I-B-VP-v3: Closing the spatial-product extension ledger Vacuum polarization, stress tensor, and induced current on a prescribed conical spacetime with Aharonov–Bohm holonomy

We give a unified construction and extension ledger for a massless scalar field on the regular conical product prescribed in I-B. The spatial operator, its self-adjoint extension, the state prescription, the curvature coupling, and the flat-bundle holonomy are explicit inputs. The extension class consists of transverse self-adjoint extensions tensored with the ordinary longitudinal Laplacian; it includes rotation-breaking finite-dimensional boundary conditions but excludes general non-product extensions.

For the Friedrichs extension, an explicit heat-kernel estimate proves infrared finiteness of the massless spectral covariance on compact tests. Direct Euclidean subtraction leaves a jointly real-analytic bisolution at every regular diagonal point, including image-boundary parameters. Analytic continuation gives the local Hadamard form used for point splitting.

For one complex field, with $\Phi =({\varphi }_1+\mathrm{i}{\varphi }_2)/\sqrt{2}$, $T=T[{\varphi }_1]+T[{\varphi }_2]$, signature $(-,+,+,+)$, and mixed components ordered as $(t,r,\phi ,z)$, define

The Friedrichs results are

The neutral real tensor is half the untwisted complex tensor. The current uses the complex first jet of the regular two-point function; replacing that function by its real part would erase it.

For every nonnegative extension in the declared product class, a finite-rank Krein formula gives an infrared-finite spectral state and an analytic regular-part correction. We supply the formula also on boundary charts of the extension space. Positivity is equivalent to positivity of the finite boundary relation. Negative transverse eigenvalues produce unstable longitudinal bands; a stationary finite-second-moment state is then excluded by a commutator-growth argument. Two projected alternatives are recorded, together with their nonlocality and Hadamard defects.

Scale-invariant irregular modes admit closed forms expressed through analytic branches of the Friedrichs polynomials at shifted arguments. At half holonomy an additional ${\mathbb{CP}}^1$ family has explicit angular-dependent observables. Scale-dependent nonnegative extensions have convergent integral representations and proved asymptotics. Generic rotation-breaking extensions with unequal deficient orders can carry a radial current.

The familiar Friedrichs coefficients are not claimed as new. Historical formula comparisons and all inherited numerical residuals are retained with their evidence grades. No numerical execution, apex renormalization, backreaction, interacting construction, or downstream programme closure is asserted.

Vacuum polarization on an idealized cosmic-string background is established subject matter HK1986,Linet1987,FS1987,Dowker1987,GL1994. The separate question of a boundary condition at the removed apex is essential: the regular metric alone does not select a self-adjoint extension KayStuder,DR2017.

This manuscript consolidates the supplied records -B-VP-v2.tex, -B-VP-AB.tex, and -B-VP-SAE.tex RecordVP2,RecordAB,RecordSAE. Those files are retained records; this consolidation neither deletes them nor assumes that their version names form an unambiguous numerical sequence. The first supplied file already identifies itself as version 2, whereas the AB and SAE records refer to an earlier -B-VP.tex as version 1. We therefore identify predecessor records by filename.

to read this version. Sections sec:host–sec:friedrichs-results contain the Friedrichs construction and all three observables. Sections sec:boundary–sec:matrix treat the product extension ledger. Section sec:ledger is the unified status table. The appendices preserve the reported computations and corrections without using them as proof.

The AB source's items C1–C7 are retained in substance:

1. The induced current is added to the Wick square and stress tensor.

2. Explicit comparisons with the flux literature are included.

3. Reports A1–A5 include character-weighted quotient images and high-precision contour comparisons.

4. The Friedrichs formulas hold for every $0<\alpha <1$, not only quotient angles. Their analytical proof does not use finite sampling.

5. The AB source revises its explanation of the earlier $10^{-17}$-scale residuals: it reports a cast of image terms to IEEE double complex and an omitted diffraction term at a rounded quotient parameter. These remain reported diagnostics because the scripts and raw logs were not supplied.

