Green-Hyperbolic Operators and Causal Propagators

For a normally hyperbolic field equation on a globally hyperbolic spacetime, compactly supported sources have unique future-supported and past-supported solutions. These are the retarded and advanced Green operators. Their difference is not another inverse: it is a homogeneous causal propagator whose image is exactly the space of spacelike-compact solutions. For the broader class of Green-hyperbolic operators, these inverse identities and support properties are the definition, and requiring the formally dual operator as well is what makes uniqueness, duality, and the exact solution sequence available.

This page establishes that theorem-level structure, constructs the propagators for an ultrastatic Klein–Gordon field, and shows precisely what fails when an unprescribed timelike boundary replaces global hyperbolicity. It does not use a Green operator to choose a quantum state: positivity and the symmetric part of a two-point function remain separate data.

Required background. Globally hyperbolic spacetimes and the Loc categories supplies Cauchy surfaces and causal futures; microlocal calculus for quantum fields supplies distributional kernels and their singular directions.

Helpful background. Time-slice axiom and relative Cauchy evolution uses the quotient construction below; isotony, additivity, duality, and primitive causality gives its algebraic setting; Green operators, causal propagators, and state-dependent two-point functions develops the physical curved-spacetime application.

The chapter overview supplies three complementary guides: the dependency map places Green hyperbolicity before propagation of singularities, the hypothesis–conclusion table separates inverse identities from support and state assumptions, and the failure map identifies unprescribed boundary data as the relevant obstruction.

For a direct route through this page, begin with the support-selected inverses, then read the uniqueness theorem and exact sequence. The ultrastatic construction supplies the main calculation, while the timelike-boundary example tests the hypothesis.

Support-selected inverses

Let $(M,g)$ be a time-oriented globally hyperbolic spacetime and let $E_1,E_2\to M$ be finite-rank vector bundles. Write $\Gamma^\infty(M,E)$ for smooth sections and $\Gamma_c^\infty(M,E)$ for compactly supported smooth sections. For $S\subset M$, the sets $J^+(S)$ and $J^-(S)$ are its causal future and causal past, including $S$ itself. Consider a linear differential operator

$$P:\Gamma^\infty(M,E_1) \longrightarrow \Gamma^\infty(M,E_2).$$

A retarded Green operator is a linear map

$$G_{\mathrm{ret}}: \Gamma_c^\infty(M,E_2) \longrightarrow \Gamma^\infty(M,E_1)$$

such that, for $f\in\Gamma_c^\infty(M,E_2)$ and $h\in\Gamma_c^\infty(M,E_1)$,

$$\begin{aligned} P G_{\mathrm{ret}}f&=f, &G_{\mathrm{ret}}Ph&=h,\\ \operatorname{supp}(G_{\mathrm{ret}}f) &\subset J^+(\operatorname{supp}f). \end{aligned}$$

The first identity says that the output solves the sourced equation. The second says that the map is also a left inverse on compactly supported inputs; it rules out adding an arbitrary homogeneous solution. The support condition says that the source-dependent response occurs only in the causal future of the source. An advanced Green operator $G_{\mathrm{adv}}$ obeys the same inverse identities with

$$\operatorname{supp}(G_{\mathrm{adv}}f) \subset J^-(\operatorname{supp}f).$$

Thus “retarded” means future-supported response and “advanced” means past-supported response throughout QFT.org. Bare symbols $G_+$ and $G_-$ are unsafe because the literature attaches the words differently: Bär calls his future-supported $G_+$ advanced and his past-supported $G_-$ retarded. In this page’s notation,

$$G_{\mathrm{ret}}=G_+^{\text{Bär}}, \qquad G_{\mathrm{adv}}=G_-^{\text{Bär}}.$$

The definitions and the source convention appear in Bär 2015, Definitions 3.1–3.2 (Open PDF).

