I-B-G: Intrinsic and holonomy-twisted Green operators on a prescribed conical spacetime Self-adjoint extensions, diffraction, finite propagation, and supported transfer from flat-completion laboratories

The regular conical product prescribed in I-B is not globally hyperbolic. Consequently, an intrinsic wave evolution requires an additional choice at the missing apex. We construct retarded and advanced inverses of the Klein–Gordon equation from a specified semibounded self-adjoint spatial extension, including flat complex line bundles with prescribed holonomy. We classify the rotation-diagonal, spatial-product extensions considered here and give their exact positivity criterion, distinguishing transverse bound states from the continuous bands they generate after adding the longitudinal direction.

For the Friedrichs extension we prove two-sided Green identities, an exact sequence on the extension-dependent spectral smooth domain, and finite propagation with respect to the intrinsic spatial distance. We derive a Sommerfeld–Carslaw–Dowker representation consisting of finitely many geometric terms and an apex-diffracted term. The latter begins at the broken-path length through the apex. Angular lift factors, local parallel trivialisations, image boundaries, and the advanced sign are specified explicitly.

A supported transfer theorem identifies the intrinsic scalar operator with the flat-completion operator before the first additional arrival. Its use in the Enter companion sheets also requires their independent retained-shadow and laboratory-margin hypotheses. A finite supplied-report catalogue is constructed without interpreting finite numerical agreement as a distributional identity or as acquisition of a continuum field. The reported numerical discrepancies from the supplied draft are retained separately as unverified numerical data. General non-Friedrichs propagation, non-diagonal extension families, spinor and gauge constraints, uniform flat limits, states, and interacting products are not constructed.

The spacetime companions X-A-ST, X-B-ST, X-C-ST, X-D-ST, V-B, IX-A-ST, IX-B-ST, IX-C-ST, and III-B use a supported developing presentation of the prescribed I-B host CheungIB,CheungXAST,CheungXBST,CheungXCST,CheungXDST, CheungVB,CheungIXAST,CheungIXBST,CheungIXCST,CheungIIIB. Their global free scalar and Green operators belong to a separately declared Minkowski completion. A developing chart does not itself supply an intrinsic conical inverse.

This paper supplies such an inverse after an apex extension has been specified. It does not change the hypotheses or conclusions of those companions. The connection to their supported scalar calculations is a separate theorem (Theorem thm:transfer and Corollary cor:expressions), not a reinterpretation of their original global analytical domains.

The analytical ingredients are classical: self-adjoint spectral theory; Bessel transforms; the Sommerfeld and Carslaw construction; Dowker's cone and flux calculations; and the geometric diffraction framework of Cheeger and Taylor ReedSimonII,Watson,Sommerfeld1896,Carslaw1899,Carslaw1920, Dowker1977,Dowker1987,CheegerTaylor1,CheegerTaylor2. Kay and Studer discuss the apex extension problem for cones and idealised cosmic strings KayStuder; the related radial inverse-square operators are treated explicitly by Dereziński and Richard DerezinskiRichard. These references provide mathematical context, not a derivation of the prescribed Enter geometry.

The Green-operator terminology is consistent with Bär–Ginoux–Pfäffle BGP. Their globally hyperbolic existence theorem is not invoked on the regular cone. Here existence comes from a declared spatial extension, and the exact sequence is proved on a declared spectral domain.

“open” means here. An item marked Open is not resolved in this paper. This is not a claim that the corresponding problem is unknown in the literature. In particular, no assertion about the absence of results in an entire manuscript corpus is needed for any proof.

-identifications. The geometry, the apex condition, and any independent Aharonov–Bohm phase are inputs. The construction gives no Einstein dynamics, spacetime emergence, conical quantum state, Hadamard covariance, time-ordered product, or interacting algebra. The finite reporting interface introduced below does not measure an unknown continuum field.

Fix

The regular spatial cone and the regular spacetime are

The metric completion adds the apex worldsheet

but $\mathcal{W}$ is not part of $M$. The opening angle is $2\pi \alpha =2\pi -\mu$. These are the prescribed background data of I-B CheungIB.

