\gamma }^{\mu }\chi \right]=0,\hfill \\ \hfill & (i{\gamma }^{\mu }{\nabla }_{\mu }-m)\chi =0.\hfill \end{array}$$

(23)

The $\nabla \diamond$ terms cancel exactly. The rescaled spinor $\chi$ satisfies the standard Dirac equation throughout ${\Omega }_{\mathrm{loc}}$. The boundary term in Equation (21) represents spinor normalization across the decoherence layer [22]: it is a matching condition, not a symmetry violation. This is analogous to the normalization conditions imposed on mode functions at a junction surface in quantum field theory in curved spacetime [12].

3.3. Klein–Gordon Field

For a real scalar field $\varphi$ of mass m with curvature coupling $\xi$, the windowed action is:

$$S_{\mathrm{KG}}=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\left[-\frac{1}{2}{g}^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -\frac{1}{2}(m^2+\xi R){\varphi }^2\right].$$

(24)

Variation with respect to $\varphi$ yields:

$$\diamond \left(x\right){\square }_g\varphi +({\nabla }_{\mu }\diamond \left(x\right)){\nabla }^{\mu }\varphi -\diamond \left(x\right)(m^2+\xi R)\varphi =0.$$

(25)

Recovery inside the window. Where ${\nabla }_{\mu }\diamond =0$, the middle term drops and Equation (25) reduces to $({\square }_g-m^2-\xi R)\varphi =0$, the standard Klein–Gordon equation in curved spacetime [12,27].

Conformal scalar case. The minimally coupled ($\xi =0$) and conformally coupled ($\xi =1/6$ in four dimensions) cases are both recovered inside the window. For a massless conformally coupled scalar in flat spacetime ($R=0$, $\diamond \to 1$), the equation reduces to $\square \varphi =0$, recovering the d’Alembertian wave equation. Thus, the window preserves the conformal properties of the Klein-Gordon equation within the localization region.

3.4. Maxwell Theory and Gauge Fields

With $F_{\mu \nu }={\nabla }_{\mu }{A}_{\nu }-{\nabla }_{\nu }{A}_{\mu }$, the windowed Maxwell action is:

$$S_{\mathrm{EM}}=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\left(-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+J^{\mu }{A}_{\mu }\right).$$

(26)

Varying with respect to $A_{\nu }$:

$$\begin{array}{cc}\hfill {\nabla }_{\mu }(\diamond \left(x\right)F^{\mu \nu })& =\diamond \left(x\right)J^{\nu },\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \diamond \left(x\right){\nabla }_{\mu }{F}^{\mu \nu }+({\nabla }_{\mu }\diamond \left(x\right))F^{\mu \nu }& =\diamond \left(x\right)J^{\nu }.\hfill \end{array}$$

(28)

Recovery inside the window. Where ${\nabla }_{\mu }\diamond =0$, Equation (21) gives ${\nabla }_{\mu }{F}^{\mu \nu }=J^{\nu }$, the standard Maxwell equation in curved spacetime.

Windowed continuity equation. Taking the covariant divergence of Equation (27):

$${\nabla }_{\nu }(\diamond \left(x\right)J^{\nu })=0.$$

(29)

This is the gauge-current Ward identity for the windowed Maxwell theory, discussed further in Section 4.

3.5. The Full Standard Model Action

For a generic SM fermion $\psi$, the full gauge-covariant derivative is:

$${\mathcal{D}}_{\mu }\psi =\left({\nabla }_{\mu }-i{g}_s{G}_{\mu }^A{T}^A-ig{W}_{\mu }^I{\tau }^I-i{g}^{\prime }Y{B}_{\mu }\right)\psi ,$$

(30)

where $G_{\mu }^A$ are the $SU{\left(3\right)}_c$ gluon fields with coupling $g_s$ and generators $T^A$; $W_{\mu }^I$ are the $SU{\left(2\right)}_L$ weak gauge bosons with coupling g and generators ${\tau }^I$; and $B_{\mu }$ is the $U{\left(1\right)}_Y$ hypercharge boson. The windowed SM action in curved spacetime is [3,6]:

$$S_{\mathrm{SM}}=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\left({\mathcal{L}}_{\mathrm{gauge}}+{\mathcal{L}}_{\mathrm{ferm}}+{\mathcal{L}}_{\mathrm{Higgs}}+{\mathcal{L}}_{\mathrm{Yuk}}\right),$$

(31)

with Lagrangian components [3,4,6]:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{gauge}}& =-\frac{1}{4}{G}_{\mu \nu }^A{G}_A^{\mu \nu }-\frac{1}{4}{W}_{\mu \nu }^I{W}_I^{\mu \nu }-\frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu },\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{ferm}}& =\sum _f{\overline{\psi }}_{f}i{\gamma }^{\mu }{\mathcal{D}}_{\mu }{\psi }_f,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Higgs}}& ={\left(D_{\mu }H\right)}^{†}\left(D^{\mu }H\right)-{\mu }^2{H}^{†}H-\lambda {\left(H^{†}H\right)}^2,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Yuk}}& =-\left({\overline{Q}}_L{Y}_u\tilde{H}{u}_R+{\overline{Q}}_L{Y}_{d}H{d}_R+{\overline{L}}_L{Y}_{e}H{e}_R\right)+\mathrm{h}.\mathrm{c}.,\hfill \end{array}$$

(35)

where $\tilde{H}=i{\sigma }^2{H}^{*}$. Gauge-fixing and Faddeev–Popov ghost terms are omitted for clarity; they window-gate in the same manner and do not alter the domain-restriction interpretation [6].