To state Green hyperbolicity, use the natural bundle-dual pairing and the spacetime volume density. The formally dual operator

$$P^\dagger: \Gamma^\infty(M,E_2^*) \longrightarrow \Gamma^\infty(M,E_1^*)$$

is characterized by

$$\int_M\langle\psi,P\phi\rangle\,\mathrm d\mathrm{vol}_g = \int_M\langle P^\dagger\psi,\phi\rangle\,\mathrm d\mathrm{vol}_g$$

whenever the two supports have compact intersection. The operator $P$ is Green hyperbolic when both $P$ and $P^\dagger$ possess retarded and advanced Green operators. In the special case $E_1=E_2=E$, if a nondegenerate symmetric or Hermitian bundle metric identifies $E$ with $E^*$ and $P$ is formally self-adjoint, it is enough to construct the two maps for $P$ itself.

Existence, uniqueness, and examples

Green hyperbolicity is a property of the full operator on the specified global spacetime, not a label read from one local coefficient. The fundamental existence theorem supplies the main family of examples: every normally hyperbolic operator on a globally hyperbolic spacetime has unique retarded and advanced Green operators Bär, Ginoux, and Pfäffle 2007, Corollary 3.4.3 (Open PDF). In the site’s mostly-minus convention, a normally hyperbolic operator on one bundle has the local form

$$P=\Box_g+A^a\nabla_a+B, \qquad \Box_g=\nabla_a\nabla^a,$$

so its principal part is the metric wave operator times the bundle identity. Klein–Gordon operators, connection wave operators, and the wave operator on differential forms belong to this class.

There are important Green-hyperbolic operators that are not normally hyperbolic. A broad first-order sufficient condition is prenormal hyperbolicity. First-order operators $P,Q:\Gamma^\infty(M,E)\to\Gamma^\infty(M,E)$ form a complementary pair when $PQ$ is normally hyperbolic; then $QP$ is normally hyperbolic as well. If $H_{\mathrm{ret/adv}}$ are the Green operators of $PQ$, then

$$G^P_{\mathrm{ret/adv}} =QH_{\mathrm{ret/adv}}$$

are the unique Green operators of $P$. For example, if $D$ is a Dirac operator and $A$ is a smooth bundle endomorphism, take $P=D+A$ and $Q=D-A$. In the (+---) convention,

$$(D+A)(D-A) = \Box+\frac{\mathrm{scal}_g}{4}-[D,A]-A^2,$$

whose second-order principal part is normally hyperbolic. This construction, including both inverse identities, is proved in Mühlhoff 2011, Definition 1, Lemma 2, Example 1, and Theorem 1.

The massive Proca operator $\delta\mathrm d+m^2$ is another Green-hyperbolic but non-normally-hyperbolic example: it is related by a differential intertwiner to the normally hyperbolic operator $\mathrm d\delta+\delta\mathrm d+m^2$. The square-root and Proca constructions are given in Bär 2015, Examples 3.3, 3.5, and 3.16 and Corollary 3.15 (Open PDF).

For normally hyperbolic operators, global Cauchy well-posedness, energy estimates, and finite propagation prove existence and uniqueness. For a general Green-hyperbolic operator, however, one should not silently import that PDE proof: Green hyperbolicity need not be recognizable from the principal symbol and need not be equivalent to a well-posed Cauchy problem. Its general uniqueness theorem instead uses the Green maps of the formally dual operator and causal support duality.

The support language makes that argument transparent. A closed set is past compact if its intersection with $J^-(x)$ is compact for every $x\in M$; equivalently, it lies to the future of some Cauchy surface. Future compact is the time-reversed condition. The two Green maps extend uniquely to full inverses

$$\begin{aligned} G_{\mathrm{ret}}&: \Gamma_{\mathrm{pc}}^\infty(M,E_2) \longrightarrow \Gamma_{\mathrm{pc}}^\infty(M,E_1),\\ G_{\mathrm{adv}}&: \Gamma_{\mathrm{fc}}^\infty(M,E_2) \longrightarrow \Gamma_{\mathrm{fc}}^\infty(M,E_1). \end{aligned}$$

Consequently there is no nonzero homogeneous solution with past-compact or future-compact support, and the compact-input Green operators are unique. See Bär 2015, Theorem 3.8 and Corollaries 3.9–3.12 (Open PDF).