Choose an origin $T_{\Sigma }$ and put

Thus

All propagation times and lengths below are measured in these $t$-units. A time bound in the $T$-coordinate is obtained by division by $c$.

Let

on the analytical completion $\hat{M}={\mathbb{R}}^{1+3}$. On a supported developing region the map

satisfies

Equivalently, if $j={\mathrm{dev}}^{-1}$ is the chart embedding from the developed laboratory to the host, then $j^{\ast }g=-\eta$. This is the signature ledger used in X-A-ST and its successors CheungXAST (:signature-ledger, :host) CheungXBST,CheungXCST,CheungXDST (:host).

We use the scalar equation operator

On the completion it is ${\square }_{\eta }+m^2$. If a companion uses the action Hessian $-P_m$, its inverse is $-G_{\pm }$, not $G_{\pm }$. This distinction preserves the normalization of X-B-ST and XI-B CheungXBST,CheungXIB.

Throughout, $\mu$ is the deficit, not the opening angle. In particular, $\alpha =\mu /(2\pi )$ would describe a different parametrisation. The Bessel orders in this paper are always $|n+\beta |/\alpha$ with $\alpha =1-\mu /(2\pi )$.

For $x=(r,\phi ,z)$ and $x^{\prime }=(r^{\prime },{\phi }^{\prime },z^{\prime })$, let

represent $\alpha (\phi -{\phi }^{\prime })$ modulo $2\pi \alpha$, and write $\Delta z=z-z^{\prime }$. Define

The intrinsic distance of $N$ is

It is realised by a spatial geodesic lying entirely in $N$. Consequently,

For every compact $K\subset M$, ${J_M}^{\pm }(K)$ is closed in $M$.

Proof. First consider the completed two-dimensional cone. It is a proper length space: bounded closed sets are compact in polar coordinates, with all angular coordinates identified at $r=0$. A length-minimising curve therefore exists, by the usual compactness argument for equi-Lipschitz curves.

The developed straight segment with angular separation $\psi$ avoids the origin because $|\psi |\le \pi \alpha <\pi$. It has length

A minimiser cannot pass through the apex, since every such curve has length at least $r+r^{\prime }$. Away from the apex a minimiser develops to a straight segment. Its continuous angular variation has absolute value less than $\pi$ and is congruent to $\psi$ modulo $2\pi \alpha$. The smallest possible absolute variation is $|\psi |$. Since the chord length is nondecreasing in the absolute angle on $[0,\pi ]$, the displayed segment is minimising.

Taking the product with ${\mathbb{R}}_z$ gives (eq:R0): for every spatial curve, the integral triangle inequality bounds its length below by the Euclidean norm of its cone-length contribution and its total $z$-displacement. Equality is obtained by traversing the cone minimiser and the $z$-segment with constant relative speeds.

A future causal curve in the ultrastatic metric satisfies $|\dot{x}{|}_q\le \dot{t}$, hence $t-t^{\prime }\ge d(x,x^{\prime })$. Conversely, a distance-realising spatial curve traversed at speed at most one gives a causal curve whenever this inequality holds. This proves (eq:causal).

Finally, if $p_j\in J_M^{+}(K)$ and $p_j\to p\in M$, choose $k_j\in K$ witnessing the relation. A subsequence has $k_j\to k\in K$; continuity of $d$ preserves the inequality. The past case is identical.

Inside a convex developing chart, a straight developed spatial segment of length $R_0$, together with elapsed time $R_0$, is null for the host metric. This is the local certificate recorded in I-B CheungIB (:local-causal). It does not by itself establish a global formula across a missing worldsheet. Proposition prop:distance is a separate global argument for the specific cone with $0<\alpha <1$; its causal curves never pass through $\mathcal{W}$.

The spacetime $M$ is not globally hyperbolic.

Proof. Fix $r_{\ast }>0$, ${\phi }_{\ast }$, $z_{\ast }$, and $a>r_{\ast }$. The points

have in their causal diamond the sequence

Indeed, the required radial travel length is $r_{\ast }-r_j<a$. The sequence has no convergent subsequence in $M$. Thus this causal diamond is not compact.

