Em03-D A Local System–Probe Realization of a Hawking-Packet Observable Compact coupling, infrared-controlled bulk approximation, and a bounded Weyl effect

We construct a compactly coupled scalar probe scheme for one bounded outgoing Hawking-packet quadrature effect on the Schwarzschild exterior. The result is restricted to a massless s-wave packet and uses full angular-shell access.

Complete stationary scattering identifies the packet with a two-channel bulk coefficient. We construct real, compactly supported bulk tests approaching that coefficient in the Unruh covariance norm. A time-derivative filter permits a quantitative spatial Cauchy-data truncation without assuming that the thermal low-frequency weight is harmless. Separately, the explicit zero-energy s-wave solutions give a local infrared bound for arbitrary compact reduced tests.

A second massless scalar is coupled to the system by a smooth compact bilinear interaction. We calculate the exact induced Weyl effect and bound its deviation from the selected compact bulk test by a convergent advanced-Green-operator construction. Residual probe noise is retained. A finite-dimensional symplectic squeezing of the probe preparation suppresses that noise while keeping the probe readout a bounded effect; dividing an effect by its attenuation coefficient is not used.

The resulting expectation comparison is uniform on an explicitly defined class of regular system states with dominated second moments. It specializes to the inherited Unruh baseline and has a constructive resource-dependent accuracy schedule. The coupling region expands with the packet approximation, the readout test grows at weak coupling, and a conjugate probe variance diverges along the noise-suppression schedule.

This is a local field–probe observable correspondence, not a pointlike detector, a packet-number instrument, coherent state transfer, or channel equivalence with the effective apparatus of Em03-C. No collapse geometry, microscopic evaporation dynamics, or quantum-gravity correction is constructed.

The construction follows

Its target is one effect, not the two-memory apparatus of Em03-C Em03C. For a normalized outgoing packet $h$, set

Since the real part of a unitary has spectrum in $[-1,1]$,

We prove a state-class comparison of the form

Here $R$ is a spatial approximation parameter and $ϵ>0$ is a coupling parameter. The probe state, coupling profile, and readout test are specified below.

The scheme uses a full spherical shell. For each finite $R$ its coupling is compactly supported strictly outside the horizon. No fixed shell is claimed to realize arbitrary packet accuracy.

The selected system state is the massless scalar Unruh prescription of Em03-A Em03A. Its horizon input is thermal in normalized Killing frequency, its infinity input is vacuum, and the two inputs are independent and centered quasifree. The outgoing packet mean is

The subscript $0$ denotes the s-wave greybody probability.

The one-mode Gaussian characteristic function used here follows from this prescription and the finite-mode construction in Em03-C. It does not require a density operator for the entire stationary continuum.

The effective Markov port, capture controls, and finite registers of Em03-B and Em03-C are not premises of the local construction.

For the massless scalar on the Schwarzschild exterior, the past and future horizon and infinity channels give complete stationary scattering. In the s-wave reduction, the incoming generalized eigenfunctions define a unitary spectral transform

intertwining the radial operator with multiplication by ${\omega }^2$. The flux-normalized stationary scattering matrix is unitary.

We use the complete wave-scattering result of Dimock Dimock, together with the ordinary one-dimensional spectral resolution for a real short-range potential DeiftTrubowitz. Baskin–Wang provides a complementary geometric account of the two radiation ends and their energy normalization BaskinWang. The exact mode and test normalizations used below are displayed explicitly.

A formally self-adjoint normally hyperbolic operator on a globally hyperbolic spacetime has the advanced and retarded Green operators and Cauchy evolution used below. Adding a smooth compactly supported symmetric zeroth-order matrix coupling gives a well-posed coupled theory and a symplectic scattering map. It induces the corresponding automorphism of the Weyl algebra.