The causal propagator and the exact sequence

Define the causal propagator using the site-wide sign

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}.$$

It is not an inverse of $P$. The two inverse identities cancel:

$$PEf=0, \qquad EPh=0.$$

Its support is nevertheless controlled,

$$\operatorname{supp}(Ef) \subset J(\operatorname{supp}f) \equiv J^+(\operatorname{supp}f) \cup J^-(\operatorname{supp}f).$$

A section is spacelike compact when its support is contained in $J(K)$ for some compact $K\subset M$. Equivalently on a globally hyperbolic spacetime, its support meets every Cauchy surface in a compact set. Hence

$$E:\Gamma_c^\infty(M,E_2) \longrightarrow \Gamma_{\mathrm{sc}}^\infty(M,E_1).$$

The central theorem is the full exact sequence

$$\begin{aligned} 0&\longrightarrow \Gamma_c^\infty(M,E_1) \xrightarrow{P}\Gamma_c^\infty(M,E_2) \xrightarrow{E}\Gamma_{\mathrm{sc}}^\infty(M,E_1)\\ &\xrightarrow{P}\Gamma_{\mathrm{sc}}^\infty(M,E_2) \longrightarrow 0. \end{aligned}$$

The last zero is substantive: $P$ maps onto every spacelike-compact source. Exactness makes four precise statements.

1. No compactly supported homogeneous solution. If $Ph=0$ and $h$ is compactly supported, then $h=G_{\mathrm{ret}}Ph=0$.

2. The kernel of $E$ consists exactly of equations of motion. If $Ef=0$, then

   $$k=G_{\mathrm{ret}}f=G_{\mathrm{adv}}f$$

   has support in the compact causal diamond

   $$J^+(\operatorname{supp}f) \cap J^-(\operatorname{supp}f).$$

   Thus $k\in\Gamma_c^\infty(M,E_1)$ and $Pk=f$. The reverse inclusion follows from $EP=0$.

3. Every spacelike-compact homogeneous solution comes from a compact source. Let $Pu=0$ and $\operatorname{supp}u\subset J(K)$. Split $u=u_-+u_+$ so that $u_-$ is supported to the past of a compact time slab and $u_+$ to its future. Then

   $$f=Pu_+=-Pu_-$$

   is compactly supported. Uniqueness in the past- and future-compact classes gives

   $$G_{\mathrm{ret}}f=u_+, \qquad G_{\mathrm{adv}}f=-u_-, \qquad Ef=u.$$

4. Every spacelike-compact source has a spacelike-compact solution. Let $q\in\Gamma_{\mathrm{sc}}^\infty(M,E_2)$ and choose a temporal cutoff $\chi$ adapted to a compact $K$ with $\operatorname{supp}q\subset J(K)$, with $\chi=0$ in the far past and $\chi=1$ in the far future. Split

   $$q_{\mathrm{pc}}=\chi q, \qquad q_{\mathrm{fc}}=(1-\chi)q, \qquad q=q_{\mathrm{pc}}+q_{\mathrm{fc}},$$

   where the first term is strictly past compact, meaning that its support lies in $J^+(K_-)$ for some compact $K_-$, and the second is strictly future compact, with support in $J^-(K_+)$ for some compact $K_+$. The extended Green maps give

   $$u =G_{\mathrm{ret}}q_{\mathrm{pc}} +G_{\mathrm{adv}}q_{\mathrm{fc}} \in\Gamma_{\mathrm{sc}}^\infty(M,E_1), \qquad Pu=q.$$

   Indeed, causal support puts the first summand in $J^+(K_-)$ and the second in $J^-(K_+)$. Their union lies in $J(K_-\cup K_+)$, so $u$ is spacelike compact. This is where both the cutoff and the support estimates enter.