This supplies directly the diagnosis used in I-B and the ST chart contracts CheungIB,CheungIXAST,CheungXBST.

Choose lifts of $\phi ,{\phi }^{\prime }$ to $\mathbb{R}$, and set

Unlike $\psi$, $\vartheta$ is not reduced modulo $2\pi \alpha$. Put

The pair is called generic if

These excluded loci are the geometric image boundaries.

If $\vartheta =\psi$, then $0\in \mathcal{K}(\psi )$, and

At the angular cut $|\psi |=\pi \alpha$, two shortest geometric terms can have equal length. Thus it is not correct globally to assert $R_k>R_0$ for every $k\ne 0$. The strict inequality does hold for these additional terms in an angularly small chart defined in Section sec:transfer.

With the displayed developing map,

The inverse gluing generator is therefore

Calling $J_{\mu }$ the positively oriented holonomy, as in the I-B convention, amounts to choosing the corresponding gluing or inverse-meridian orientation CheungIB (:holonomy). Both generators fix $t,z$; they have no translational part. Reversing the chosen generator reverses every holonomy sign below. The distinction is recorded rather than hidden in the notation.

For $\beta \in [0,1)$, represent a section of a flat complex line bundle by a function on the angular cover satisfying

Its norm is taken over one angular period. Denote the resulting Hilbert space by ${\mathcal{H}}_{\beta }={L^2}_{\beta }(N,\mathrm{d}V)$. In this lifted parallel representation the flat covariant derivative is the ordinary derivative.

Let $A_{0,\beta }=-{\Delta }_q$ on compactly supported smooth sections. Angular separation gives

The radial expression on $L^2((0,\infty ),r\mathrm{d}r)$ is

The expression ${\ell }_{\nu }$ is limit circle at zero exactly when $0\le \nu <1$, and is limit point at infinity. The transverse minimal operator has deficiency indices $(d_{\beta },d_{\beta })$, where

In particular:

1. $d_0=1$, with deficient mode $n=0$.

2. For $0<\beta <1$, only $n=0,-1$ can be deficient. They are deficient respectively when $\beta <\alpha$ and $1-\beta <\alpha$.

3. Two deficient modes occur precisely when $1-\alpha <\beta <\alpha$, which requires $\alpha >1/2$.

The full minimal operator on $N$ has infinite deficiency indices if $d_{\beta }>0$, and is essentially self-adjoint if $d_{\beta }=0$.

Proof. At zero the solutions at spectral parameter zero are $r^{\nu },r^{-\nu }$ for $\nu >0$, and $1,\log r$ for $\nu =0$. Both are square integrable with weight $r$ exactly for $\nu <1$. At infinity the expression is limit point, for example after conjugating by $f\mapsto r^{1/2}f$ to the half-line Schrödinger expression

Weyl's alternative gives the radial assertion ReedSimonII,DerezinskiRichard. Counting the angular modes gives the transverse indices.

Fourier transformation in $z$ gives the radial expressions shifted by ${k_z}^2$. When a deficient transverse mode exists, its boundary data may vary over an infinite-dimensional space of longitudinal functions. If no such mode exists, the direct integral of the unique self-adjoint fibre closures is the closure of the full operator.

We consider only extensions of the form

where $B_{\perp }$ is a self-adjoint extension of the transverse minimal operator and commutes with rotations. The transverse boundary parameters are independent of $k_z$.

This restriction is stronger than rotation and $z$-translation invariance alone. Such invariance can also permit measurable $k_z$-dependent boundary conditions. Those extensions, longitudinally nonlocal conditions, and non-product extensions are not classified here.