No Lagrangian component in Equations (32)–(35) is algebraically altered. The window $\diamond \left(x\right)$ restricts the domain over which each term is physically instantiated. Within ${\Omega }_{\mathrm{loc}}$, where $\diamond =1$, the action (31) is identical to the standard curved-spacetime SM action. Outside ${\Omega }_{\mathrm{loc}}$, no field contributes localized stress-energy and no physical time evolution is defined.

4. Windowed Noether Analysis and Ward Identities

4.1. Setup: Global Internal Symmetries

Consider the windowed action (13) for a generic field $\Phi$ with Lagrangian $\mathcal{L}(\Phi ,\nabla \Phi ,g_{\mu \nu })$. Suppose the ungated theory admits a continuous global internal symmetry $\Phi \to \Phi +\delta \Phi$ with associated on-shell conserved Noether current $J^{\mu }$:

$${\nabla }_{\mu }{J}^{\mu }=0.$$

(36)

Applying the same infinitesimal transformation to the windowed action (13) and following the standard Noether procedure [4,28] yields the windowed Ward identity:

$${\nabla }_{\mu }(\diamond J^{\mu })=0.$$

(37)

Expanding via the product rule:

$${\nabla }_{\mu }{J}^{\mu }=-({\nabla }_{\mu }ln\diamond )J^{\mu }.$$

(38)

The right-hand side of Equation (38) is zero whenever ${\nabla }_{\mu }\diamond =0$, that is, everywhere except in ${\mathcal{B}}_{ϵ}$.

Five critical points follow directly:

1.

Interior conservation. Inside ${\Omega }_{\mathrm{loc}}$ where $\diamond =1$, standard conservation ${\nabla }_{\mu }{J}^{\mu }=0$ is recovered exactly.

2.

Boundary localization. Apparent non-conservation is confined to ${\mathcal{B}}_{ϵ}$ where ${\nabla }_{\mu }\diamond \ne 0$.

3.

True conserved current. The globally conserved quantity is $\diamond J^{\mu }$, not $J^{\mu }$ alone.

4.

Standard limit. As $\diamond \to 1$ globally, ${\nabla }_{\mu }ln\diamond \to 0$ everywhere and standard conservation is recovered without modification.

5.

Open-system analogy. The structure of Equation (38) is mathematically identical to effective current non-conservation arising in open quantum systems after tracing over environmental degrees of freedom [1]. In the present context, the origin is geometric rather than environmental, but the Ward-identity structure is the same.

4.2. Worked Example: Electromagnetic Current Conservation

The electromagnetic Ward identity provides a concrete illustration. Inside ${\Omega }_{\mathrm{loc}}$, where $\diamond =1$, the standard QED Ward–Takahashi identity holds in its usual form [3,6]:

$$k_{\mu }{\mathcal{M}}^{\mu }=0,$$

(39)

where $k^{\mu }$ is the photon momentum and ${\mathcal{M}}^{\mu }$ is a QED amplitude. This identity follows from $U\left(1\right)$ gauge invariance of the action. At the level of the equations of motion, its covariant form is ${\nabla }_{\mu }{J}_{\mathrm{EM}}^{\mu }=0$, or equivalently ${\nabla }_{\nu }\left({\nabla }_{\mu }{F}^{\mu \nu }\right)={\nabla }_{\nu }{J}^{\nu }=0$ by antisymmetry.

For the windowed Maxwell theory, from Equations (27) and (29), the windowed identity is:

$${\nabla }_{\nu }(\diamond J_{\mathrm{EM}}^{\nu })=0,$$

(40)

which reproduces ${\nabla }_{\nu }{J}_{\mathrm{EM}}^{\nu }=0$ in the interior and acquires a boundary term proportional to $({\nabla }_{\nu }\diamond )J_{\mathrm{EM}}^{\nu }$ in ${\mathcal{B}}_{ϵ}$. Taking the covariant divergence of Equation (28) explicitly:

$${\nabla }_{\nu }\left[\diamond {\nabla }_{\mu }{F}^{\mu \nu }+({\nabla }_{\mu }\diamond )F^{\mu \nu }\right]={\nabla }_{\nu }(\diamond J^{\nu }),$$

(41)

and the left side gives ${\nabla }_{\nu }(\diamond J^{\nu })$ after applying the Bianchi identity for F, confirming consistency.

Current apparently leaks across the boundary layer of the localization window, but the total windowed current $\diamond J^{\mu }$ is exactly conserved. This is the field-theoretic analogue of a quantum open system: tracing over environmental degrees of freedom produces apparent non-conservation in the subsystem current, while the total system plus environment current is conserved [1]. Here the “environment” is the decoherence boundary layer ${\mathcal{B}}_{ϵ}$.

4.3. Gauge Symmetry Is Not Broken

Because $\diamond \left(x\right)$ is a gauge singlet, it carries no charge under any SM gauge group. It multiplies gauge-invariant Lagrangian densities and therefore cannot break gauge symmetry.

In the path-integral formulation, Ward–Takahashi identities follow from gauge invariance of the functional measure [4,6]. The window $\diamond \left(x\right)$ enters as a fixed background function in the integrand; it does not transform under gauge transformations and therefore does not modify the measure. Accordingly, all Ward–Takahashi identities of the ungated theory continue to hold within ${\Omega }_{\mathrm{loc}}$. The window introduces only boundary-supported contact terms that vanish outside ${\mathcal{B}}_{ϵ}$.

Gauge-fixing and Faddeev–Popov ghost terms transform under BRST symmetry in the standard way [6]. Because ⋄ is BRST inert, the ghost sector windows identically to the matter sector and no ghost-number anomaly is introduced.