The first three statements are the content and proof mechanism of Bär 2015, Theorem 3.22 (Open PDF); the full sequence, including terminal surjectivity, is stated and proved in Bär 2017, Theorem 3.6.21, equation (3.9), pp. 138–139 (Open PDF). In particular, $E$ induces a linear isomorphism

$$\frac{\Gamma_c^\infty(M,E_2)} {P\Gamma_c^\infty(M,E_1)} \xrightarrow{\ \cong\ } \operatorname{Sol}_{\mathrm{sc}}(P), \qquad [f]\longmapsto Ef,$$

where $\operatorname{Sol}_{\mathrm{sc}}(P)$ denotes smooth spacelike-compact solutions of $Pu=0$.

An ultrastatic Klein–Gordon construction

Take

$$M=\mathbb R\times\Sigma, \qquad g=\mathrm dt^2-h,$$

where $(\Sigma,h)$ is a compact Riemannian manifold without boundary. For a minimally coupled scalar of mass $m>0$,

$$P=\Box_g+m^2=\partial_t^2+A, \qquad A=-\Delta_h+m^2.$$

Here $\Delta_h=\operatorname{div}_h\operatorname{grad}_h$ is nonpositive. More precisely, $A$ denotes the positive self-adjoint closure of $-\Delta_h+m^2$, initially defined on $C^\infty(\Sigma)$, with domain $D(A)=H^2(\Sigma)$. Since $m>0$, the functional-calculus operator

$$S_A(\tau)=A^{-1/2}\sin\!\bigl(\tau A^{1/2}\bigr)$$

is bounded. Compactness gives an orthonormal eigenbasis

$$A e_j=\omega_j^2 e_j, \qquad \omega_j>0.$$

For $f\in C_c^\infty(\mathbb R\times\Sigma)$, write

$$f_j(t)=\int_\Sigma \overline{e_j(x)}f(t,x)\,\mathrm d\mathrm{vol}_h(x).$$

Mode-by-mode Duhamel evolution gives

$$\begin{aligned} (G_{\mathrm{ret}}f)(t,x) &= \sum_j e_j(x) \int_{-\infty}^{t} \frac{\sin\!\bigl(\omega_j(t-s)\bigr)}{\omega_j} f_j(s)\,\mathrm ds,\\ (G_{\mathrm{adv}}f)(t,x) &= -\sum_j e_j(x) \int_{t}^{\infty} \frac{\sin\!\bigl(\omega_j(t-s)\bigr)}{\omega_j} f_j(s)\,\mathrm ds. \end{aligned}$$

Smoothness of $f$ and elliptic estimates make its mode coefficients decrease faster than any power of $\omega_j$, so both series converge with all derivatives. The sign of the advanced formula is essential. For one mode, the retarded kernel is

$$S_{\mathrm{ret},j}(t-s) = \theta(t-s) \frac{\sin\!\bigl(\omega_j(t-s)\bigr)}{\omega_j}.$$

It is continuous at $t=s$, while its first derivative jumps from zero to one. Therefore

$$(\partial_t^2+\omega_j^2)S_{\mathrm{ret},j}(t-s) =\delta(t-s),$$

which verifies $PG_{\mathrm{ret}}f=f$ and fixes the normalization. Integrating by parts in $s$ for a compactly supported test section verifies the other identity $G_{\mathrm{ret}}Ph=h$; the advanced check is identical.

The mode formulas show the time ordering, but each eigenfunction extends across all of $\Sigma$, so no individual mode displays spatial causality. Finite propagation for the normally hyperbolic operator supplies the stronger cancellation statement

$$\operatorname{supp}(G_{\mathrm{ret/adv}}f) \subset J^\pm(\operatorname{supp}f).$$

This distinction prevents a common circular argument: spectral calculus constructs and normalizes the maps, while the hyperbolic energy estimate proves their sharp spacetime support. The general existence, causal support, and continuity results are Bär, Ginoux, and Pfäffle 2007, Theorems 3.3.1 and 3.4.7 and Proposition 3.4.8 (Open PDF).

