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Green hyperbolic partial differential equation

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Contents

Idea

A partial differential equation (PDE) is Green hyperbolic (Bär 14, def. 3.2, Khavkine 14, def. 2.2) if it behaves like a normally hyperbolic differential equation on a globally hyperbolic spacetime in that it has unique advanced and retarded Green functions.

Duhamel’s principle essentially establishes the equivalence between hyperbolic differential equations with a well-posed Cauchy problem and Green hyperbolic systems. (Khavkine 14, p. 12)

Definition

Let Σ\Sigma be a time-oriented Lorentzian manifold of dimension p+1p+1, or more generally a conal manifold with conal causal structure.

E fb Σ \array{ E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }

be a smooth vector bundle. Write

E˜ *E * Σ Σ p+1T *Σ \tilde E^\ast \;\coloneqq\; E^\ast \otimes_\Sigma \wedge_\Sigma^{p+1} T^\ast \Sigma

for the densitized dual vector bundle, hence the tensor product of vector bundles of the dual vector bundle with the differential n-form-bundle.

Definition

(compactly sourced causal support)

Given a vector bundle EΣE \overset{}{\to} \Sigma over a manifold Σ\Sigma with causal structure

Write Γ Σ()\Gamma_{\Sigma}(-) for spaces of smooth sections, and write

Γ cp() compact support Γ Σ,±cp() compactly sourced future/past support Γ Σ,scp() spacelike compact support Γ Σ,(f/p)cp() future/past compact support Γ Σ,tcp() timelike compact support \array{ \Gamma_{cp}(-) & \text{compact support} \\ \Gamma_{\Sigma,\pm cp}(-) & \text{compactly sourced future/past support} \\ \Gamma_{\Sigma,scp}(-) & \text{spacelike compact support} \\ \Gamma_{\Sigma,(f/p)cp}(-) & \text{future/past compact support} \\ \Gamma_{\Sigma,tcp}(-) & \text{timelike compact support} }

for the linear subspaces on those smooth sections whose support is

  1. (cpcp) inside a compact subset

  2. (±cp\pm cp) inside the closed future cone/closed past cone, respectively, of a compact subset,

  3. (scpscp) inside the closed causal cone of a compact subset, which equivalently means that the intersection with every (spacelike) Cauchy surface is compact (Sanders 13, theorem 2.2),

  4. (fcpfcp) inside the past of a Cauchy surface (Sanders 13, def. 3.2),

  5. (pcppcp) inside the future of a Cauchy surface (Sanders 13, def. 3.2),

  6. (tcptcp) inside the future of one Cauchy surface and the past of another (Sanders 13, def. 3.2)

(Bär 14, section 1, Khavkine 14, def. 2.1)

Definition

(formally adjoint differential operators)

Two differential operators

P,P *:Γ Σ(E)Γ Σ(E˜ *) P, P^\ast \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\tilde E^\ast)

are called formally adjoint differential operators via a bilinear differential operator

(1)K:Γ Σ(E)Γ Σ(E)Γ Σ( pT *Σ) K \;\colon\; \Gamma_\Sigma(E) \otimes \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\wedge^{p} T^\ast \Sigma)

such that for all Φ 1,Φ 2Γ Σ(E)\Phi_1, \Phi_2 \in \Gamma_\Sigma(E) we have

P(Φ 1)Φ 2Φ 1P *(Φ 2)=dK(Φ 1,Φ 2) P(\Phi_1) \cdot \Phi_2 - \Phi_1 \cdot P^\ast(\Phi_2) \;=\; d K(\Phi_1, \Phi_2)

(Bär 14, 3. (1) Khavkine 14, def. 2.4)

Definition

(advanced and retarded Green functions and causal Green function)

Let Σ\Sigma be a smooth manifold with causal structure, let EΣE \to \Sigma be a smooth vector bundle and let P:Γ Σ(E)Γ Σ(E˜ *)P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast) be a differential operator on its space of smooth sections.