4.4. Worked Example: Windowed Stress-Energy Tensor for a Free Scalar

For the free massive scalar $\varphi$ with windowed action (24) in the minimally coupled case where $\xi =0$ , the Hilbert stress-energy tensor is :

$$T_{\mu \nu }=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta S_{\mathrm{KG}}}{\delta g^{\mu \nu }}}=\diamond \left(x\right)\left[{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -\frac{1}{2}{g}_{\mu \nu }\left({(\nabla \varphi )}^2+m^2{\varphi }^2\right)\right].$$

(42)

Inside ${\Omega }_{\mathrm{loc}}$, this is precisely the standard stress-energy tensor for a free scalar field in curved spacetime [12]. Outside ${\Omega }_{\mathrm{loc}}$, $\diamond =0$ and the stress-energy vanishes identically: the field sources nothing. In ${\mathcal{B}}_{ϵ}$, ⋄ interpolates smoothly, producing a boundary-layer contribution of order $O(ϵ/L)$ relative to the bulk for any macroscopic scale $L\gg ϵ$.

Taking the covariant divergence of Equation (42):

$$\begin{array}{cc}\hfill {\nabla }^{\mu }{T}_{\mu \nu }& =({\nabla }^{\mu }\diamond )\left[{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -\frac{1}{2}{g}_{\mu \nu }({(\nabla \varphi )}^2+m^2{\varphi }^2)\right]\hfill \\ \hfill & +\diamond \left[{\square }_g\varphi {\nabla }_{\nu }\varphi +{\nabla }_{\mu }\varphi {\nabla }_{\nu }{\nabla }^{\mu }\varphi -{\nabla }_{\nu }\left(\frac{1}{2}({(\nabla \varphi )}^2+m^2{\varphi }^2)\right)\right].\hfill \end{array}$$

(43)

Using the windowed equation of motion (25) to eliminate ${\square }_g\varphi$ inside the window, the second line vanishes inside ${\Omega }_{\mathrm{loc}}$ and the divergence is sourced only by the ${\nabla }^{\mu }\diamond$ term in ${\mathcal{B}}_{ϵ}$. This confirms Equation (??) explicitly for the free scalar.

5. Covariant Closure, Ward Identities, the Contracted Bianchi Identities, and the Requirement for the Compensator

The windowed Ward identities derived in Section 4 are exact and complete as statements about the internal symmetries of the windowed matter sector: $\diamond J^{\mu }$ is conserved, gauge and Lorentz symmetries are unbroken, and apparent non-conservation of $J^{\mu }$ is confined to the boundary layer ${\mathcal{B}}_{ϵ}$. These results, however, are not isolated facts. They are one face of a single underlying requirement that runs through every sector of the windowed theory: exact covariant closure of the full localized-plus-boundary system. Making that requirement explicit reveals why the windowed Ward identities and the compensator $T_{\mu \nu }^{\mathrm{comp}}$ introduced in Section 6 are parallel consequences of the same product-rule identity rather than independent constructions.

The internal-symmetry side of this principle is already established. For any Noether current $J^{\mu }$ of the ungated theory, the windowed action yields

$${\nabla }_{\mu }\left(\diamond J^{\mu }\right)=0,$$

(44)

so that the globally conserved quantity is $\diamond J^{\mu }$, not $J^{\mu }$ alone. Apparent non-conservation of $J^{\mu }$ in ${\mathcal{B}}_{ϵ}$ is a boundary flux, not a symmetry violation. At the quantum level, the same structure governs the Ward–Takahashi identities: because $\diamond \left(x\right)$ is a fixed gauge-singlet scalar that does not alter the functional measure (Section 4.3), it introduces only boundary-supported contact terms proportional to $\nabla \diamond$. The classical Ward identity and the quantum Ward–Takahashi identity are therefore both expressions of the same localized conservation law, differing only in whether the current is evaluated on shell or off shell.

The gravitational side introduces an additional structural requirement that the internal-symmetry analysis does not see. The localized stress-energy tensor

$$T_{\mu \nu }^{\mathrm{loc}}\equiv \diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}$$

(45)

is generically not divergence-free:

$${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{loc}}=({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}\ne 0\mathrm{in}{\mathcal{B}}_{ϵ}.$$

(46)

For internal currents this boundary non-conservation is the complete story: the windowed current $\diamond J^{\mu }$ absorbs it and global conservation is restored. For the stress-energy tensor the situation is more demanding. The Einstein tensor satisfies the contracted Bianchi identity

$${\nabla }^{\mu }{G}_{\mu \nu }=0$$

(47)

identically and without exception, so the total source on the right-hand side of the Einstein field equations must be globally divergence-free. The windowed matter stress-energy $T_{\mu \nu }^{\mathrm{loc}}$ alone fails this requirement. A boundary-supported compensating contribution is therefore not optional: it is required by the geometry of general relativity applied to any compact-support matter source. Specifically,

$${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{comp}}=-({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}$$

(48)

must hold so that ${\nabla }^{\mu }(T_{\mu \nu }^{\mathrm{loc}}+T_{\mu \nu }^{\mathrm{comp}})=0$ globally. This is the compensator equation whose consequences are developed in Section 6.