6. The bibliography retains the source's access grades, with their meaning stated in Section sec:evidence. The Smith chapter's 1989/1990 citation variation is retained as a bibliographic note.

7. The deficient twisted modes and their extension families are made explicit.

The SAE source's items D1–D6 are likewise retained:

1. Boundary data, a Krein correction, positivity, unstable extensions, projected constructions, scale-invariant closed forms, and scale-dependent integral representations are included.

2. The AKO and Filgueiras–Moraes comparisons and the SAE verification reports are retained.

3. Nonnegative non-Friedrichs product extensions move from “Open” in the predecessor ledger to constructed spectral states with evaluated observables or convergent evaluation formulas.

4. The established Friedrichs formulas are unchanged. Shifted arguments refer to an entire geometric branch, not to the globally periodic physical absolute-value mode sum.

5. Rotation-breaking $U(2)$ boundary conditions are admitted only after dropping rotation invariance, while retaining the spatial-product assumption.

6. Bibliographic access marks remain provenance statements; they are not a replacement for a formula-level comparison or an execution certificate.

A submission cannot preserve a false implication merely because it appeared in a predecessor. The following corrections are incorporated and their affected reports remain visible:

1. Infrared finiteness is proved. Nonnegativity alone is not used to infer a massless covariance. The Friedrichs heat estimate and the non-Friedrichs Krein bound supply the necessary estimates.

2. Image boundaries are included. The total Euclidean remainder is treated by joint ellipticity. No diagonal parameter is discarded as a measure-zero exception.

3. Current subtraction is stated correctly. The derivative difference of the direct parametrix does not vanish along arbitrary approaches to coincidence. The current is defined using the subtracted complex kernel. For the time splitting used to compute its azimuthal component, the direct contribution is identically zero.

4. Boundary charts are genuine boundary relations. An “infinite eigenvalue” is replaced by an explicit subspace and a compressed Hermitian relation. The finite-matrix small-radius limit is not asserted for every boundary chart.

5. Small-radius remainder orders are corrected. The first correction for $0<\nu <1/2$ is retained, but its remainder has additional regimes at $\nu =1/3$. The logarithmic case $\nu =1/2$ is supplied.

6. Asymptotic zeros are respected. A relative error is not asserted when a displayed leading stress coefficient vanishes.

7. Transverse and full spectra are separated. A negative transverse eigenvalue generates a full longitudinal continuous band, not a full-space $L^2$ eigenfunction.

8. State selection is not universal state uniqueness. The distinguished zero-mean spectral Fock state is specified. No classification of all algebraic ground states is claimed.

9. Projection changes the field algebra. The two projected constructions are not states with the original local canonical commutator. Their defects are stated on their actual domains.

10. Computation is not silently promoted to proof. All inherited execution claims are SUPPLIED_ NUMERIC{} and UNCERTIFIED{} as rigorous total-error bounds.

These corrections do not alter the Friedrichs polynomials, its current, the scale-invariant correction formulas, or the finite-chart Krein coefficient.

The Enter companion host is prescribed, not derived in this paper. I-B distinguishes a finite residual density matrix from a covariant tensor-valued distribution: the field, observable, state, renormalization prescription, and distributional interpretation must be supplied separately CheungIB. We supply those ingredients for a free test field. None is extracted from an Enter readout.

The destructive-readout interpretation of Enter CheungEnter,CheungPhil,CheungEn09 and the reporting discipline of XVI-O1-ST CheungO1ST constrain interpretation but do not construct the continuum covariance. The finite instrument and its limitation are reproduced in Section sec:finite.

of I-B-G. The supplied Friedrichs foundation identifies -B-G-v2.tex, with source labels :host, :product, :positivity, :green, :speed, :closed, :structure, :window, :transfer, and :transfer-contract CheungIBG. The AB and SAE records also refer to an earlier title and theorem numbering. This paper uses source labels when discussing the companion and reproduces the operator, kernel, boundary, and positivity calculations needed for its principal results. Finite propagation and supported transfer are used only in their separately stated scope.

comparisons are not imported constructions. X-A-ST, X-B-ST, and X-C-ST concern supported scalar completion operations CheungXAST,CheungXBST,CheungXCST. XI-E separates a named Hadamard product domain from selection of a positive state CheungXIE. Oai282-A separates finite tests from continuum coverage CheungOai282A. These comparisons do not import a gauge domain, interacting product, contact classification, or dynamics law.

of “Open”. An item marked Open is not resolved by this manuscript. This is not a claim that it is unknown in the literature.