Subtracting the two maps removes the step functions:

$$(Ef)(t,x) = \sum_j e_j(x) \int_{-\infty}^{\infty} \frac{\sin\!\bigl(\omega_j(t-s)\bigr)}{\omega_j} f_j(s)\,\mathrm ds.$$

Because $\Sigma$ itself is compact, $M=J(\Sigma)$; consequently every smooth solution has spacelike-compact support. The exact sequence therefore identifies the full smooth Klein–Gordon solution space in this ultrastatic example, not merely a proper support-restricted subspace.

From the exact sequence to the free-field phase space

Now specialize to a real bundle $E$ with a nondegenerate symmetric fiber pairing and a formally self-adjoint $P:\Gamma(E)\to\Gamma(E)$. The adjoint relation between the Green maps is

$$G_{\mathrm{ret}}^\dagger=G_{\mathrm{adv}}, \qquad E^\dagger=-E.$$

Hence

$$\sigma([f],[h]) = \int_M\langle f,Eh\rangle\,\mathrm d\mathrm{vol}_g$$

defines an antisymmetric form on $\Gamma_c^\infty(M,E)/P\Gamma_c^\infty(M,E)$. It is well defined because changing either representative by an element in the image of $P$ gives zero after formal integration by parts and $PE=EP=0$. It is nondegenerate: if $\sigma([f],[h])=0$ for every $h$, then $Ef=0$ as a distribution, so exactness implies $f=Pk$ with compactly supported $k$ and therefore $[f]=0$.

For the real Klein–Gordon field, this abstract form has a concrete Cauchy-surface expression. With future unit normal $n$ and the site convention

$$\Omega_\Sigma(\phi,\psi) = \int_\Sigma \left( \phi\nabla_n\psi-\psi\nabla_n\phi \right)\mathrm d\Sigma,$$

one has

$$\sigma([f],[h]) =-\Omega_\Sigma(Ef,Eh).$$

The minus sign comes from $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$; reversing the definition of $E$ reverses it. This identification follows from Green’s identity and is proved in Bär, Ginoux, and Pfäffle 2007, Lemma 4.7.7, p. 150 (Open PDF). A general Green-hyperbolic operator need not possess this particular scalar current or even a canonical Cauchy formulation.

This is the classical phase space underlying the free bosonic CCR algebra. With the site convention for $E$, the smeared-field commutator can be written in either equivalent form,

$$\begin{aligned} [\Phi(f),\Phi(h)] &=-i\,\sigma([f],[h])\,\mathbf 1\\ &=i\,\Omega_\Sigma(Ef,Eh)\,\mathbf 1 \qquad\text{(Klein–Gordon).} \end{aligned}$$

This fixes the antisymmetric part of a two-point function. It does not supply a positive symmetric part, a complex structure, a vacuum, or a Hadamard state. Those are genuinely additional choices and conditions, developed in Green operators, causal propagators, and state-dependent two-point functions.

The same support mechanism explains the free-field time-slice property. Let $O$ be an open neighborhood of a Cauchy surface. For compactly supported $f$, choose a smooth $\chi$ that changes from zero to one so that its transition inside $J(\operatorname{supp}f)$ lies in $O$, and set

$$k = \chi G_{\mathrm{adv}}f + (1-\chi)G_{\mathrm{ret}}f.$$

Causal support and global hyperbolicity make $k$ compactly supported: only the intersection of the transition region with $J(\operatorname{supp}f)$ must be compact, not an entire slab over a noncompact Cauchy surface. Outside that transition region, $Pk=f$, so

$$f_O=f-Pk$$

is supported in $O$ and represents the same quotient class as $f$. Thus sources in an arbitrarily thin Cauchy neighborhood generate the same free solution algebra. This argument requires the two inverse identities as well as causal support; finite propagation by itself is not enough.