The three identities—the windowed Ward identity (44), the Ward–Takahashi identity, and the Bianchi-driven compensator equation (48)—are parallel manifestations of one principle: localization via $\diamond \left(x\right)$ converts every bulk conservation law into a bulk-plus-boundary conservation law, and covariant closure demands that the boundary term generated by $\nabla \diamond$ be accounted for in every sector of the theory. The Ward identity is necessary but not sufficient for full covariant closure of the windowed theory. The Bianchi condition is the additional gravitational constraint that forces the compensator into existence. This structure is not new to the present framework: it is already implicit in the Israel junction conditions [14] (the sharp-window limit of Eq. (48)) and in the Ashtekar–Krishnan dynamical horizon flux balance laws [15] (the smooth-window generalization to nonstationary horizons). The present treatment makes the common origin of both explicit.

Once $T_{\mu \nu }^{\mathrm{comp}}$ is established, a further question arises: does it leave a macroscopic gravitational imprint after the localization interval has ended and the baryonic fields have dispersed? For ordinary, weakly localized fields, the compensator is a surface-to-volume suppressed boundary term and vanishes under coarse-graining. Section 7 establishes this as a formal Lemma, recovering standard QFT and GR in all macroscopically accessible regimes. The Lemma also identifies precisely the high-localization-density regime in which regularity fails, yielding a macroscopic remnant $T_{\mu \nu }^{\mathrm{Rem}}$ that is globally conserved and pressureless.

6. Boundary Conservation, $T_{\mu \nu }^{\mathrm{comp}}$, and the Structure of $T_{\mu \nu }^{\mathrm{nl}}$

6.1. The Boundary Divergence Problem

The contracted Bianchi identity requires ${\nabla }^{\mu }{G}_{\mu \nu }=0$ identically, which in turn requires ${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{tot}}=0$ for the total source of the EFE. The windowed matter stress-energy does not satisfy this globally when ⋄ has compact support. Writing $T_{\mu \nu }^{\mathrm{SM}}$ for the SM stress-energy satisfying ${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{SM}}=0$ inside the window:

$${\nabla }^{\mu }(\diamond T_{\mu \nu }^{\mathrm{SM}})=({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}+\diamond \underset{=0\mathrm{inside}}{\underbrace{{\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{SM}}}}=J_{\nu },$$

(49)

where $J_{\nu }\equiv ({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}$ is a source supported in ${\mathcal{B}}_{ϵ}$.

6.2. The Compensator $T_{\mu \nu }^{\mathrm{comp}}$

For the contracted Bianchi identity to hold, a compensator stress-energy $T_{\mu \nu }^{\mathrm{comp}}$ must be introduced satisfying:

$${\nabla }^{\mu }\left(\diamond T_{\mu \nu }^{\mathrm{SM}}+T_{\mu \nu }^{\mathrm{comp}}\right)=0\Rightarrow {\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{comp}}=-J_{\nu }=-({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}.$$

(50)

The term $T_{\mu \nu }^{\mathrm{comp}}$ is sourced by the window gradient and therefore has support concentrated in ${\mathcal{B}}_{ϵ}$. It does not obey the contracted Bianchi identity in isolation; rather, it acts to restore that identity globally when $T_{\mu \nu }^{\mathrm{SM}}$ fails to do so locally at the window boundary [24].

As a toy model, suppose the field-theoretic construction of $T_{\mu \nu }^{\mathrm{comp}}$ is illustrated using a fermionic field. Then, a singlet fermion N with SM-generated mass $m_N=m_0+y_{N}H$ has Lagrangian [12,29]:

$$\begin{array}{cc}\hfill {\mathcal{L}}_N& =\sqrt{-g}\overline{N}\left(i{\gamma }^a{e}_a^{\mu }{\nabla }_{\mu }-m_N\right)N,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill m_N& =m_0+y_{N}H.\hfill \end{array}$$

(52)

The windowed action and resulting compensator stress-energy are:

$$\begin{array}{cc}\hfill S_N& =\int d^{4}x\diamond \left(x\right){\mathcal{L}}_N,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill T_{\mu \nu }^{\mathrm{comp}}& =-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta S_N}{\delta g^{\mu \nu }}}.\hfill \end{array}$$

(54)

Note that the window function appears in Equation (53) only because the present subsection constructs a fermionic toy model of a possible compensator sector. This should not be read as a claim that all compensator constructions must be windowed in this specific way. The important result is that the compensator sector may be represented as arising from a generating field, here denoted by N, whose associated stress-energy provides an explicit realization for the development below.

6.3. Worked Example: Schwarzschild Geometry

The compensator mechanism can be demonstrated explicitly in a static spherically symmetric spacetime. Consider the Schwarzschild metric:

$$d{s}^2=-\left(1-{\displaystyle \frac{2M}{r}}\right)d{t}^2+{\left(1-{\displaystyle \frac{2M}{r}}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2,$$

(55)

and suppose a matter distribution is localized exterior to radius $r_0>2M$ via the smooth radial window:

$$\diamond \left(r\right)=\frac{1}{2}\left[1+tanh\left({\displaystyle \frac{r-r_0}{\delta }}\right)\right],$$

(56)

with $\delta \ll r_0$ the boundary-layer thickness. For any static stress-energy $T_{\mu \nu }^{\mathrm{SM}}\left(r\right)$, the covariant divergence of the windowed tensor in the radial direction yields:

$${\nabla }^{\mu }{T}_{\mu r}^{\mathrm{loc}}=\left({\partial }_r\diamond \right)T_{rr}^{\mathrm{SM}},$$

(57)

where ${\partial }_r\diamond =\frac{1}{2\delta }{\mathrm{sech}}^2\left(\frac{r-r_0}{\delta }\right)$ is confined to the boundary layer $|r-r_0|\lesssim \delta$. Consistency with the contracted Bianchi identity then requires a compensator satisfying:

$${\nabla }^{\mu }{T}_{\mu r}^{\mathrm{comp}}=-\left({\partial }_r\diamond \right)T_{rr}^{\mathrm{SM}}.$$