Fix

The regular spacetime is

with

The metric completion adds $\mathcal{W}=\{r=0\}$, but $\mathcal{W}\not\subset M$. We use $\hslash =c=1$, and $t$ is length-valued time.

The massless equation is

On $M$, $R=0$. No distributional apex curvature coupling is included.

Let

For $\beta \in [0,1)$, the flat Hermitian line bundle $L_{\beta }$ is represented on the angular cover by

In a lifted parallel frame its connection is ordinary differentiation. Put

The spatial Hilbert space is ${\mathcal{H}}_{\beta }={L^2}_{\beta }(N,\mathrm{d}V)$.

The main ledger concerns

where $B_{\perp }$ is a self-adjoint extension of the transverse cone Laplacian in ${L^2}_{\beta }({C_{\alpha }}^{\circ },\alpha r\mathrm{d}r\mathrm{d}\phi )$. It is complex linear. Rotation invariance is imposed for the diagonal subledger but not for the full finite-dimensional boundary relation.

General $z$-dependent, longitudinal-momentum-dependent, and non-product extensions are outside this class.

The Friedrichs extension is denoted $B_F$. It is the closure of the nonnegative energy form initially defined on compactly supported smooth sections.

A neutral real field uses $\beta =0$. For the charged field,

Thus $⟨{\Phi }^{†}\Phi ⟩$ has the same scalar covariance normalization as one real Wick square, whereas the complex stress tensor has twice the corresponding real differential expression. The real components need not be separate globally trivial bundles.

The antiparticle sector uses the conjugate bundle and conjugate extension. A real sign bundle at $\beta =1/2$ is not what is called one charged complex field here.

For

a periodic gauge may be chosen locally with $eA=\beta \mathrm{d}\phi$, up to an integer gauge shift. The twisted parallel representation satisfies

For a flux $\mathcal{F}$, our convention is $\beta =\mathrm{frac}(e\mathcal{F}/2\pi )$.

On a developing chart,

If a companion completion uses

then

The equation operator and action Hessian have opposite signs. This distinction matters when comparing inverse conventions.

Euclidean continuation uses

With $t=-\mathrm{i}\tau$, mixed stress components obey

The static energy density is

For chosen lifts define

Set

An image boundary is a locus

If $\vartheta =\psi +2\pi \alpha \ell$, with $\psi \in (-\pi \alpha ,\pi \alpha ]$, a lifted kernel satisfies

A reduced-angle expression must retain this bundle factor. All diagonal calculations use one local parallel frame.

Relative to $\mathrm{d}V$, the Friedrichs heat kernel is

For each compact regular spatial set $K$,

At $\beta =0$, ${\nu }_{\ast }=0$.

Proof. The Friedrichs radial transform uses $J_{\nu }(kr)$ with measure $k\mathrm{d}k$. Weber's integral is

Combining the normalized angular modes with the longitudinal heat kernel gives (eq:heat) Watson,DLMF.

On $K$, the power series gives, uniformly for $s\ge 1$,

The orders grow linearly in $|n|$, so their sum is bounded by $C_K{s}^{-{\nu }_{\ast }}$. The remaining prefactor is $O(s^{-3/2})$.

For every $v\in {C_c}^{\infty }(N;L_{\beta })$, ${B_F}^{-1/4}v\in {\mathcal{H}}_{\beta }$. On a fixed compact support this map is continuous in the compact smooth test topology. There is no $L^2$ zero eigenfunction.

Proof. The spectral theorem gives

For $s\le 1$, contraction bounds the integrand by $s^{-1/2}{\Vert v\Vert }^2$. For $s\ge 1$, Proposition prop:heat bounds it by

Both integrals converge and give continuity. Zero form energy would imply a locally parallel section; no nonzero parallel section is square integrable on $N$.