The rotation characters $e^{\mathrm{i}(n+\beta )\gamma }$ are distinct. Thus $B_{\perp }$ is diagonal in $n$. Fixing a reference length ${\ell }_0>0$, the deficient radial domains have boundary coefficients of the form

The coefficients can equivalently be defined by limiting Wronskians. The corresponding Green boundary form shows that the self-adjoint one-mode conditions are:

• $\nu =0$: the Friedrichs condition $a=0$, or

  $${\mathsf{L}}_L:f(r)=a\log (r/L)+o(1),0<L<\infty ;$$

• $0<\nu <1$: the Friedrichs condition $a=0$, or

  $${\mathsf{E}}_{\lambda }:b=\lambda a,\lambda \in \mathbb{R}.$$

  We also write $\lambda =\infty$ for Friedrichs.

Indeed, for $0<\nu <1$ the boundary form is a nonzero real multiple of ${\overline{a}}_f{b}_g-{\overline{b}}_f{a}_g$, so its one-dimensional self-adjoint subspaces have exactly this real projective parametrisation. The logarithmic case is analogous.

Under Assumption ass:product, the following hold.

1. The Friedrichs extension $A_{F,\beta }$ of $-{\Delta }_q$ is nonnegative. Its form domain is the closure of compactly supported smooth sections in

   $${\Vert u\Vert }^2+{\Vert {\nabla }_{\beta }u\Vert }^2.$$

2. In the logarithmic mode, ${\mathsf{L}}_L$ has exactly one transverse negative eigenvalue

   $$-{{\kappa }_L}^2,{\kappa }_L=\frac{2e^{-{\gamma }_E}}{L},$$

   with radial eigenfunction $K_0({\kappa }_{L}r)$.

3. In a mode $0<\nu <1$, ${\mathsf{E}}_{\lambda }$ has a transverse negative eigenvalue exactly when $\lambda <0$. It is unique in that mode and is $-{\kappa }^2$, where

   $$\lambda =\frac{\Gamma (-\nu )}{\Gamma (\nu )}{\left(\frac{\kappa }{2}\right)}^{2\nu }.$$

   For $\lambda \in [0,\infty ]$ the transverse extension is nonnegative.

4. Every transverse ledger extension has essential spectrum $[0,\infty )$, with only the finitely many negative eigenvalues just described below it. Let

   $${\kappa }_{\ast }=\max (\{0\}\cup \{{\kappa }_j:-{{\kappa }_j}^2\text{ is a transverse negative eigenvalue}\}).$$

   Then the full product operator satisfies

   $$\mathrm{spec}(B)={\mathrm{spec}}_{\mathrm{ess}}(B)=[m^2-{\kappa }_{\ast }^2,\infty ),B\ge 0⟺m\ge {\kappa }_{\ast }.$$

   The full operator has no $L^2(N)$ point spectrum.

Proof. The first assertion is the Friedrichs construction applied to the nonnegative covariant Dirichlet form ReedSimonII.

At a negative radial energy $-{\kappa }^2$, square integrability at infinity forces the solution to be a multiple of $K_{\nu }(\kappa r)$. For $\nu =0$,

if and only if $L=2e^{-{\gamma }_E}/\kappa$. This gives one and only one negative eigenvalue for each finite $L$.

For $0<\nu <1$,

The ratio of the two boundary coefficients is (eq:kappa). Since $\Gamma (-\nu )<0$, its right-hand side is a bijection from $\kappa >0$ onto $\lambda <0$.

The Friedrichs radial spectrum is $[0,\infty )$, as follows from the Bessel spectral transform. Changing finitely many endpoint conditions gives a finite-rank transverse resolvent difference, so it does not change the transverse essential spectrum. The preceding square-integrable solutions account for all negative eigenvalues. There is no transverse zero eigenfunction: the solution permitted at zero cannot also be square integrable at infinity. These facts also follow from the explicit half-line analysis in DerezinskiRichard.

Fourier transformation in $z$ turns $B$ into the direct integral of $B_{\perp }+{k_z}^2+m^2$. Each transverse bound state therefore generates a continuous band $[m^2-{{\kappa }_j}^2,\infty )$, not a full-space bound eigenfunction. The transverse continuum generates $[m^2,\infty )$. Their union is (eq:full-spectrum). For any fixed spectral value, the equation ${k_z}^2+\lambda +m^2=E$ holds on a Lebesgue-null set of $k_z$ for each transverse spectral value $\lambda$; the direct-integral spectral representation therefore has no full-space eigenvectors.