(58)

In the sharp limit $\delta \to 0$, the window becomes a step function and the compensator reduces to a thin spherical shell at $r=r_0$ with surface stress-energy $T_{\mu \nu }^{\mathrm{comp}}=S_{\mu \nu }\delta (r-r_0)$. This is precisely the structure of the Israel junction conditions [14]. The bulk EFE on either side of the shell remain unmodified. The same compensator structure appears in dynamical horizon flux balance laws, where boundary-supported stress-energy restores conservation across nonstationary horizons [15]. These examples demonstrate that $T_{\mu \nu }^{\mathrm{comp}}$ is not an exotic addition but is already implicit in well-established gravitational constructions.

6.4. The Decomposition of $T_{\mu \nu }^{\mathrm{nl}}$

With $T_{\mu \nu }^{\mathrm{comp}}$ defined, the structure of the EFE can now be stated precisely. The full non-local boundary term $T_{\mu \nu }^{\mathrm{nl}}$ appearing in the companion paper decomposes as:

$$T_{\mu \nu }^{\mathrm{nl}}=T_{\mu \nu }^{\mathrm{comp}}+T_{\mu \nu }^{\mathrm{Rem}},$$

(59)

where $T_{\mu \nu }^{\mathrm{comp}}$ is the boundary-layer compensator defined by Equation (50) and $T_{\mu \nu }^{\mathrm{Rem}}$ is the macroscopic remnant stress-energy that $T_{\mu \nu }^{\mathrm{comp}}$ becomes after coarse-graining in the high-localization-density regime, derived in Section 7. The EFE is therefore:

$$G_{\mu \nu }+g_{\mu \nu }\Lambda =8\pi G\left[\diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}+T_{\mu \nu }^{\mathrm{comp}}+T_{\mu \nu }^{\mathrm{Rem}}\right].$$

(60)

Neither $T_{\mu \nu }^{\mathrm{comp}}$ nor $T_{\mu \nu }^{\mathrm{Rem}}$ carries a window factor. $T_{\mu \nu }^{\mathrm{comp}}$ is sourced where ${\nabla }_{\mu }\diamond \ne 0$, which is the boundary layer ${\mathcal{B}}_{ϵ}$, at the outer edge of and beyond ${\Omega }_{\mathrm{loc}}$. $T_{\mu \nu }^{\mathrm{Rem}}$ is globally extended. Applying $\diamond \left(x\right)$ to either term would suppress it in the region where it is required to be nonzero, causing the Bianchi identity to fail globally.

In the normal low-localization-density regime, $T_{\mu \nu }^{\mathrm{Rem}}=0$ by the Vanishing Lemma of Section 7, and Equation (60) reduces to:

$$G_{\mu \nu }+g_{\mu \nu }\Lambda =8\pi G\left[\diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}+T_{\mu \nu }^{\mathrm{comp}}\right],$$

(61)

with $T_{\mu \nu }^{\mathrm{comp}}$ macroscopically invisible (also by the Vanishing Lemma) and the EFE sourced effectively only by $T_{\mu \nu }^{\mathrm{SM}}$ inside the window. Inside ${\Omega }_{\mathrm{loc}}$ where $\diamond =1$ and $T_{\mu \nu }^{\mathrm{comp}}=0$, this reduces to the standard EFE.

7. The Vanishing Lemma and the Remnant Stress-Energy $T_{\mu \nu }^{\mathrm{Rem}}$

7.1. The Coarse-Graining Prescription

What happens to $T_{\mu \nu }^{\mathrm{comp}}$ outside a localization interval? Once localization ceases, fields cease to be operationally meaningful, but for spacetime to remain locally coherent, that is for the contracted Bianchi identity to be locally satisfied after localization, some stress-energy must maintain causal coherence. That is to say, $T_{\mu \nu }^{\mathrm{comp}}$ must become delocalized. The framework accounts for this by coarse-graining $T_{\mu \nu }^{\mathrm{comp}}$ over scales larger than the microscopic coherence length $ϵ$.

Coarse-graining averages over microscopic degrees of freedom below the astrophysically or experimentally relevant resolution length. Rapidly oscillating phases and off-diagonal momentum correlations, which generate the pressure terms in $T_{\mu \nu }^{\mathrm{comp}}$, cancel in a phase-incoherent ensemble averaged over such scales. What survives is the slowly varying, large-scale residual, the macroscopic remnant of past intense localization:

$${\langle T_{\mu \nu }^{\mathrm{comp}}\rangle }_{\mathrm{coarse}}⟶T_{\mu \nu }^{\mathrm{Rem}}.$$

(62)

The angle brackets ${\langle ·\rangle }_{\mathrm{coarse}}$ denote the coarse-graining average over microscopic degrees of freedom; the arrow indicates that $T_{\mu \nu }^{\mathrm{Rem}}$ is defined as the macroscopic limit of this average. When the compensator is regular and oscillatory, this average vanishes, as established by the Vanishing Lemma below. When the baryonic localization is sufficiently intense and non-adiabatic, as during a major astrophysical localization event, the average can yield a nonzero $T_{\mu \nu }^{\mathrm{Rem}}$. The central question for the EFE is therefore whether ordinary field evolution produces a nonzero $T_{\mu \nu }^{\mathrm{Rem}}$.