What a timelike boundary changes

The simplest explicit failure test is the massless wave operator on the Minkowski half-space

$$M_+=\{(t,x):x\ge0\}, \qquad P=\partial_t^2-\partial_x^2.$$

On full $1+1$ dimensional Minkowski space, the retarded kernel is

$$R_0(\tau,\xi) = \frac12\theta(\tau)\theta(\tau^2-\xi^2), \qquad P R_0=\delta(\tau)\delta(\xi).$$

For a source point $x'>0$, the method of images gives two different retarded kernels on $M_+$. Acting on $C_c^\infty(M_+^\circ)$, where $M_+^\circ=\{x>0\}$ is the interior, they are

$$\begin{aligned} R_D(t,x;t',x') &=R_0(t-t',x-x')-R_0(t-t',x+x'),\\ R_N(t,x;t',x') &=R_0(t-t',x-x')+R_0(t-t',x+x'). \end{aligned}$$

The image source lies at $-x'$, outside the interior, so both kernels satisfy the same sourced wave equation for $x>0$ and both are future supported. At the boundary,

$$R_D\rvert_{x=0}=0, \qquad \partial_xR_N\rvert_{x=0}=0.$$

They are therefore the Dirichlet and Neumann retarded propagators. Odd and even extension across $x=0$ verifies both inverse identities on interior test sources, and time reversal gives the corresponding advanced pair. Their difference,

$$R_N-R_D=2R_0(t-t',x+x'),$$

is a nonzero homogeneous reflected wave in the interior. The differential expression and interior source alone do not choose between them.

This is the precise obstruction hidden by the phrase “remove global hyperbolicity.” A timelike boundary requires an operator domain or boundary condition; different admissible choices can define different dynamics and different causal propagators. On a standard static spacetime with timelike boundary, an admissible self-adjoint realization $A_\Theta$ of the spatial elliptic operator whose spectrum is bounded below does recover unique advanced and retarded solutions for that chosen boundary condition Dappiaggi, Drago, and Ferreira 2019, Theorem 30 and equations (21)–(22) (Open PDF). Without that extra datum, the half-space example licenses only local interior propagation before the first boundary encounter—not a canonical global Green operator or the exact sequence above.

Global hyperbolicity is therefore a powerful sufficient hypothesis, not a converse classification: some nonglobally hyperbolic settings admit Green operators after extra choices, while others admit none or admit nonunique ones.

What Green hyperbolicity does not prove

Green hyperbolicity settles a specific linear, support-sensitive inverse problem. It does not by itself establish any of the following.

• A general Cauchy theorem. Some Green-hyperbolic operators are not hyperbolic systems in a principal-symbol sense. Their Green maps may exist even when a conventional Cauchy formulation is not the right characterization.
• A Feynman inverse or a quantum state. Retarded and advanced support fix $E$, but positivity, the Hadamard condition, and the symmetric covariance remain additional.
• Gauge reduction. A gauge-invariant field equation may need gauge fixing before a Green-hyperbolic operator appears, and the resulting solution pairing can retain global cohomological degeneracies.
• Decay or scattering. Finite propagation controls where influence can travel, not how rapidly solutions decay or whether asymptotic completeness holds.
• Interacting dynamics. The exact sequence is linear. Perturbative products and nonlinear equations require further analytic and renormalization input.

These limits are not weaknesses of the theorem. They identify the exact downstream questions for which one must add microlocal spectrum, positivity, gauge, boundary, or nonlinear hypotheses.

Common pitfalls

Inferring the name from a plus or minus sign. Different sources reverse the labels attached to $G_\pm$. Read the support inclusion first; on this site, retarded means future supported.

Checking only $PG=1$. A right inverse may still differ by a homogeneous solution. A Green operator also obeys $GP=1$ on compactly supported sections and the appropriate causal support condition.