For $m=0,\beta =0$, Friedrichs is the only nonnegative extension in this ledger. For $m>0$, a logarithmic extension is nonnegative precisely when $m\ge {\kappa }_L$. The same mass-shift qualification applies to every $\lambda <0$ power-law mode. It would be incorrect to label all such massive operators “not nonnegative” merely because their transverse extension has a negative eigenvalue.

The geometry does not select $L$, $\lambda$, or the Friedrichs choice. The latter is the finite-energy closure of compactly supported data. Kay–Studer's discussion KayStuder supplies physical context, not an additional premise selecting an extension.

Let $B$ be any extension in the restricted ledger. By Theorem thm:positivity, $B$ is bounded below. Choose $b>0$ so that $B+b\ge 1$, and define

This is a Fréchet space; its topology is independent of the admissible shift $b$. Put

The first space has its usual inductive-limit topology over compact time supports; it is not asserted to be a Fréchet space.

For real $\lambda$ set

The spectral multiplier is bounded for each fixed $t$ because $B$ is bounded below. Functional calculus, rather than convergence of an operator power series on arbitrary smooth vectors, is the definition.

For $f\in {{\mathcal{D}}_{\mathrm{tc}}}_B$, define

Thus, in time-convolution notation,

The minus sign in the first advanced integral is essential.

For every restricted ledger extension $B$:

1. Compactly supported smooth spacetime sections belong continuously to ${{\mathcal{D}}_{\mathrm{tc}}}_B$, and

   $${G_{\pm }}^B:{{\mathcal{D}}_{\mathrm{tc}}}_B⟶{\mathcal{E}}_B$$

   are continuous.

2. With $P_B={{\partial }_t}^2+B$,

   $$P_B{G}_{\pm }^{B}f={G_{\pm }}^B{P}_{B}f=f(f\in {{\mathcal{D}}_{\mathrm{tc}}}_B).$$

   On compactly supported smooth sections, $P_B=P_m$.

3. $G_{+}^{B}f$ vanishes before the time support of $f$, and $G_{-}^{B}f$ vanishes after it.

4. The sequence

   $$0⟶{{\mathcal{D}}_{\mathrm{tc}}}_B\stackrel{P_B}{\to }{{\mathcal{D}}_{\mathrm{tc}}}_B\stackrel{E_B}{\to }{\mathcal{E}}_B\stackrel{P_B}{\to }{\mathcal{E}}_B⟶0$$

   is exact.

5. In the Hermitian spacetime pairing,

   $$⟨g,G_{+}^{B}f⟩=⟨G_{-}^{B}g,f⟩,⟨g,E_{B}f⟩=-⟨E_{B}g,f⟩(f,g\in {{\mathcal{D}}_{\mathrm{tc}}}_B).$$

   These are formal adjoint identities on the stated test domain, not assertions of self-adjointness or skew-adjointness of unbounded spacetime operators.

6. Every ${G_{\pm }}^{B}f$ is smooth on the regular spacetime $M$.

Proof. The spatial differential expression maps compactly supported smooth sections to themselves. Every such section therefore lies in every power domain of $B$. The same statement with compact-support seminorm estimates proves the asserted continuous inclusion.

On compact time intervals the multipliers $w_t$ and ${\partial }_t{w}_t$ are bounded uniformly on the spectrum; higher time derivatives are controlled by finitely many additional graph seminorms. Differentiation under the integral in (eq:Green) consequently gives

on ${{\mathcal{H}}^{\infty }}_B$. Duhamel differentiation proves the right inverse identities. Integration by parts in $s$, using compact time support, proves the left inverse identities. The support-in-time assertions are immediate.