7.2. The Vanishing Lemma

A formal result establishes that for regular quantum fields, those whose stress-energy is finite, varies only on microscopic scales, and carries no persistent nonperturbative boundary structure, the coarse-grained compensator vanishes macroscopically.

Lemma (Vanishing of coarse-grained compensator). Let $T_{\mu \nu }^{\mathrm{SM}}$ satisfy ${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{SM}}=0$ and let $T_{\mu \nu }^{\mathrm{comp}}$ satisfy ${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{comp}}=-({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}$. For any coarse-graining scale $L\gg ϵ$,

$${\overline{T_{\mu \nu }^{\mathrm{comp}}}}_L\stackrel{L/ϵ\to \infty }{\to }0.$$

(63)

Proof. The divergence of $T_{\mu \nu }^{\mathrm{comp}}$ is supported only in the boundary layer ${\mathcal{B}}_{ϵ}$ of thickness $ϵ$ where ${\nabla }^{\mu }\diamond \ne 0$. The fraction of any coarse-graining volume occupied by ${\mathcal{B}}_{ϵ}$ scales as $ϵ/L$. Since $T_{\mu \nu }^{\mathrm{SM}}$ is finite on ${\mathcal{B}}_{ϵ}$, the coarse-grained source term scales as ${\left(\overline{\nabla \diamond T}\right)}_L\sim L^{-1}{T}_{\mu \nu }^{\Sigma }$, where $T_{\mu \nu }^{\Sigma }$ is the stress-energy evaluated on the boundary. Taking $L\to \infty$ yields Equation (63). □

This Lemma establishes that the compensator sector carries no macroscopic stress-energy for ordinary quantum fields. A nonzero macroscopic remnant arises only in regimes of extreme localization density at a causally irreversible boundary where the regularity assumption fails. A causally irreversible boundary is one for which no external coarse-graining scale can fully contain the boundary layer on any accessible timescale.

7.3. Interpretation and Experimental Relevance

The Lemma establishes that for all SM field content—quarks, leptons, gauge bosons, Higgs—the coarse-grained boundary correction is macroscopically invisible. The EFE in the bulk are sourced by $\diamond T_{\mu \nu }^{\mathrm{SM}}$, which reduces to $T_{\mu \nu }^{\mathrm{SM}}$ inside the window and zero outside. Standard GR is recovered in the macroscopic limit, consistent with all precision tests of the EFE at solar system and cosmological scales.

Relation to precision QED tests. The Lemma ensures that no spurious boundary-layer stress-energy contaminates the calculation of QED corrections to atomic energy levels or to the electron anomalous magnetic moment. Within the localization interval defined by a scattering experiment or atomic transition, the window is effectively global ($\diamond \approx 1$), standard QED applies without modification, and $T_{\mu \nu }^{\mathrm{comp}}$ is negligible at the scale of any precision electromagnetic measurement. The extraordinary agreement between QED predictions and experiment [3] is therefore fully consistent with the windowed framework.

Relation to electroweak precision tests. The SM electroweak sector has been tested to sub-percent precision at LEP, Tevatron, and the LHC [3]. The Lemma guarantees that windowing introduces no correction to S, T, and U oblique parameters, or to any other precision electroweak observable, at scales $L\gg ϵ$. The boundary layer ${\mathcal{B}}_{ϵ}$, set by the decoherence timescale of the detector [2,22], is far below the resolution of any current or foreseeable electroweak measurement.

7.4. When the Lemma Fails: The High-Density Regime

The Lemma fails when the localization rate density $\lambda$ (events per unit 4-volume) exceeds the critical threshold identified in the companion paper (with c explicit):

$$\lambda \ll {\lambda }_{*}(\ell )\equiv {\displaystyle \frac{c}{{\ell }^4}},$$

(64)

where ℓ is the characteristic localization scale. When $\lambda \lesssim {\lambda }_{*}$, localization events are sparse enough that causal reconciliation completes and $T_{\mu \nu }^{\mathrm{comp}}$ is macroscopically negligible. When $\lambda \gtrsim {\lambda }_{*}$, localization events are too dense for causal reconciliation, $T_{\mu \nu }^{\mathrm{SM}}$ is not bounded on ${\mathcal{B}}_{ϵ}$, the regularity assumption fails, and the coarse-grained compensator does not vanish:

$$\lambda \gtrsim {\lambda }_{*}(\ell )\Rightarrow T_{\mu \nu }^{\mathrm{Rem}}={\langle T_{\mu \nu }^{\mathrm{comp}}\rangle }_{\mathrm{coarse}}\ne 0.$$

(65)

In this regime, $T_{\mu \nu }^{\mathrm{Rem}}$ is nonzero and the full decomposition $T_{\mu \nu }^{\mathrm{nl}}=T_{\mu \nu }^{\mathrm{comp}}+T_{\mu \nu }^{\mathrm{Rem}}$ is required in the EFE.

7.5. Conservation and Formal Structure of $T_{\mu \nu }^{\mathrm{Rem}}$

Only the stress-energy that survives the process of delocalization remains physically meaningful. The remnant stress-energy $T_{\mu \nu }^{\mathrm{Rem}}$ left after coarse-graining obeys the contracted Bianchi identity globally, unlike $T_{\mu \nu }^{\mathrm{SM}}$ and $T_{\mu \nu }^{\mathrm{comp}}$, which are only proposed to do so over an interval of localization when combined:

$${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{Rem}}=0.$$

(66)

Because stress-energy localized and then delocalized by this process can only be accessed again by repeating a similar process that formed it, this stress-energy is essentially a geometric remnant of past intense localization events.