The PDE input is the standard Green-operator and Cauchy theory BGP. The local system–probe construction and its scattering interpretation are those of Fewster–Verch FewsterVerch. Its Weyl-algebra version is described by Fewster–Jubb–Ruep FewsterJubbRuep. We give the particular test transformation and estimates required here.

The affine-horizon and Killing-infinity product prescription defines a quasifree Hadamard Unruh state on the Schwarzschild exterior.

This is the Dappiaggi–Moretti–Pinamonti construction DMP, already declared in Em03-A. Its global microlocal proof is not reproduced.

None of these external statements supplies the compact bulk-to-packet approximation proved in Section sec:packetbridge.

boundary. The manuscript supplies analytical constructions. Compilation, formal verification, numerical evaluation, calibration, and experimental realization are not asserted.

In physical units,

The time generator is normalized at infinity.

For the PDE estimates we measure lengths in units of $r_{\mathrm{h}}$:

The packet rescales as

This preserves both packet normalization and its annihilator. After this paragraph, tildes are suppressed:

The occupation integrals are unchanged by the rescaling. All constants in the estimates are dimensionless in this convention.

The physical coupling associated with a dimensionless profile $ϵ\rho$ is ${r_{\mathrm{h}}}^{-2}ϵ\rho$. A dimensionless scalar smearing is converted to its physical smearing with the corresponding factor ${r_{\mathrm{h}}}^{-3}$. Thus the profile choices below do not equate quantities of different physical dimensions.

The exterior is globally hyperbolic, with $t=\mathrm{constant}$ Cauchy surfaces. Indeed, in the conformally related optical metric, a causal curve has $|\mathrm{d}x/\mathrm{d}t|\le 1$. On every bounded $x$-interval the angular optical geometry is regular and compact. An inextendible causal curve cannot escape the spatial geometry at finite $t$, and therefore crosses every time slice once.

We use

For both the real system field $\varphi$ and real probe field $\psi$, the free action is

Both fields are massless.

Let $Y_0=(4\pi {)}^{-1/2}$. Writing

in the selected angular sector gives

For a real reduced test $f(t,x)$, define its bulk representative

Then

This follows from $\mathrm{d}{\mathrm{vol}}_g=r^{2}F\mathrm{d}t\mathrm{d}x\mathrm{d}\Omega$. Compact reduced support gives compact bulk support outside the horizon.

The causal propagator convention is

It is consistent with

For $\mathsf{D}={{\partial }_t}^2+\mathsf{L}$, the retarded solution has the usual positive sine kernel. The advanced-minus-retarded propagator has the corresponding negative sine kernel.

Let $f_{+}(x,\omega )\sim e^{\mathrm{i}\omega x}$ at $+\infty$ and $f_{-}(x,\omega )\sim e^{-\mathrm{i}\omega x}$ at $-\infty$ be Jost solutions. Define

These are the horizon-incident and infinity-incident modes. The infinity-incident mode has an outgoing infinity reflection amplitude denoted ${\rho }_{\mathrm{I}}(\omega )$.

The incoming spectral transform is

It is understood in its unitary $L^2$ extension.

The canonical field expansion is

Consequently a compact real reduced test has coefficient

so that

Annihilators are conjugate-linear in their tests:

Define the real symplectic form and Weyl convention by

The Unruh second-moment norm is

For real linear observables,

The norm dominates the ordinary two-channel $L^2$ norm.

This section is specific to the s-wave potential. No higher-angular threshold estimate is silently imported.

The potential $V$ is smooth, nonnegative, and satisfies

Its zero-energy Jost solutions are

and

There is an ${\omega }_c>0$ for which

Proof. Since $\mathrm{d}x=F^{-1}\mathrm{d}r$,

Near the horizon, $x$ grows logarithmically in $r-1$; at infinity, $x=r+O(\log r)$. These facts prove the first-moment bound.

Substitution in $\mathsf{L}f=0$ verifies the two solutions. They tend to one at their respective normalization ends. Using ${\partial }_{x}r=F$ and $F^{\prime }=r^{-2}$ gives their Wronskian $-1$.