Then a linear map

G P,±:Γ Σ,cp(E˜ *)Γ Σ,±cp(E) \mathrm{G}_{P,\pm} \;\colon\; \Gamma_{\Sigma, cp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, \pm cp}(E)

from spaces of sections of compact support to spaces of sections of causally sourced future/past support (def. ) is called an advanced or retarded Green function for PP, respectively, if

  1. for all ΦΓ Σ,cp(E 1)\Phi \in \Gamma_{\Sigma,cp}(E_1) we have

    (2)G P,±P(Φ)=Φ G_{P,\pm} \circ P(\Phi) = \Phi

    and

    (3)PG P,±(Φ)=Φ P \circ G_{P,\pm}(\Phi) = \Phi
  2. the support of G P.±(Φ)G_{P.\pm}(\Phi) is in the closed future cone or closed past cone of the support of Φ\Phi, respectively.

If the advanced/retarded Green functions G P±G_{P\pm} exists, then the difference

G PG P,+G P, \mathrm{G}_P \coloneqq \mathrm{G}_{P,+} - \mathrm{G}_{P,-}

is called the causal Green function.

(Bär 14, def. 3.2, cor. 3.10)

Definition

(Green hyperbolic differential operator)

Let EfbΣE \overset{fb}{\to} \Sigma be a smooth vector bundle over a smooth manifold Σ\Sigma with causal structure.

A linear hyperbolic differential operator.

P:Γ Σ(E)Γ Σ(E˜ *) P \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_{\Sigma}(\tilde E^\ast)

is called Green hyperbolic with respect to the given causal structure if PP as well as its formal adjoint differential operator P *P^\ast (def. ) admit advanced and retarded Green functions (def. ).

(Bär 14, def. 3.2, Khavkine 14, def. 2.2)

(Bär 14, cor. 3.12

Properties

Formally adjoint Green hyperbolic operators

Proposition

(causal Green functions of formally adjoint Green hyperbolic differential operators are formally adjoint)

Let

P,P *:Γ Σ(E)Γ Σ(E˜ *) P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast)

be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ). Then also their causal Green functions G P\mathrm{G}_P and G P *G_{P^\ast} (def. ) are formally adjoint differential operators, up to a sign:

(G P) *=G P *. \left( \mathrm{G}_P \right)^\ast \;=\; - \mathrm{G}_{P^\ast} \,.

(Khavkine 14, (24), (25))

Continuous linear functionals on the solution space

Definition

(Fréchet topological vector space on spaces of smooth sections of a smooth vector bundle)

Let EfbΣE \overset{fb}{\to} \Sigma be a smooth vector bundle. On its real vector space Γ Σ(E)\Gamma_\Sigma(E) of smooth sections consider the seminorms indexed by a compact subset KΣK \subset \Sigma and a natural number NN \in \mathbb{N} and given by

Γ Σ(E) p K N [0,) Φ maxnN(supxK| nΦ(x)|), \array{ \Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,, }

where on the right we have the absolute values of the covariant derivatives of Φ\Phi for any fixed choice of connection on EE and norm on the tensor product of vector bundles (T *Σ) Σ n ΣE(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E .

This makes Γ Σ(E)\Gamma_\Sigma(E) a Fréchet topological vector space.

For KΣK \subset \Sigma any closed subset then the sub-space of sections

Γ Σ,K(E)Γ Σ(E) \Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)

of sections whose support is inside KK becomes a Fréchet topological vector spaces with the induced subspace topology, which makes these be closed subspaces.

Finally, the vector spaces of smooth sections with prescribed causal support (def. ) are inductive limits of vector spaces Γ Σ,K(E)\Gamma_{\Sigma,K}(E) as above, and hence they inherit topological vector space structure by forming the corresponding inductive limit in the category of locally convex topological vector spaces. For instance

Γ Σ,cp(E)limKΣKcompactΓ Σ,K(E) \Gamma_{\Sigma,cp}(E) \;\coloneqq\; \underset{\underset{ {K \subset \Sigma} \atop {K\, \text{compact}} }{\longrightarrow}}{\lim} \Gamma_{\Sigma,K}(E)

etc.

(Bär 14, 2.1, 2.2)

Definition

(distributional sections)

Let EfbΣE \overset{fb}{\to} \Sigma be a smooth vector bundle over a smooth manifold with causal structure.