To show this formally, let $\mathcal{I}\left(x\right)$ be the function defining the information density of a spacetime remnant:

$$\mathcal{I}\left(x\right)={\int }_{J^{-}\left(x\right)}{d}^4{x}^{\prime }\sqrt{-g^{\prime }}{\mathcal{L}}_{\mathrm{Rem}}\left(x^{\prime }\right).$$

(67)

An effective action for that potential is given by Equation (68), with a retarded kernel $\mathcal{K}(x,x^{\prime })$ enforcing causal propagation. Varying this action yields an effective stress-energy by the same process as used in Equation (54) but without the interval of localization [30]:

$$\begin{array}{cc}\hfill S_{\mathrm{Rem}}& ={\displaystyle \frac{1}{2}}\int d^{4}x{d}^4{x}^{\prime }\sqrt{-g}\sqrt{-g^{\prime }}\mathcal{K}(x,x^{\prime })\mathcal{I}\left(x\right)\mathcal{I}\left(x^{\prime }\right),\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill T_{\mu \nu }^{\mathrm{Rem}}& =-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta S_{\mathrm{Rem}}}{\delta g^{\mu \nu }}}.\hfill \end{array}$$

(69)

7.6. Comparing Remnant and Compensator Stress-Energy

Showing that Equation (62) holds from Equations (54) and (69) is fairly straightforward using Parker’s approach to derive an effective action and stress-energy tensor [27].

If $Z[g,H]$ is the generating function for N, then:

$$Z[g,H]=\int \mathcal{D}N\mathcal{D}\overline{N}exp\left(i{S}_N[g,N,H]\right).$$

(70)

In Equation (70), $\mathcal{D}N$ and $\mathcal{D}\overline{N}$ are the functional path integrals over all fermionic field configurations, with $\mathcal{D}N\equiv {\prod }_{x}dN\left(x\right)$. The corresponding effective action is:

$$S_{\mathrm{eff}}[g,H]=-ilnZ[g,H].$$

(71)

Taking the variation of Equation (71) and using Z to find the relation between Z and $T_{\mu \nu }^{\mathrm{Rem}}$:

$${\displaystyle \frac{\delta S_{\mathrm{Rem}}}{\delta g^{\mu \nu }}}=-i{\displaystyle \frac{1}{Z}}{\displaystyle \frac{\delta Z}{\delta g^{\mu \nu }}}=-i{\displaystyle \frac{1}{Z}}\int \mathcal{D}N\mathcal{D}\overline{N}\left(i{\displaystyle \frac{\delta S_N}{\delta g^{\mu \nu }}}\right)e^{i{S}_N}=〈{\displaystyle \frac{\delta S_N}{\delta g^{\mu \nu }}}〉,$$

(72)

where $\langle ·\rangle$ denotes the path-integral expectation value over N. Equation (72) shows that the relation in Equation (62) is valid.

Coarse-graining over unresolved velocities and phases forces the stress-energy to approximate a dust-like form. To see why: $T_{\mu \nu }^{\mathrm{Rem}}$ represents the coarse-grained residual of a phase-incoherent ensemble. The off-diagonal and isotropic pressure components arise from correlated microscopic velocities and phases, which average to zero over the coarse-graining volume when phase coherence is absent. What survives is the energy-density part, which is insensitive to phase cancellation. Pressure terms are therefore suppressed, yielding [27]:

$$T_{\mu \nu }^{\mathrm{Rem}}\approx {\rho }_{\mathrm{Rem}}{u}_{\mu }{u}_{\nu },p_{\mathrm{Rem}}\approx 0.$$

(73)

Here ${\rho }_{\mathrm{Rem}}$ is the effective energy density of the remnant, $u^{\mu }$ is its four-velocity (normalized as $g_{\mu \nu }{u}^{\mu }{u}^{\nu }=-1$), and the tensor structure ${\rho }_{\mathrm{Rem}}{u}_{\mu }{u}_{\nu }$ is the standard stress-energy form of a pressureless perfect fluid (dust). The condition $p_{\mathrm{Rem}}\approx 0$ means the remnant does not support pressure gradients, hydrostatic equilibrium, or ram-pressure forces: it is gravitationally active but thermally inert, interacting with the metric but not with the thermodynamic processes that equilibrate baryonic gas. The equation of state $w\equiv p/\rho \approx 0$ matches the kinematic definition of cold pressureless matter. This is a purely kinematic consequence of delocalization, not a claim about a specific microscopic species. The magnitude ${\rho }_{\mathrm{Rem}}$ is not fixed by the present arguments; it depends on the localization history and rate density, and is left for future quantitative treatment.

8. Summary of the Windowed Field Equations

The physical content of the localization-based picture is expressed by two equations. The windowed SM action:

$$S_{\mathrm{SM}}=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\left({\mathcal{L}}_{\mathrm{gauge}}+{\mathcal{L}}_{\mathrm{ferm}}+{\mathcal{L}}_{\mathrm{Higgs}}+{\mathcal{L}}_{\mathrm{Yuk}}\right),$$

(74)

and the Einstein field equations with the full decomposition of the non-local boundary term:

$$G_{\mu \nu }+g_{\mu \nu }\Lambda =8\pi G\left[\diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}+T_{\mu \nu }^{\mathrm{comp}}+T_{\mu \nu }^{\mathrm{Rem}}\right].$$

(75)

In Equation (75): $\diamond \left(x\right)T_{\mu \nu }^{\mathrm{SM}}$ is the windowed matter source, equal to $T_{\mu \nu }^{\mathrm{SM}}$ inside ${\Omega }_{\mathrm{loc}}$ and zero outside; $T_{\mu \nu }^{\mathrm{comp}}$ is the boundary-layer compensator sourced by $-({\nabla }^{\mu }\diamond )T_{\mu \nu }^{\mathrm{SM}}$, which restores local conservation at the window boundary and is macroscopically negligible for regular fields; and $T_{\mu \nu }^{\mathrm{Rem}}$ is the globally-conserved macroscopic remnant that arises only in the high-localization-density regime where $\lambda \gtrsim {\lambda }_{*}(\ell )$. Neither $T_{\mu \nu }^{\mathrm{comp}}$ nor $T_{\mu \nu }^{\mathrm{Rem}}$ carries a window factor, because both terms must be nonzero at and beyond the boundary of ${\Omega }_{\mathrm{loc}}$.