The Jost Volterra equations have the zero-frequency kernel $y-x$ in the right equation and its left-end analogue. The first-moment bound gives convergent series and continuity at $\omega =0$, including derivatives on compact $x$-sets. The nonzero Wronskian therefore gives the final assertion.

Choose one such ${\omega }_c$. For $X>0$, set

For $j=\mathrm{H},\mathrm{I}$, $|x|\le X$, and $\omega >0$,

Proof. For $0\le \omega \le {\omega }_c$, use $|\sin (\omega z)/\omega |\le |z|$ in the Jost Volterra series. It gives $|f_{\pm }(x,\omega )|\le e^{m_X}$ on the compact region. Together with $|\tau |\le 4\omega$, this proves the low-frequency bound.

For $\omega \ge {\omega }_c$, the Volterra series gives

Current conservation gives $|\tau |\le 1$. This proves the remaining bound.

For a reduced test supported at $|x|\le X$, define

The $L^1$ norms are over ${\mathbb{R}}^2$.

Every compact smooth real reduced test satisfies

In particular, its thermal infrared integral is finite.

Proof. At $0<\omega \le {\omega }_c$, use (eq:testcoefficient) and Lemma lem:modebound:

At $\omega \ge {\omega }_c$, integrate once by parts in $t$:

Use $1+2n_{\mathrm{H}}(\omega )\le 1+2/(\beta \omega )$ for both channel bounds. Summing the two channel estimates and performing the elementary integrals gives (eq:compactcovariancebound).

This local threshold calculation is not a replacement of unselected radiation by vacuum. Both incoming channels appear in the estimate, with the actual horizon thermal weight.

Fix a normalized

Stationary scattering gives

Thus

This is the coefficient of the outgoing scattering observable in the bulk incoming coordinates. It is not a finite-radius pointwise limit of an unsmeared quantum field.

Unitarity gives

Consequently

There is no packet-count rate factor $1/(2\pi )$ in this formula.

For $C\ge 1$, let ${\mathfrak{S}}_C$ consist of normal states in the regular Unruh representation of the completed two-channel Weyl algebra for which

on its real weighted coefficient domain. The second moment includes any nonzero mean.

The baseline belongs to ${\mathfrak{S}}_1$. The class is not a statement about all Hadamard states.

It is nontrivial beyond the baseline. A Weyl displacement by $l$ changes $X(k)$ by a scalar of magnitude $|s(l,k)|\le 2\Vert l{\Vert }_0\Vert k{\Vert }_{\mathrm{U}}$. The displaced Unruh state therefore belongs to ${\mathfrak{S}}_{1+4\Vert l{\Vert }_0^2}$.

For $\omega \in {\mathfrak{S}}_C$ and weighted coefficients $k,l$,

In particular,

Proof. Put $d=k-l$. The Weyl relation expresses $W(k)$ as $W(l)W(d)$ times a scalar phase of angle of magnitude $|s(k,l)|/2$. Cauchy–Schwarz for the state and spectral calculus give

The phase costs at most $|s(k,l)|/2$. This proves the first inequality. Since

taking real parts and dividing by two proves the second.

The estimate uses exact Weyl relations rather than a formal unbounded-operator Baker–Campbell–Hausdorff manipulation.

For real Cauchy data $(q,p)$, define the linear observable

Its incoming coefficient is

For compact smooth data this follows directly from the canonical field expansion. Its indicated spectral extensions follow by continuity on their moment domains.

Let

It is an antiunitary involution commuting with real functions of $\omega$. It incorporates the reality convention of the incoming scattering modes; componentwise conjugation must not be substituted for it.

Set

and define

These give real $q,p$, satisfy $k(q,p)=k_{-}$, and have spectral support in the compact positive band. They belong to every integer Sobolev space, by spectral functional calculus and elliptic regularity for $\mathsf{L}$.