The vector spaces of smooth sections with restricted support from def. structures of topological vector spaces via def. . We denote the topological dual spaces by

Γ Σ(E˜ *)(Γ Σ,cp(E)) * \Gamma'_{\Sigma}(\tilde{E}^*) \coloneqq (\Gamma_{\Sigma,cp}(E))^*

etc.

This is the space of distributional sections of the bundle E˜ *\tilde{E}^*.

With this notations, smooth compactly supported sections of the same bundle, regarded as the non-singular distributions, constitute a dense subset

Γ Σ,cp(E˜ *)denseΓ Σ(E˜ *). \Gamma_{\Sigma,cp}(\tilde{E}^*) \underset{\text{dense}}{\hookrightarrow} \Gamma'_{\Sigma}(\tilde{E}^*) \,.

Imposing the same restrictions to the supports of distributions as in def. , we have the following subspaces of distributional sections:

Γ Σ,cp(E˜ *),Γ Σ,±cp(E˜ *),Γ Σ,scp(E˜ *),Γ Σ,fcp(E˜ *),Γ Σ,pcp(E˜ *),Γ Σ,tcp(E˜ *)Γ Σ(E˜ *). \Gamma'_{\Sigma,cp}(\tilde E^\ast) , \Gamma'_{\Sigma,\pm cp}(\tilde E^\ast) , \Gamma'_{\Sigma,scp}(\tilde E^\ast) , \Gamma'_{\Sigma,fcp}(\tilde E^\ast) , \Gamma'_{\Sigma,pcp}(\tilde E^\ast) , \Gamma'_{\Sigma,tcp}(\tilde E^\ast) \subset \Gamma'_{\Sigma}(\tilde E^\ast) .

(Sanders 13, Bär 14)

Proposition

(causal Green functions of Green hyperbolic differential operators are continuous linear maps)

Given a Green hyperbolic differential operator PP (def. ), the advanced, retarded and causal Green functions of PP (def. ) are continuous linear maps with respect to the topological vector space structure from def. and also have a unique continuous extension to the spaces of sections with larger support (def. ) as follows:

G P,+ :Γ Σ,pcp(E˜ *)Γ Σ,pcp(E), G P, :Γ Σ,fcp(E˜ *)Γ Σ,fcp(E), G P :Γ Σ,tcp(E˜ *)Γ Σ(E), \begin{aligned} \mathrm{G}_{P,+} &\;\colon\; \Gamma_{\Sigma, pcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, pcp}(E) , \\ \mathrm{G}_{P,-} &\;\colon\; \Gamma_{\Sigma, fcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, fcp}(E) , \\ \mathrm{G}_{P} &\;\colon\; \Gamma_{\Sigma, tcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma}(E) , \end{aligned}

such that we still have the relation

G P=G P,+G P, \mathrm{G}_P = \mathrm{G}_{P,+} - \mathrm{G}_{P,-}

and

PG P,±=G P,±P=id P \circ \mathrm{G}_{P,\pm} = \mathrm{G}_{P,\pm} \circ P = id

and

suppG P,±(α˜ *)J ±(suppα˜ *). supp \mathrm{G}_{P,\pm}(\tilde{\alpha}^*) \subseteq J^\pm(supp \tilde{\alpha}^*) \,.

(Bär 14, thm. 3.8, cor. 3.11)

Proposition

(topological duality with causally restricted supports)

Let EfbΣE \overset{fb}{\to} \Sigma be a smooth vector bundle over a smooth manifold with causal structure.

Then there are the following isomorphisms of topological vector spaces between dual spaces of spaces of sections and spaces of distributional sections, with restricted supports (def. ):