For a nonnegative spatial extension whose compact tests belong to $\mathrm{dom}(B^{-1/4})$, define

and the positive-frequency kernel

The zero-mean quasifree spectral Fock state has smeared covariance $⟨{\mathcal{K}}_{B}f,{\mathcal{K}}_{B}g⟩$.

For a complex field the particle and antiparticle constructions use $B$ and its conjugate extension. Charge-changing two-point functions vanish.

The algebra is the extension-selected test-field CCR algebra. Its commutator is the causal kernel of that extension. No compact-test exactness assertion $\ker E=P{C}_c^{\infty }(M)$ is needed. That separate question is recorded as open in the supplied I-B-G :compact-exact.

The Friedrichs prescription is a continuous positive covariance on compact smooth spacetime tests and defines a spectral ground state. With

the real and charged commutators are respectively

The state is invariant under $t,z$ translations, rotations, and proper time-oriented $(t,z)$ boosts. Its covariance has scaling degree two:

Proof. Lemma lem:infrared bounds the one-particle vector on fixed compact supports. The Gram form proves positivity and continuity; the spectral Fock construction gives a nonnegative Hamiltonian Wald1994. The difference of time orientations is

Differentiating in the second time variable at equal times gives the canonical $+\mathrm{i}\mathrm{id}$ sign.

Translations and rotations commute with the operator. For boosts use

The inner mass-shell measure is the boost-invariant $1+1$-dimensional positive-frequency measure of mass $k$. The infrared estimate justifies compact testing. Rescaling $k,p$ proves the dilation identity.

For Friedrichs, intrinsic finite propagation gives local commutativity for intrinsically spacelike-separated supports CheungIBG (:speed). This is not a statement about an arbitrary completion.

The zero-mean spectral state is distinguished by the prescription, not by a theorem classifying every algebraic ground state. Additional zero-frequency covariance terms or appropriate coherent shifts are not ruled out merely by positivity of the extension.

For Euclidean points set

The heat-integrated kernel

is a fundamental distribution for $L_E$ in each variable. For $\eta >0$,

where $a=\eta /\alpha$, $b=\phi -{\phi }^{\prime }$. For $\beta =0$,

At $\alpha =1,\beta =0$, this is $(4{\pi }^2{{\sigma }_E}^2{)}^{-1}$.

Proof. Large-time convergence follows from Proposition prop:heat; the additional Euclidean factor gives $s^{-2-{\nu }_{\ast }}$. Integrating the heat equation proves the fundamental-distribution identities. Substitute $\zeta =r{r}^{\prime }/(2s)$ and use

The constants become $(8{\pi }^2\alpha r{r}^{\prime }{)}^{-1}$. Absolute convergence justifies the Friedrichs sum for $\eta >0$. Splitting $n\ge 0$ and $n\le -1$ gives the two geometric series Watson,DLMF.

Define

For $\zeta >0$, at generic angles,

The identity extends after joint distributional testing through image boundaries. Near such a boundary ${\vartheta }_j$,

The weight decays exponentially at large $u$. For $\beta =0$,

Proof. Set $A=\pi /\alpha$, $b=u/\alpha$, $c=\vartheta /\alpha$, and

Splitting positive and negative frequencies gives

The denominators give the asserted joint local bound. For $0<\beta <1$, a decay rate is $\min (\beta ,1-\beta )/\alpha$. At $\beta =0$, the zero-order sine vanishes and rate $1/\alpha$ suffices.

Insert the integral representation

DLMF (Eq. 10.32.4). Twisted Fourier summation of the first term gives the image indicator and its phases. An Abel factor $e^{-ϵ{\nu }_n}$ justifies summation first. The joint bound is locally integrable in $(u,\vartheta )$; it and the large-$u$ decay justify the distributional limit. Combining fractions at $\beta =0$ gives the last formula.

At generic pairs,

Across image boundaries the right-hand side denotes the total joint distribution fixed by the heat kernel.