Choose a real $\chi \in {C_c}^{\infty }((-2,2))$, equal to one on $[-1,1]$. For $R\ge 1$, put

Let $s_R$ solve

Choose a smooth time function ${\eta }_T$, equal to zero for $t\le -T$ and one for $t\ge T$, with $T>0$. Define

Because

the reduced test is smooth and compact, with

The full support includes the angular sphere $S^2$.

The reduced field observable of $\mathsf{D}({\eta }_T{s}_R)$ is $\mathcal{L}(q_R,p_R)$. The coefficient of $F_{R,T}$ is

The time-slice identity is exact for every $T>0$.

Proof. Pair the test with a homogeneous field solution and integrate the ${\eta }_T^{\prime \prime }$ term once by parts:

The integrated symplectic pairing is constant in time. Since $\int {\eta }_T^{\prime }=1$, it equals $\mathcal{L}(q_R,p_R)$.

Alternatively the advanced and retarded solutions of $\mathsf{D}({\eta }_T{s}_R)$ are $({\eta }_T-1)s_R$ and ${\eta }_T{s}_R$, respectively, confirming the sign convention. In (eq:testcoefficient), applying $-{{\partial }_t}^2$ to the test multiplies its coefficient by ${\omega }^2$. This proves (eq:kR).

The time filter is not an assertion that a compact time window exactly selects a band-limited packet. It first acts on a band-limited auxiliary coefficient $k_{-}$, then controls the low-frequency leakage caused by spatial Cauchy truncation.

Write

Define

All fractional powers are those of the positive self-adjoint radial operator.

For a bound using only ordinary spatial derivatives, put

Then

The compact tests (eq:compactpackettest) satisfy

For every $\omega \in {\mathfrak{S}}_C$,

Proof. From (eq:Cauchycoefficient),

Use

and $1+2n_{\mathrm{H}}\le 1+2/(\beta \omega )$. The resulting integral is bounded by

Spectral unitarity gives precisely (eq:packeterror).

For $\lambda \ge 0$, ${\lambda }^{3/2},\lambda \le 1+{\lambda }^2$ and ${\lambda }^{5/2}\le 1+{\lambda }^3$. Also

These facts prove (eq:ordinarytailbound).

Multiplication by $\chi (x/R)$ tends to the identity in every integer Sobolev norm on fixed Sobolev functions. This follows from the $L^2$ derivative tails and the bounds ${{\partial }_x}^j\chi (x/R)=O(R^{-j})$. In particular, $\delta p_R\to 0$ in $H^2$ and $\delta q_R\to 0$ in $H^3$. The potential and its required derivatives are bounded, so $D_p(R)+D_q(R)\to 0$.

Finally, $W_{\varphi }(F_{R,T})=W(k_R)$ in the common regular representation. Apply Lemma lem:Weylcomparison with $l=k_{\ast }$.

This proves the packet bridge from bulk Cauchy evolution and complete scattering. It does not assume the effective temporal port of Em03-C.

The spatial truncation accounts for both radiation ends. Its support moves toward spatial infinity and toward the horizon as $R$ increases. It is therefore not a fixed finite-radius, outgoing-only detector construction.

Fix $R,T$ and abbreviate

Choose a spacelike-compact free probe solution $v_{\mathrm{bulk}}$ equal to the constant solution $1$ on a neighborhood of $K$. Such a solution can be constructed explicitly.

At $t_0=-T-1$, prescribe physical probe Cauchy position equal to one on the spatial projection of $J^{-}(K)$ at that time, extend it smoothly with compact radial support, and prescribe zero velocity. The constant function solves $P1=0$. Finite propagation and Cauchy uniqueness imply equality with one on $K$.

Write

Thus $v=\sqrt{4\pi }r(x)$ on $K$.

Choose ${\eta }_{+}$ equal to zero for $t\le T+1$ and one for $t\ge T+2$, and put

This is a real compact test in the future processing slab. Its advanced solution satisfies

and therefore equals one on $K$.