Within ${\Omega }_{\mathrm{loc}}$ where $\diamond =1$, $T_{\mu \nu }^{\mathrm{comp}}=0$, and $T_{\mu \nu }^{\mathrm{Rem}}=0$ for regular fields, Equation (75) reduces identically to the standard EFE. The cosmological constant $\Lambda$ appears as an integration constant, as in Jacobson’s original derivation [31].

The three conditions under which each established framework applies follow from the ratio of the spin-connection curvature scale to the matter gradient scale. In Equations (76)–(78), ${\Sigma }^{ab}=\frac{1}{4}[{\gamma }^a,{\gamma }^b]$ denotes the spinor representation generators of the Lorentz group, distinct from the stress-energy tensor, and $\frac{1}{2}{\left({\omega }_{ab}\right)}_{\mu }{\Sigma }^{ab}$ measures the strength of geometric curvature coupling relative to the matter field gradient ${\partial }_{\mu }$:

$$\begin{array}{cc}\hfill \frac{1}{2}{\left({\omega }_{ab}\right)}_{\mu }{\Sigma }^{ab}\gg {\partial }_{\mu }& \Rightarrow \mathrm{GR}\mathrm{is}\mathrm{the}\mathrm{appropriate}\mathrm{framework},\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill \frac{1}{2}{\left({\omega }_{ab}\right)}_{\mu }{\Sigma }^{ab}\ll {\partial }_{\mu }& \Rightarrow \mathrm{Standard}\mathrm{QFT}\mathrm{is}\mathrm{the}\mathrm{appropriate}\mathrm{framework},\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill \frac{1}{2}{\left({\omega }_{ab}\right)}_{\mu }{\Sigma }^{ab}\sim {\partial }_{\mu }& \Rightarrow \mathrm{Equations}\left(74\right)--\left(75\right)\mathrm{apply}.\hfill \end{array}$$

(78)

Condition (78) is rare, extreme, and highly localized, which accounts both for the empirical success of GR and QFT within their respective domains and for the difficulty of probing the transition between them experimentally.

9. Conclusions

This paper has applied the localization window $\diamond \left(x\right)$ systematically to the field content of the Standard Model and established the following results.

First, the windowed action prescription produces equations of motion for Dirac fermions, Klein–Gordon scalars, Maxwell gauge fields, and the full SM Lagrangian that reduce identically to their standard curved-spacetime counterparts inside the localization region. All boundary corrections are confined to a layer of thickness $ϵ\sim c{\tau }_{\mathrm{dec}}$ set by the decoherence timescale, and are suppressed by $ϵ/L$ relative to the bulk for any macroscopic scale L.

Second, Noether analysis of the windowed action yields windowed Ward identities ${\nabla }_{\mu }(\diamond J^{\mu })=0$. Gauge invariance under $SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ and local Lorentz invariance are preserved exactly within the window. Apparent non-conservation is a kinematic boundary effect, structurally identical to open-system flux terms from decoherence theory [1] and to the contact terms appearing in Ward–Takahashi identities for subsystem currents in QFT [6]. Worked examples demonstrate this for electromagnetic current conservation and for spinor normalization across a decoherence boundary.

Third, the non-local boundary term $T_{\mu \nu }^{\mathrm{nl}}$ of the companion paper decomposes precisely as $T_{\mu \nu }^{\mathrm{nl}}=T_{\mu \nu }^{\mathrm{comp}}+T_{\mu \nu }^{\mathrm{Rem}}$. The compensator $T_{\mu \nu }^{\mathrm{comp}}$ is the boundary-layer term required locally by the contracted Bianchi identity. Neither component carries a window factor, because both must be nonzero at and beyond the boundary of the localization window. The sharp-window limit of $T_{\mu \nu }^{\mathrm{comp}}$ recovers the Israel junction conditions exactly [14], and the smooth-window generalization is structurally identical to the Ashtekar–Krishnan dynamical horizon flux balance laws [15].

Fourth, the Vanishing Lemma establishes that for all regular SM fields, $T_{\mu \nu }^{\mathrm{comp}}$ vanishes upon coarse-graining and $T_{\mu \nu }^{\mathrm{Rem}}=0$. Standard field evolution leaves no macroscopic stress-energy remnant sourcing the EFE. This result is fully consistent with precision tests of QED [3], the electroweak sector, and GR at all currently accessible scales. A nonzero $T_{\mu \nu }^{\mathrm{Rem}}$ arises only when the localization rate density exceeds ${\lambda }_{*}(\ell )=c/{\ell }^4$; in that regime it is globally conserved, pressureless, and obeys the contracted Bianchi identity independently.

Future work will develop quantitative predictions from the boundary-layer structure and examine whether the windowed Ward identities lead to observable signatures in quantum optical and condensed-matter systems, where decoherence boundaries are experimentally accessible [2,22].