Γ Σ,cp(E) * Γ Σ(E˜ *), Γ Σ,+cp(E) * Γ Σ,fcp(E˜ *), Γ Σ,cp(E) * Γ Σ,pcp(E˜ *), Γ Σ,scp(E) * Γ Σ,tcp(E˜ *), Γ Σ,fcp(E) * Γ Σ,+cp(E˜ *), Γ Σ,pcp(E) * Γ Σ,cp(E˜ *), Γ Σ,tcp(E) * Γ Σ,scp(E˜ *), Γ Σ(E) * Γ Σ,cp(E˜ *). \begin{aligned} \Gamma_{\Sigma,cp}(E)^* &\simeq \Gamma'_{\Sigma}(\tilde E^\ast) , \\ \Gamma_{\Sigma,+cp}(E)^* &\simeq \Gamma'_{\Sigma,fcp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,-cp}(E)^* &\simeq \Gamma'_{\Sigma,pcp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,scp}(E)^* &\simeq \Gamma'_{\Sigma,tcp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,fcp}(E)^* &\simeq \Gamma'_{\Sigma,+cp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,pcp}(E)^* &\simeq \Gamma'_{\Sigma,-cp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,tcp}(E)^* &\simeq \Gamma'_{\Sigma,scp}(\tilde E^\ast) , \\ \Gamma_{\Sigma}(E)^* &\simeq \Gamma'_{\Sigma,cp}(\tilde E^\ast) . \end{aligned}

(Sanders 13, thm. 4.3, Bär 14, lem. 2.14)

Proposition

(exact sequence of Green hyperbolic differential operator)

Let Γ Σ(E)PΓ Σ(E˜ *)\Gamma_\Sigma(E) \overset{P}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast) be a Green hyperbolic differential operator (def. ) with causal Green function G\mathrm{G} (def. ). Then the sequences

0Γ Σ,cp(E)PΓ Σ,cp(E˜ *)G PΓ Σ,scp(E)PΓ Σ,scp(E˜ *)0, 0 \to \Gamma_{\Sigma,cp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,cp}(\tilde E^\ast) \overset{\mathrm{G}_P}{\longrightarrow} \Gamma_{\Sigma,scp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,scp}(\tilde E^\ast) \to 0 ,
0Γ Σ,tcp(E)PΓ Σ,tcp(E˜ *)G PΓ Σ(E)PΓ Σ(E˜ *)0 0 \to \Gamma_{\Sigma,tcp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,tcp}(\tilde E^\ast) \overset{\mathrm{G}_P}{\longrightarrow} \Gamma_{\Sigma}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma}(\tilde E^\ast) \to 0

of these operators restricted to functions with causally restricted supports as indicated (def. ) are exact sequences of topological vector spaces and continuous linear maps between them.

Under passing to dual spaces and using the isomorphisms of spaces of distributional sections (def. ) from prop. this yields the following dual exact sequence of topological vector spaces and continuous linear maps between them:

(4)0Γ Σ,tcp(E)P *Γ Σ,tcp(E˜ *)G P *Γ Σ(E)P *Γ Σ(E˜ *)0, 0 \to \Gamma'_{\Sigma,tcp}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma,tcp}(\tilde E^\ast) \overset{-\mathrm{G}_{P^*}}{\longrightarrow} \Gamma'_{\Sigma}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma}(\tilde E^\ast) \to 0 ,
(5)0Γ Σ,cp(E)P *Γ Σ,cp(E˜ *)G P *Γ Σ,scp(E)P *Γ Σ,scp(E˜ *)0 0 \to \Gamma'_{\Sigma,cp}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma,cp}(\tilde E^\ast) \overset{-\mathrm{G}_{P^*}}{\longrightarrow} \Gamma'_{\Sigma,scp}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma,scp}(\tilde E^\ast) \to 0

In particular this means that there is a linear isomorphism between the space ker scp(P)ker_{scp}(P) of spatially compact solutions to the differential equation and the quotient space of the compactly supported dual sections by the image of PP:

(6)Γ Σ,cp(E˜ *)/im(P)AG PAker scp(P). \Gamma_{\Sigma,cp}(\tilde E^\ast)/im(P) \underoverset{\simeq}{\phantom{A}\mathrm{G}_P\phantom{A}}{\longrightarrow} ker_{scp}(P) \,.

The following proof is a slight refinement of (Khavkine 14, prop. 2.1). The refinement consists of reducing the check of exactness to the construction of a contracting homotopy of the complex into itself (cf. the answer to MO208985, where this refinement has appeared in a simplified context).