For completeness, the homogeneous Cauchy problem is unique in ${\mathcal{E}}_B$: applying bounded spectral projections of $B$ reduces it to a bounded-operator Cauchy problem; the projections tend strongly to the identity. Its solution with data $u_0,u_1\in {{\mathcal{H}}^{\infty }}_B$ is

If $P_{B}u=0$ and $u$ has compact time support, then $u=G_{+}^B{P}_{B}u=0$, so the first arrow in (eq:exact) is injective. The inverse identities give $E_B{P}_B=0$ and $P_B{E}_B=0$. If $E_{B}f=0$, then $u=G_{+}^{B}f=G_{-}^{B}f$ has time support in the compact interval spanning ${\mathrm{supp}}_{t}f$, and $P_{B}u=f$. This proves exactness at the second space.

If $P_{B}u=0$, choose $\chi \in C^{\infty }(\mathbb{R})$ equal to zero for $t<0$ and one for $t>1$, and set

Cauchy uniqueness gives

hence $E_{B}f=u$. Finally, for arbitrary $v\in {\mathcal{E}}_B$,

is a smooth spectral-domain solution of $P_{B}u=v$. This proves the last surjectivity.

Since $W_B(t{)}^{\ast }=W_B(t)$ and $W_B(-t)=-W_B(t)$, Fubini's theorem on the compact time supports of the paired tests gives the adjoint identities.

Interior elliptic regularity gives continuous local embeddings

Together with the spectral time regularity, these imply smoothness on $M$.

The exact sequence (eq:exact) depends on $B$. It is not an exactness theorem on the usual compact-spacetime test space. On that space we have proved only

The reverse inclusion is not established here. A time-compact spectral solution need not have support bounded away from the removed worldsheet. This distinction must not be erased when comparing with Minkowski equation quotients or free-field algebras.

Let $B=A_{F,\beta }+m^2$. For every compactly supported smooth spacetime section $f$,

The causal relation is the intrinsic relation (eq:causal), not an auxiliary completion relation.

Proof. We prove the domain-of-dependence statement for the homogeneous equation and then use Duhamel's formula.

Let $u(0)=u_0,\dot{u}(0)=u_1$, with smooth spectral data vanishing on $B_d(x_0,\varrho )$. For $t\ge 0$, choose a smooth nonincreasing ${\varphi }_{ϵ}$, equal to one on $(-\infty ,-ϵ]$ and zero on $[0,\infty )$, and put

The distance is Lipschitz and $|\nabla d|\le 1$ almost everywhere.

Multiplication by bounded Lipschitz functions with bounded gradient preserves the Friedrichs form domain $Q_F$. Indeed, approximate a form-domain section by compactly supported smooth sections, apply the Leibniz estimate, and smooth the multiplier on the compact regular supports. Thus all integrations by parts below are identities in the closed form, including when the support of ${\chi }_t$ approaches the apex.

Use the augmented local energy

The extra $|u{|}^2$ term also handles the massless case. The form identity for $B$ yields

Since ${\varphi }_{ϵ}^{\prime }\le 0$,

Also $2|u\dot{u}|\le |u{|}^2+|\dot{u}{|}^2\le e$. Hence

Gronwall's inequality gives $\mathcal{E}(t)=0$. Letting $ϵ\downarrow 0$ proves that $u(t)$ vanishes on $B_d(x_0,\varrho -t)$. Time reversal gives the negative-time statement.

It follows that $W_B(t)$ propagates spatial support by at most $|t|$. Duhamel's formula then gives (eq:finite-speed); closedness of ${J_M}^{\pm }(\mathrm{supp}f)$ was proved in Proposition prop:distance.

The argument uses the closed Friedrichs form rather than an integration over $r>ϵ$ followed by an unjustified boundary limit. It does not prove finite propagation for the general ${\mathsf{L}}_L$ or ${\mathsf{E}}_{\lambda }$ domains. The special ${\mathsf{E}}_0$ case is treated separately below. The result is consistent with the general conic diffraction theory of Cheeger–Taylor CheegerTaylor1,CheegerTaylor2.

In this section $B=A_{F,\beta }+m^2$, and all spatial kernels are relative to $\mathrm{d}V$.