Omitting the terminal zero. The full exact sequence does assert that every spacelike-compact source lies in the image of $P$ on spacelike-compact sections. This uses the extensions of the two Green maps to past-compact and future-compact sources; it is stronger than the source-to-homogeneous-solution statement alone.

Confusing compact with spacelike compact. A spacelike-compact solution may extend for all time. What remains compact is its intersection with each Cauchy surface, or equivalently its containment in $J(K)$ for one compact $K$.

Confusing a causal propagator with a retarded inverse. $G_{\mathrm{ret}}f$ solves $Pu=f$ and lives to the future of the source. By contrast, $Ef$ solves the homogeneous equation and generally has both future and past support.

Reading spatial causality from a single eigenmode. An ultrastatic eigenfunction is spatially global. Sharp causal support emerges from the complete mode sum and the finite-propagation theorem.

Treating the causal propagator as a state. $E$ fixes the commutator. A state must additionally provide a positive two-point distribution with the required antisymmetric part and ultraviolet regularity.

Exercises

1. Translate the Green-operator labels. A source calls a future-supported operator $G_+$ “advanced,” a past-supported operator $G_-$ “retarded,” and defines $G=G_+-G_-$. Translate all three objects into QFT.org notation.

Solution

The support, not the word, fixes the translation:

$$G_{\mathrm{ret}}=G_+, \qquad G_{\mathrm{adv}}=G_-, \qquad E=G_{\mathrm{ret}}-G_{\mathrm{adv}}=G_+-G_-.$$

No physical sign changes once the naming map is applied consistently.

2. Identify the kernel of the causal propagator. Let $f\in\Gamma_c^\infty(M,E_2)$ and suppose $Ef=0$. Prove directly that $f\in P\Gamma_c^\infty(M,E_1)$.

Solution

The hypothesis gives

$$G_{\mathrm{ret}}f=G_{\mathrm{adv}}f=:k.$$

If $K=\operatorname{supp}f$, then

$$\operatorname{supp}k \subset J^+(K)\cap J^-(K),$$

which is compact by global hyperbolicity. Thus $k$ is compactly supported, and either Green identity gives $Pk=f$. Conversely, $EPh=0$ for every compactly supported $h$.

3. Read both spacelike-compact arrows. First let $Pu=0$ and $\operatorname{supp}u\subset J(K)$ for compact $K$. Choose a temporal cutoff $\chi$ that is zero in the far past and one in the far future. Set $u_+=\chi u$ and $u_-=(1-\chi)u$, and show that a compact source produces $u$ through $E$. Then let $q$ be a spacelike-compact source and construct a spacelike-compact $v$ satisfying $Pv=q$.

Solution

Because $Pu=0$,

$$f=Pu_+=-Pu_-$$

is supported where $\mathrm d\chi\ne0$. Its intersection with $J(K)$ is compact, so $f\in\Gamma_c^\infty(M,E_2)$. The section $u_+$ is past compact and solves $Pu_+=f$; uniqueness in that support class gives $G_{\mathrm{ret}}f=u_+$. Similarly, $u_-$ is future compact and solves $Pu_-=-f$, so $G_{\mathrm{adv}}f=-u_-$. Therefore

$$Ef=u_++u_-=u.$$

For the second claim, split $q=q_{\mathrm{pc}}+q_{\mathrm{fc}}$ with a temporal cutoff, where the pieces are respectively strictly past compact and strictly future compact. Then

$$v =G_{\mathrm{ret}}q_{\mathrm{pc}} +G_{\mathrm{adv}}q_{\mathrm{fc}}$$

is spacelike compact and satisfies $Pv=q$. This is the terminal surjectivity encoded by the last zero in the exact sequence.

4. Check the oscillator normalization. For $\omega>0$, verify distributionally that

$$S_{\mathrm{ret}}(\tau) = \theta(\tau)\frac{\sin(\omega\tau)}{\omega}$$

satisfies $(\partial_\tau^2+\omega^2)S_{\mathrm{ret}}=\delta$.