Proof

Let Σ p ,Σ p +Σ\Sigma_p^-, \Sigma_p^+ \subset \Sigma be two Cauchy surfaces, with Σ p \Sigma_p^- in the past of Σ p +\Sigma_p^+. Let also {χ +,χ }\{\chi_+,\chi_-\} be a partition of unity subordinate to the cover {J +(Σ p ),J (Σ p +)}\{J^+(\Sigma_p^-), J^-(\Sigma_p^+)\} of Σ\Sigma, that is, smooth functions such χ ++χ =1\chi_+ + \chi_- = 1, while χ +=0\chi_+ = 0 on the past of Σ p \Sigma_p^- and χ =0\chi_- = 0 on the future of Σ p +\Sigma_p^+.

We can use these functions to define the following contracting homotopy of our complex into itself:

0 Γ Σ,cp(E) P Γ Σ,cp(E˜ *) G Γ Σ,scp(E) P Γ Σ,scp(E˜ *) 0 id χG id P χ id G χ id 0 Γ Σ,cp(E) P Γ Σ,cp(E˜ *) G Γ Σ,scp(E) P Γ Σ,scp(E˜ *) 0 \begin{array}{ccccccccccc} 0 & \to & \Gamma_{\Sigma,cp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,cp}(\tilde E^\ast) & \overset{\mathrm{G}}{\longrightarrow} & \Gamma_{\Sigma,scp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,scp}(\tilde E^\ast) & \to & 0 \\ & & {\downarrow} id & {\swarrow} {}_\chi\mathrm{G} & {\downarrow} id & {\swarrow} P_\chi & {\downarrow} id & {\swarrow} \mathrm{G}_\chi & {\downarrow} id \\ 0 & \to & \Gamma_{\Sigma,cp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,cp}(\tilde E^\ast) & \overset{\mathrm{G}}{\longrightarrow} & \Gamma_{\Sigma,scp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,scp}(\tilde E^\ast) & \to & 0 \end{array}

The homotopy maps are defined as follows:

χG[α˜ *] =χ +G [α˜ *]+χ G +[α˜ *], P χ[ψ] =P[χ +ψ]χ +P[ψ]=P[χ ψ]+χ P[ψ], G χ[α˜ *] =G +[χ +α˜ *]+G [χ α˜ *]. \begin{aligned} {}_\chi\mathrm{G}[\tilde{\alpha}^*] &= \chi_+ \mathrm{G}_-[\tilde{\alpha}^*] + \chi_- \mathrm{G}_+[\tilde{\alpha}^*] , \\ P_\chi[\psi] &= P[\chi_+\psi] - \chi_+ P[\psi] = -P[\chi_-\psi] + \chi_- P[\psi] , \\ \mathrm{G}_\chi[\tilde{\alpha}^*] &= \mathrm{G}_+[\chi_+\tilde{\alpha}^*] + \mathrm{G}_-[\chi_-\tilde{\alpha}^*] . \end{aligned}

The contracting identities

χGP =id, P χG+P χG =id, GP χ+G χP =id, PG χ =id, \begin{aligned} {}_\chi\mathrm{G} \circ P &= id , \\ P\circ {}_\chi\mathrm{G} + P_\chi \circ \mathrm{G} &= id , \\ \mathrm{G}\circ P_\chi + \mathrm{G}_\chi \circ P &= id , \\ P \circ \mathrm{G}_\chi &= id , \end{aligned}

are simply a matter of direct calculation.

The identity morphism of our complex to itself induces an isomorphism on its cohomology. On the other hand, since this morphism itself is induced by a homotopy, it must be in fact be the zero map on cohomology. This is only possible when all cohomologies vanish and our complex is exact. The continuity of the differential operators PP and P *P^* is standard. The continuity of the Green function acting on smooth functions was already noted in prop. .

To see the second exact sequences, observe that differential operators extend continuously to distributions in a standard way. The only nontrivial check is on the Green functions. Their continuity is discussed in (Sanders 13, sec. 5) and (Baer 14, lem. 4.1). The exactness follows from the same argument as in the previous argument (since a contracting homotopy dualizes to a contracting homotopy). The exactness of a sequence similar to the one above also appears as (Bär 14, thm. 4.3). The continuity of the extensions of Green functions to distribution follows from standard arguments, which was checked for instance in (Bär 14, lem. 4.1). The standard argument consists of noting that the adjoint of a continuous linear map is also continuous, both for the weak and strong topologies on distributions (Treves 67, prop.-cor. 19.5).