Proof. Apply Lemma lem:angular inside the four-dimensional heat integral and use

At generic pairs the integrals converge absolutely. The joint local bound gives the tested boundary prescription.

At every $X_0$ with $r_0>0$, choose a sufficiently small Euclidean developing neighbourhood $V_E$ and parallel frame. Then

is jointly real-analytic on $V_E\times V_E$, including its diagonal, Hermitian, and a bisolution of the flat Euclidean Laplacian.

No condition such as $1/(2\alpha )\notin \mathbb{N}$ is required.

Proof. In developing Cartesian coordinates the two terms are fundamental distributions of the same local operator in each variable. Their delta sources cancel. The difference is therefore annihilated, distributionally, by ${\Delta }_X$ and ${\Delta }_{X^{\prime }}$, hence by the elliptic operator ${\Delta }_X+{\Delta }_{X^{\prime }}$ on the product. Elliptic regularity gives smoothness and analytic elliptic regularity gives joint real analyticity MorreyNirenberg,Hormander. Hermiticity follows from the self-adjoint heat kernel.

This treats the total distribution and never differentiates a moving image-sector indicator in isolation.

Near a generic diagonal pair, $H_{F,E}$ is precisely (eq:contour) with the $k=0$ term removed. At a boundary parameter its analytic representative is the one in Theorem thm:analytic.

For $\alpha =1/N$ and $N\beta \in \mathbb{Z}$, every $\sin (\pi {\nu }_n)$ vanishes and the kernel is the full finite character-weighted quotient image kernel. For even $N$, the strict image set at zero angle omits the two boundary representatives of one quotient image. The full quotient sum or an equivalent limiting prescription must be used; simply dropping them is incorrect. For incompatible characters diffraction need not vanish.

Every regular Lorentzian point has a sufficiently small developing neighbourhood on which

where $H_F$ is smooth and locally analytic. It is obtained by continuing the time difference of $H_{F,E}$.

Proof. For positive Euclidean separation the spectral kernel is $e^{-a\sqrt{B_F}}/(2\sqrt{B_F})$. Its boundary value at $a=ϵ+\mathrm{i}(t-t^{\prime })$ is $W_F$, by the spectral theorem and Lemma lem:infrared. The analytic remainder has a holomorphic extension near the regular diagonal. After shrinking the neighbourhood it continues smoothly in the time difference. The direct term gives the displayed boundary value. The local geometry and connection are flat and the field is massless, so the local logarithmic Hadamard coefficient vanishes DecaniniFolacci.

A global Radzikowski characterization on this non-globally-hyperbolic host is not proved Radzikowski. Extension-dependent geometric and diffractive fronts require a separate analysis CheegerTaylor,CheungIBG. A broken path through the completed apex is not a regular null geodesic of $M$. No statement is made at $r=0$.

On regular developing neighbourhoods use local covariant Hadamard point splitting, the usual scaling and smooth mass-dependence conditions, and zero Minkowski-vacuum normalization. The flat connection is an external background.

No state-dependent or holonomy-dependent nonlocal subtraction, and no apex-supported counterterm, is included.

This is the local framework of Wald1977,HollandsWald2001,HollandsWald2005,DecaniniFolacci. Its local formulas are applied on regular charts, not as a globally hyperbolic existence theorem for all of $M$.

Let brackets denote coincidence and let $H$ be the regular complex two-point kernel in a parallel frame. Put

For one real field use $h=\mathrm{Re}H$, and for complex stress use $h=2\mathrm{Re}H$. For a smooth symmetric real bisolution define

Indices are parallel transported before coincidence.

For the charged field,

The two slots carry opposite connection characters.

Under Assumption ass:ren:

1. The Wick-square ambiguity $a{m}^2+bR$, the usual stress ambiguities, and local background-gauge-curvature terms vanish on the regular part. The possible current ambiguity proportional to ${\nabla }^{\nu }{F}_{\nu \mu }$ also vanishes.