For $u>0$, define

This series converges absolutely.

Set

and

Then

At a generic angle this extends continuously to $u=0$. For $\beta =0$,

The function ${\Phi }_{\beta }$ is jointly locally integrable in $(u,\vartheta )$ up to $u=0$, including image boundaries, and decays exponentially as $u\to \infty$.

Proof. Split the series into $n\ge 0$ and $n\le -1$, and write $\sin x=(e^{\mathrm{i}x}-e^{-\mathrm{i}x})/(2\mathrm{i})$. The resulting two geometric sums give (eq:Phi-closed); the $n=0,\beta =0$ contribution is zero. Combining conjugate geometric denominators gives (eq:Carslaw).

The possible boundary poles are precisely $\vartheta \pm \pi \in 2\pi \alpha \mathbb{Z}$. Near each such angle ${\vartheta }_j$, the closed form is bounded by a constant times

This is locally integrable in the two variables. Away from those angles, all denominators remain nonzero at $u=0$.

For $0<\beta <1$, exponential decay follows with rate at least $\min (\beta ,1-\beta )/\alpha$. For $\beta =0$, the zero mode does not contribute, and a rate $1/\alpha$ suffices.

The untwisted expression is the Carslaw weight; the twisted version is the corresponding cone-flux construction Carslaw1899,Carslaw1920,Dowker1977, Dowker1987.

We do not assert a master formula for arbitrary bounded Borel functions. The following theorem concerns the heat semigroup, the positive resolvent, and the retarded wave kernel, with the latter interpreted distributionally.

For any radial flat kernel $F(R)$ for which the following expression is defined, write

The following identities hold.

1. The massless spatial heat kernel is

   $$\begin{array}{rl}{H}_{\beta }(\tau ;x,x^{\prime })=& \frac{e^{-\Delta z^2/(4\tau )}}{\sqrt{4\pi \tau }}\frac{e^{-(r^2+r^{\prime 2})/(4\tau )}}{4\pi \alpha \tau }\\ & \times \sum _{n\in \mathbb{Z}}{e}^{\mathrm{i}(n+\beta )(\phi -{\phi }^{\prime })}{I}_{{\nu }_n}\left(\frac{r{r}^{\prime }}{2\tau }\right),\tau >0.\end{array}$$

   At generic pairs, the heat kernel of $B$ equals \[ S_[ (4)^-3/2e^{-m^2-R^2/(4)} ].

2. For $s>0$, the kernel of $(B+s^2{)}^{-1}$ is \[ S_[ {e^{-^2+m^2\,R}}{4 R} ].

3. Let $F_m^{+}(t,R)$ be the flat retarded kernel of ${{\partial }_t}^2-{\Delta }_{{\mathbb{R}}^3}+m^2$. For $R>0$,

   $$F_m^{+}(t,R)=\frac{\delta (t-R)}{4\pi R}-\theta (t-R)\frac{m{J}_1(m\sqrt{t^2-R^2})}{4\pi \sqrt{t^2-R^2}},$$

   where the second term is zero at $m=0$, and its quotient has its continuous value at the front. The intrinsic retarded kernel is

   $$G_{+}(t;x,x^{\prime })={\mathcal{S}}_{\beta }[F_m^{+}(t,\cdot )].$$

   For $m=0$, at a generic pair,

   $$G_{+}(t;x,x^{\prime })=\sum _{k\in \mathcal{K}(\vartheta )}{e}^{2\pi \mathrm{i}\beta k}\frac{\delta (t-R_k)}{4\pi R_k}-\theta (t-{\rho }_0)\frac{{\Phi }_{\beta }(u_0,\vartheta )}{4{\pi }^2\alpha r{r}^{\prime }\sinh u_0},$$

   with

   $$\cosh u_0=\frac{t^2-r^2-r^{\prime 2}-\Delta z^2}{2r{r}^{\prime }}.$$

   The last term means its locally integrable one-sided extension at $t={\rho }_0$.