Putting the above results together, it follows:

Proposition

(distributions on Green hyperbolic PDE solution space are the generalized PDE solutions)

Let P,P *:Γ Σ(E)Γ Σ(E˜ *)P, P^\ast \;\colon\; \Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast) be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ).

Then

1)

the canonical pairing (from prop. )

Γ Σ,cp(E˜ *) Γ Σ(E) α * , Φ α a *(x)Φ a(x)dvol Σ(x) \array{ \Gamma'_{\Sigma,cp}(\tilde E^\ast) &\otimes& \Gamma_\Sigma(E) &\overset{}{\longrightarrow}& \mathbb{C} \\ \alpha^\ast &,& \Phi &\mapsto& \int \alpha^\ast_a(x) \Phi^a(x)\, dvol_\Sigma(x) }

induces a continuous linear isomorphism

(ker(P)) *Γ Σ,cp(E˜ *)/im cp(P *) (ker(P))^\ast \;\simeq\; \Gamma'_{\Sigma,cp}(\tilde E^\ast)/im_{cp}(P^\ast)

2)

a continuous linear functional on the solution space

u sol(ker scp(P)) * u_{sol} \in \left(ker_{scp}(P)\right)^\ast

is equivalently a distributional section (def. )

uΓ Σ(E) u \in \Gamma'_{\Sigma}(E)

which is a generalized solution to the differential equation

P *u=0, P^\ast u = 0 \,,

and this is a continuous linear isomorphism given by pullback along the causal Green function G P\mathrm{G}_P (def. ):

(7)(ker scp(P)) *G P *=()G P{uΓ Σ(E)|P *u=0}. \left(ker_{scp}(P)\right)^\ast \underoverset{\simeq}{-\mathrm{G}_{P^\ast} = (-)\circ \mathrm{G}_P}{\longrightarrow} \left\{ u \in \Gamma'_{\Sigma}(E) \,\vert\, P^* u = 0 \right\} \,.

Similarly:

(8)(ker(P)) *G P *=()G P{uΓ Σ,scp(E)|P *u=0}. \left(ker(P)\right)^\ast \underoverset{\simeq}{-\mathrm{G}_{P^\ast} = (-)\circ \mathrm{G}_P}{\longrightarrow} \left\{ u \in \Gamma'_{\Sigma,scp}(E) \,\vert\, P^* u = 0 \right\} \,.
Proof

Observe that both ker scp(P)Γ Σ,scp(E)\ker_{scp}(P) \subset \Gamma_{\Sigma,scp}(E) and im tcp(P *)Γ Σ,tcp(E˜ *)im_{tcp}(P^*) \subset \Gamma'_{\Sigma,tcp}(\tilde{E}^*) are closed subspaces: the first by continuity of PP and the second by exactness of the sequence (4) in prop. , which implies that this, too, is the kernel of a continuous linear functional.

This implies (Treves 67, props. 35.5, 35.6) that the linear isomorphisms

(ker(P)) *Γ Σ,cp(E˜ *)/im cp(P *) (ker(P))^\ast \;\simeq\; \Gamma'_{\Sigma,cp}(\tilde E^\ast)/im_{cp}(P^\ast)

and

(9)(ker scp(P)) *Γ Σ,tcp(E˜ *)/im tcp(P *) (\ker_{scp}(P))^* \;\simeq\; \Gamma'_{\Sigma,tcp}(\tilde{E}^*) / im_{tcp}(P^*)

obtained from the underlying exact sequences of vector spaces in prop. are also continuous linear isomorphisms for the dual space topology on the left.

The first of these is the statement 1) to be proven.

From the second, by once again exploiting the exactness of the sequence (4) in prop. , we also have the chain of isomorphisms

(10)Γ Σ,tcp(E˜ *)/im tcp(P *)Γ Σ,tcp(E˜ *)/ker tcp(G P *)G P *ker(P *)Γ Σ(E). \Gamma'_{\Sigma,tcp}(\tilde{E}^*) / im_{tcp}(P^*) \simeq \Gamma'_{\Sigma,tcp}(\tilde{E}^*) / \ker_{tcp}(G_{P^*}) \underoverset{\simeq}{G_{P^\ast}}{\to} \ker(P^*) \subset \Gamma'_{\Sigma}({E}).