2. Local conservation corrections and the regular conformal trace anomaly vanish.

3. For $F_h=[h]$,

   $${\nabla }^{\mu }{T}_{\mu \nu }=0,T^{\mu }{}_{\mu }=\frac{1}{2}(6\xi -1){\square }_g{F}_h,$$
   $$T_{\mu \nu }(\xi )-T_{\mu \nu }(1/6)=\left(\xi -\frac{1}{6}\right)(g_{\mu \nu }{\square }_g-{\nabla }_{\mu }{\nabla }_{\nu })F_h.$$

   For the complex field $F_h=2F$.

4. The current defined by (eq:current-prescription) is real and conserved.

These statements do not classify completion-supported distributions or contact terms in higher products.

Proof. All admissible local geometric ambiguity and anomaly expressions vanish when $m=0$, the Riemann tensor is zero, and the external connection curvature is zero. Holonomy is not a local curvature invariant.

In a parallel Cartesian chart put

Symmetry, the bisolution equations, and Synge's rule give

Thus

Taking a divergence, trace, and coupling difference proves (iii). Hermiticity makes the current real. Applying a divergence to its coincidence expression gives $\mathrm{i}e[({\square }_X-{\square }_{X^{\prime }})H]=0$.

The symmetric real part suffices for the stress tensor but not for the current. If $H$ is replaced by $\mathrm{Re}H$ before taking the antisymmetric first derivative, the azimuthal current is lost.

In the periodic gauge,

The covariant derivatives in (eq:current-prescription) cancel this phase. The prescription is therefore gauge invariant. The parametrix must be transformed with the same parallel factor.

For the azimuthal current computed by time splitting at $\Delta \phi =0$, the direct parametrix has zero azimuthal derivative difference. No extra subtraction appears in that particular calculation.

It is false that the derivative difference of $(4{\pi }^2{{\sigma }_E}^2{)}^{-1}$ vanishes along every splitting. For example a spatial derivative of that singular function is generally nonzero and divergent. Equation (eq:current-prescription), using the regular kernel, is the splitting-independent definition.

Define

For $r=r^{\prime }$, $\Delta \phi =\Delta z=0$, and $\Delta \tau =s$,

Proof. The angular sum is

where

Also

and

Multiply by $(8{\pi }^2{r}^2{)}^{-1}$, subtract $(4{\pi }^2{s}^2{)}^{-1}$, and use $P_{\beta }=6B_2/{\alpha }^2-1$, $C_{\beta }=-30B_4/{\alpha }^4-1$. Theorem thm:analytic identifies these with the unique regular coincidence limits.

For one neutral real scalar,

For one charged complex scalar,

Their traces are respectively

Proof. The Wick squares follow from Lemma lem:expansion. For stress first apply the real differential expression to $\mathrm{Re}{H}_{F,E}$. Reflections, angular evenness of the real part, translations, boosts, and scaling imply

Radial conservation gives $c=-3b$.

For $F=P_{\beta }/(48{\pi }^2{r}^2)$,

The time-split identities give

At conformal coupling the trace vanishes, hence $a=b$. For other couplings use

Take $\beta =0$ for one real field; multiply by two for one complex stress tensor. The trace formulas follow directly.

At conformal coupling the neutral energy density is negative:

At $\alpha =1/2$, the single nontrivial image gives

Then

so $T^t{}_t=1/(96{\pi }^2{r}^4)$. This is an independent sign check.

For one charged complex field,

It is conserved and independent of $\xi$. With the fractional-part convention it is periodic in flux, odd under $\beta \mapsto 1-\beta$, and zero at $0,1/2$.

Proof. Time split at $\Delta \phi =0$. The direct azimuthal term vanishes. Differentiating (eq:modes),

where

Since ${\partial }_{\phi }-{\partial }_{{\phi }^{\prime }}=2{\partial }_{\Delta \phi }$, (eq:current-prescription) gives the result. The other components vanish by the time and longitudinal evenness and radial endpoint symmetry. Conservation follows from $j^r=0$ and angular independence. The remaining properties follow from the polynomial and mode relabeling.

If $1/(2\alpha )\notin \mathbb{N}$, then

For $\alpha >1/2$, the image sum is empty, and on a sufficiently small diagonal neighbourhood the regular remainder is wholly represented by diffraction.