4. The formulas are globally bundle-valued distribution identities with the joint-integration prescription in the proof. No arbitrary pointwise value at an image boundary is required. The advanced kernel is

   $$G_{-}(t;x,x^{\prime })=G_{+}(-t;x,x^{\prime }).$$

Proof. Step 1: the mode heat kernel. The Friedrichs radial operator is diagonalised by the Bessel transform with kernel $J_{\nu }(kr)$. Weber's exponential integral gives

Together with the angular normalization and the one-dimensional longitudinal heat kernel, this proves (eq:modes). We use the classical integral in Watson; equivalent Bessel identities are recorded in DLMF.

Step 2: the angular summation. For $\zeta >0$, set

Use the integral representation

valid for the present parameters Watson,DLMF. The first integral is the $n$-th twisted Fourier coefficient of the piecewise smooth function

Indeed, integrating over one angular period and changing variables in each summand gives the coefficient

Its imaginary part vanishes, and its real part is the first Bessel integral.

Insert the Abel factor $e^{-ϵ{\nu }_n}$. The first Fourier series tends to $h_{\zeta }$ at every generic angle and in local angular $L^1$. The second becomes

Lemma lem:Phi supplies local joint integrability, uniformly for small $ϵ$; at infinity the factor $e^{-\zeta \cosh u}$ dominates. Hence

Multiplying by the remaining factors in (eq:modes), with $\zeta =r{r}^{\prime }/(2\tau )$, proves the heat identity.

Step 3: the resolvent. Since $B\ge 0$,

The flat heat integral is $e^{-\sqrt{s^2+m^2}R}/(4\pi R)$. For generic pairs the diffraction weight is bounded at $u=0$, and the exponential in $\rho (u)$ controls infinity. After compact spatial smearing, the joint bound in Lemma lem:Phi justifies the same interchanges across image boundaries. This proves the resolvent formula.

Step 4: the global wave distribution. Define the right-hand side of (eq:wave-master) first by joint testing in time and the spatial pair. The geometric terms are the usual flat wave distributions on the indicated angular sectors, with their locally integrable sector indicators.

For the diffracted term, compact time support bounds $u$: $F_m^{+}(t,\rho (u))$ vanishes unless $\rho (u)$ is no larger than the largest positive time in the test support. On compact regular spatial sets, $\rho (u)\ge r+r^{\prime }>0$. After time testing,

is a bounded smooth function of $\rho >0$ on the relevant compact range. Lemma lem:Phi therefore makes the joint $u$-angular integration locally integrable, including at $u=0$ and an image boundary. This gives an unambiguous global distribution.

Its Laplace transform is the resolvent formula from Step 3, because

On the spectral side,

Both sides are causal distributions of at most polynomial growth after compact spatial smearing. Injectivity of the Laplace transform therefore proves (eq:wave-master) globally.

For $m=0$, integrating the delta in $u$ uses

This gives (eq:massless). At a generic pair, ${\Phi }_{\beta }$ is bounded at zero and the resulting edge is of order $(t-{\rho }_0{)}^{-1/2}$. Finally, the advanced identity follows from Definition def:Green and oddness of $W_B$.

The extension across image boundaries in the proof is a joint-distribution construction. It is not justified merely by saying that the exceptional angular pairs have measure zero: a distribution can be supported on such a set. The heat and resolvent identities, together with Laplace uniqueness, fix the global distribution.

The kernel is a section of $\mathrm{Hom}(L_{\beta ,x^{\prime }},L_{\beta ,x})$. For example,

If

then every formula reduced to $\psi$ acquires the overall factor

The same identity holds for ${\Phi }_{\beta }$. Omitting this factor would destroy bundle covariance.

On a simply connected chart, the lifted representation is a parallel trivialisation; the differential operator is the ordinary scalar Klein–Gordon operator. By contrast, the periodic gauge

has covariant derivative ${\partial }_{\phi }+\mathrm{i}\beta$, and its kernel is

In that gauge the direct term has the corresponding endpoint phase. It is not the ordinary flat kernel with unit direct weight. All flat comparisons below use a local parallel trivialisation.