Combining this with (9) yields the desired isomorphism in 2)

The same argument applied to the exact sequence (5) yields the second statement in 2)

The PP-Symplectic and PP-Peierls brackets

Definition

(PP-Symplectic and PP-Peierls bracket)

Let

P,P *:Γ Σ(E)Γ Σ(E˜ *) P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast)

be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ) via a differential operator KK (1).

Then:

  1. Consider ker scp(P)Γ Σ,scp(E)\ker_{scp}(P) \subset \Gamma_{\Sigma,scp}(E) and ker(P *)Γ Σ(E)\ker'(P^*) \subset \Gamma'_{\Sigma}(E), where ker\ker' simply means that we are considering distributional solutions. For any Cauchy surface Σ pι Σ pΣ\Sigma_p \overset{\iota_{\Sigma_p}}{\hookrightarrow} \Sigma the PP-Symplectic bracket is the bilinear map

    ker scp(P)ker(P *) {,} Σ p,K (Φ 1,Φ 2) Σ p(ι Σ p) *K(Φ 1,Φ 2) \array{ \ker_{scp}(P) \otimes \ker'(P^*) &\overset{ \left\{-,- \right\}_{\Sigma_p,K} }{\longrightarrow}& \mathbb{R} \\ (\mathbf{\Phi}_1, \mathbf{\Phi}_2) &\mapsto& \underset{\Sigma_p}{\int} (\iota_{\Sigma_p})^\ast K(\mathbf{\Phi}_1, \mathbf{\Phi}_2) }

    Note that, even though Φ 2ker(P *)\Phi_2\in \ker'(P^*) is a distribution on Σ\Sigma, we are integrating it over a codimension-11 surface Σ pΣ\Sigma_p \hookrightarrow \Sigma. In this case, the restriction to Σ p\Sigma_p is well-defined by the theorem on the pullback of a distribution (this prop. there). The only condition to check is that the conormal bundle of the embedding Σ pΣ\Sigma_p \hookrightarrow \Sigma does not intersect WF(Φ 2)WF(\Phi_2). But, since the Φ 2\Phi_2 is annihilated by P *P^*, by the theorem on the propagation of singularities (this prop. there), WF(Φ 2)WF(\Phi_2) contains only covectors that are characteristic with respect to P *P^* (those on which the principal symbol of P *P^* fails to be invertible). But by its definition, a Cauchy surface Σ p\Sigma_p must be nowhere characteristic, meaning that its conormal bundle does not intersect WF(Φ 2)WF(\Phi_2).
    Because the arguments Φ 1\Phi_1 and Φ 2\Phi_2 are annihilated respectively by PP and P *P^*, equation (1) ensures that the definition of {,} Σ p,K\{-,-\}_{\Sigma_p,K} is actually independent of the choice of Σ p\Sigma_p.

  2. The PP-Peierls bracket is the bilinear map

    Γ Σ,cp(E˜ *)Γ Σ,tcp(E˜ *) {,} Σ,G (α˜ 1 *,α˜ 2 *) Σα˜ 1 *G P *(α˜ 2 *)=ΣG˜ P(α 1 *)α˜ 2 * \array{ \Gamma_{\Sigma,cp}(\tilde E^\ast) \otimes \Gamma'_{\Sigma,tcp}(\tilde E^\ast) & \overset{ \left\{ -,- \right\}_{\Sigma, \mathrm{G}} }{ \longrightarrow } & \mathbb{R} \\ (\tilde \alpha^\ast_1, \tilde \alpha^\ast_2) &\mapsto& \underset{\Sigma}{\int} \tilde \alpha^\ast_1 \cdot \mathrm{G}_{P^*}(\tilde \alpha^\ast_2) = -\underset{\Sigma}{\int} \tilde \mathrm{G}_{P}(\alpha^\ast_1) \cdot \tilde \alpha^\ast_2 }