Solution

Away from $\tau=0$, the sine solves the homogeneous equation. The function itself has no jump because $\sin0=0$. Its first derivative is $\theta(\tau)\cos(\omega\tau)$, whose jump at the origin is one. Differentiating once more therefore contributes exactly $\delta(\tau)$; the remaining smooth terms cancel against $\omega^2S_{\mathrm{ret}}$.

5. Distinguish two boundary dynamics. Verify the boundary conditions for $R_D$ and $R_N$ above and explain why their difference is homogeneous in the interior.

Solution

The full-space kernel $R_0(\tau,\xi)$ is even in $\xi$. At $x=0$, its values at $-x'$ and $+x'$ agree, so the difference defining $R_D$ vanishes. Its $\xi$ derivative is odd, so the two derivatives in the sum defining $R_N$ cancel. Applying $P$ to the image term places its delta source at $x=-x'$, outside $x>0$; hence that term, and therefore $R_N-R_D$, is homogeneous in the interior. The two kernels solve the same interior equation but encode different boundary conditions.

6. Separate Green hyperbolicity from a ground state. Let $m=0$ in the ultrastatic example and suppose $\Sigma$ is connected, so the constant spatial mode has $\omega_0=0$. What becomes of the causal kernel in that mode, and why does this not automatically give a stationary ground-state two-point function?

Solution

The continuous limit is

$$\lim_{\omega\to0} \frac{\sin(\omega\tau)}{\omega} =\tau.$$

Thus the zero mode still has retarded, advanced, and causal Green kernels; the equation is Green hyperbolic. A stationary positive-frequency covariance would instead contain a factor such as $e^{-i\omega\tau}/(2\omega)$, which has no finite $\omega\to0$ limit. The infrared state problem is therefore additional to the causal inverse problem.

7. Prove nondegeneracy on the quotient. For self-adjoint $P$, show that $\sigma([f],[h])=\int_M\langle f,Eh\rangle\,\mathrm d\mathrm{vol}_g$ is well defined and nondegenerate.

Solution

If $f$ is replaced by $f+Pk$, formal self-adjointness gives

$$\int_M\langle Pk,Eh\rangle\,\mathrm d\mathrm{vol}_g = \int_M\langle k,PEh\rangle\,\mathrm d\mathrm{vol}_g =0.$$

Replacing $h$ by $h+P\ell$ gives zero because $EP\ell=0$. Since $E^\dagger=-E$, the form is antisymmetric. If $\sigma([f],[h])=0$ for all $h$, then $Ef=0$ distributionally; exactness gives $f=Pk$ for compactly supported $k$, so $[f]=0$.

Applications and next steps

• Propagation of singularities for hyperbolic fields adds the cotangent-space statement describing how the Green kernels carry null singularities.
• Green operators, causal propagators, and state-dependent two-point functions adds the physical distinction among causal response, commutators, and state-dependent correlators.
• Covariant symplectic structure and conserved inner products compares the quotient pairing with the Cauchy-surface current and treats boundary flux.
• Time-slice axiom and relative Cauchy evolution promotes the cutoff construction to the locally covariant algebraic framework.

References

• Bär, Christian. “Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes.” Communications in Mathematical Physics 333 (2015): 1585–1615. DOI. Open PDF.
• Bär, Christian. Geometric Wave Equations. Lecture notes, University of Potsdam, winter term 2015/16, version of April 13, 2017. Open PDF.
• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, 2007. DOI. Open PDF.
• Dappiaggi, Claudio, Nicolò Drago, and Hugo R. C. Ferreira. “Fundamental Solutions for the Wave Operator on Static Lorentzian Manifolds with Timelike Boundary.” Letters in Mathematical Physics 109 (2019): 2157–2186. DOI. Open PDF.
• Mühlhoff, Rainer. “Cauchy Problem and Green’s Functions for First Order Differential Operators and Algebraic Quantization.” Journal of Mathematical Physics 52 (2011): 022303. DOI. Open HTML.