Let

The interaction is

The coupled equations are

The postinteraction probe effect is

The complementary outcome is $I-O_{ϵ}$. Both outcomes are retained.

The profile is smooth, compact, and spherically symmetric. All angular sectors decouple in the coupled equations. The selected probe test is an s-wave test, so it induces only the selected system angular sector.

The readout effect itself is a specified final probe operation. The manuscript does not construct its laboratory electronics.

Let

Define

The local scattering morphism obeys

For a centered quasifree probe state with variance convention

the induced effect is

Its expectation in every system state is

The induced system test is compactly supported in $K$.

Proof. For future-supported free test vectors $H$, comparison of advanced coupled and uncoupled solutions gives the test transformation

Equivalently, pair the coupled solution with its advanced test solution and use Green's identity in an early Cauchy region, where both theories agree. The test classes are those modulo the respective wave operators.

The transformation preserves the symplectic form: compare the conserved coupled symplectic current on past and future Cauchy surfaces outside $K$. It therefore gives the Weyl scattering morphism ${\Theta }_{ϵ}(W(H))=W({\theta }_{ϵ}H)$. This is the local linear scattering construction stated in Section sec:scope.

Applying it to $H=(0,g/ϵ)$ yields (eq:exactscatteredtests). System and probe Weyl operators belong to separate tensor factors, so no additional mixed phase appears. Apply $\mathrm{id}\otimes \sigma$, then take the real part. Since $c_{ϵ}$ is real and nonnegative, this gives (eq:inducedeffect). Multiplication by $\rho$ gives the support assertion.

This is an equality of bounded effects in a Weyl-algebra quantization. The unbounded polynomial field algebra alone would not supply the effect (eq:probeeffect).

The s-wave coupled operator is

In particular, because $\rho =-F$ and $F=Y_{0}f/(rF(r))$,

Here the first $F$ in $\rho =-F$ is the bulk test, while $F(r)=1-r^{-1}$ is the metric coefficient.

Work on the finite time interval

For a reduced function $u$, define

Let $G_{\mathrm{A},\mathsf{D}}$ be the reduced free advanced Green operator, and define

Its source is compactly supported inside the coupling slab.

The following deliberately conservative constants suffice:

The operator $A$ satisfies

Proof. For a free inhomogeneous solution with zero final data, differentiate the sum of energies in the definition of $\mathcal{X}$. Spatial integration by parts cancels the principal wave terms. The remaining potential and commutator terms contain only $V,V^{\prime },V^{\prime \prime }$. Cauchy–Schwarz and the product rule give

with the reversed-time version for advanced evolution. The stated $c_2$ dominates all the finite product-rule coefficients.

The spatial product estimate

follows by expanding the derivatives through order two. Backward Gronwall inequality on $I$ gives

Let $v_{\mathrm{A}}=G_{\mathrm{A},\mathsf{D}}{g}_{\mathrm{red}}$. It equals $v$ on $K$ and has finite $\mathcal{X}$ norm. Put

Assume $z_{ϵ}<1$.

The reduced advanced coupled solution is

Hence the reduced system test is

where

The reduced residual probe test is

Moreover,

Proof. The two advanced equations give

Substitution gives the displayed inverse. Lemma lem:energy proves convergence in $\mathcal{X}$. The resulting solution has zero final data and satisfies the coupled equation. Cauchy uniqueness identifies it with the exact coupled advanced solution; smooth-source regularity makes it smooth. The test formulas and remainder estimate follow by substitution.

Abbreviate the two scalar coefficients in (eq:compactcovariancebound) by

Choose a rectangular coupling support bound $I\times [-X,X]$, and let $V_K=2X\Delta$. Put

For a smooth $u$ of finite $\mathcal{X}$ norm,

Proof. The one-dimensional Sobolev inequality gives, with a conservative constant,

Therefore