    Since G P\mathrm{G}_P annihilates im cp(P)im_{cp}(P) and the image of G P\mathrm{G}_P is annihilated by any distribution in im tcp(P *)im'_{tcp}(P^*) (again, imim' refers to the fact that we are considering distributional sections), the PP-Peierls bracket descends to a bilinear map on

    Γ Σ,cp(E˜ *)/im cp(P *)Γ Σ,tcp(E˜ *)/im tcp(P) {,} Σ,G \array{ \Gamma_{\Sigma,cp}(\tilde E^\ast) / im_{cp}(P^*) \otimes \Gamma'_{\Sigma,tcp}(\tilde E^\ast) / im'_{tcp}(P) & \overset{ \left\{ -,- \right\}_{\Sigma, \mathrm{G}} }{ \longrightarrow } & \mathbb{R} }
Proposition

(causal Green function transforms PP-Peierls bracket to PP-symplectic pairing)

Let

P,P *:Γ Σ(E)Γ Σ(E˜ *) P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast)

be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ) via a differential operator KK (1).

Then the causal Green function intertwines the PP-Symplectic bracket with the PP-Peierls bracket (def. ) in that for every Cauchy surface Σ pΣ\Sigma_p \hookrightarrow \Sigma and all

α˜ 1 *Γ Σ,cp(E˜ *),α˜ 2 *Γ Σ,tcp(E˜ *) \tilde \alpha_1^\ast \;\in\; \Gamma_{\Sigma,cp}(\tilde E^\ast) , \quad \tilde \alpha_2^\ast \;\in\; \Gamma'_{\Sigma,tcp}(\tilde E^\ast)

we have

(11){α˜ 1 *,α˜ 2 *} Σ,G={G P(α˜ 1 *),G P *(α˜ 2 *)} Σ p,K. \left\{ \tilde \alpha^\ast_1, \tilde \alpha^\ast_2 \right\}_{\Sigma,\mathrm{G}} \;=\; -\left\{ \mathrm{G}_P(\tilde \alpha^\ast_1), \mathrm{G}_{P^*}(\tilde \alpha^\ast_2) \right\}_{\Sigma_p,K} \,.

(Khavkine 14, lemma 2.5)

As currently defined, the pairings {,} Σ p,K\{-,-\}_{\Sigma_p,K} and {,} Σ,G\{-,-\}_{\Sigma,G} are far from being anti-symmetric. In particular, the two arguments may come from very different spaces. In the self-adjoint case P=P *P=P^*, we may ask whether the domains of the two arguments may be respectively enlarged or shrunk to be equal and such that the pairing becomes anti-symmetric. When that is possible, we get an honest symplectic or Poisson bracket. The simplest such choices are

ker scp(P)ker scp(P) {,} Σ p,K Γ Σ,cp(E˜ *)Γ Σ,cp(E˜ *) {,} Σ,G \array{ \ker_{scp}(P) \otimes \ker_{scp}(P) & \overset{ \left\{ -,- \right\}_{\Sigma_p, K} }{ \longrightarrow } & \mathbb{R} \\ \Gamma_{\Sigma,cp}(\tilde E^\ast) \otimes \Gamma_{\Sigma,cp}(\tilde E^\ast) & \overset{ \left\{ -,- \right\}_{\Sigma, \mathrm{G}} }{ \longrightarrow } & \mathbb{R} }

This problem is analogous to studying (anti-)self-adjoint extensions of a symmetric unbounded operator on Hilbert space.

Examples

Example

(Klein-Gordon operator is Green hyperbolic)

For Σ\Sigma a globally hyperbolic spacetimes then the Klein-Gordon operator P=m 2P = \Box - m^2 (i.e. the wave operator for m=0m = 0) is Green hyperbolic according to def. (e. g. Bär-Ginoux-Pfaeffle 07) and formally self-adjoint (this example). The corresponding PP-Peierls bracket (def. ) is the original Peierls bracket.

Example

(Dirac operator is Green hyperbolic)

The Dirac operator DD squaring to a Green hyperbolic operator is itself Green hyperbolic.

(Bär 14, corollary 3.15, example 3.16)

References

Last revised on August 1, 2018 at 07:53:17. See the history of this page for a list of all contributions to it.