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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)
Let $\Sigma$ be a time-oriented Lorentzian manifold of dimension $p+1$, or more generally a conal manifold with conal causal structure.
be a smooth vector bundle. Write
for the densitized dual vector bundle, hence the tensor product of vector bundles of the dual vector bundle with the differential n-form-bundle.
(compactly sourced causal support)
Given a vector bundle $E \overset{}{\to} \Sigma$ over a manifold $\Sigma$ with causal structure
Write $\Gamma_{\Sigma}(-)$ for spaces of smooth sections, and write
for the linear subspaces on those smooth sections whose support is
($cp$) inside a compact subset
($\pm cp$) inside the closed future cone/closed past cone, respectively, of a compact subset,
($scp$) 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),
($fcp$) inside the past of a Cauchy surface (Sanders 13, def. 3.2),
($pcp$) inside the future of a Cauchy surface (Sanders 13, def. 3.2),
($tcp$) 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)
(formally adjoint differential operators)
are called formally adjoint differential operators via a bilinear differential operator
such that for all $\Phi_1, \Phi_2 \in \Gamma_\Sigma(E)$ we have
(Bär 14, 3. (1) Khavkine 14, def. 2.4)
(advanced and retarded Green functions and causal Green function)
Let $\Sigma$ be a smooth manifold with causal structure, let $E \to \Sigma$ be a smooth vector bundle and let $P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)$ be a differential operator on its space of smooth sections.
Then a linear map
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 $P$, respectively, if
for all $\Phi \in \Gamma_{\Sigma,cp}(E_1)$ we have
and
the support of $G_{P.\pm}(\Phi)$ is in the closed future cone or closed past cone of the support of $\Phi$, respectively.
If the advanced/retarded Green functions $G_{P\pm}$ exists, then the difference
is called the causal Green function.
(Green hyperbolic differential operator)
Let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle over a smooth manifold $\Sigma$ with causal structure.
A linear hyperbolic differential operator.
is called Green hyperbolic with respect to the given causal structure if $P$ as well as its formal adjoint differential operator $P^\ast$ (def. ) admit advanced and retarded Green functions (def. ).
(Bär 14, def. 3.2, Khavkine 14, def. 2.2)
(advanced and retarded Green functions of Green hyperbolic differential operator are unique)
The advanced and retarded Green functions (def. ) of a Green hyperbolic differential operator (def. ) are unique.
(causal Green functions of formally adjoint Green hyperbolic differential operators are formally adjoint)
Let
be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ). Then also their causal Green functions $\mathrm{G}_P$ and $G_{P^\ast}$ (def. ) are formally adjoint differential operators, up to a sign:
(Fréchet topological vector space on spaces of smooth sections of a smooth vector bundle)
Let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_\Sigma(E)$ of smooth sections consider the seminorms indexed by a compact subset $K \subset \Sigma$ and a natural number $N \in \mathbb{N}$ and given by
where on the right we have the absolute values of the covariant derivatives of $\Phi$ for any fixed choice of connection on $E$ and norm on the tensor product of vector bundles $(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E$.
This makes $\Gamma_\Sigma(E)$ a Fréchet topological vector space.
For $K \subset \Sigma$ any closed subset then the sub-space of sections
of sections whose support is inside $K$ 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 $\Gamma_{\Sigma,K}(E)$ 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
etc.
Let $E \overset{fb}{\to} \Sigma$ 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
etc.
This is the space of distributional sections of the bundle $\tilde{E}^*$.
With this notations, smooth compactly supported sections of the same bundle, regarded as the non-singular distributions, constitute a dense subset
Imposing the same restrictions to the supports of distributions as in def. , we have the following subspaces of distributional sections:
(Sanders 13, Bär 14)
(causal Green functions of Green hyperbolic differential operators are continuous linear maps)
Given a Green hyperbolic differential operator $P$ (def. ), the advanced, retarded and causal Green functions of $P$ (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:
such that we still have the relation
and
and
(topological duality with causally restricted supports)
Let $E \overset{fb}{\to} \Sigma$ 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. ):
(Sanders 13, thm. 4.3, Bär 14, lem. 2.14)
(exact sequence of Green hyperbolic differential operator)
Let $\Gamma_\Sigma(E) \overset{P}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast)$ be a Green hyperbolic differential operator (def. ) with causal Green function $\mathrm{G}$ (def. ). Then the sequences
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:
In particular this means that there is a linear isomorphism between the space $ker_{scp}(P)$ of spatially compact solutions to the differential equation and the quotient space of the compactly supported dual sections by the image of $P$:
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).
Let $\Sigma_p^-, \Sigma_p^+ \subset \Sigma$ be two Cauchy surfaces, with $\Sigma_p^-$ in the past of $\Sigma_p^+$. Let also $\{\chi_+,\chi_-\}$ be a partition of unity subordinate to the cover $\{J^+(\Sigma_p^-), J^-(\Sigma_p^+)\}$ of $\Sigma$, that is, smooth functions such $\chi_+ + \chi_- = 1$, while $\chi_+ = 0$ on the past of $\Sigma_p^-$ and $\chi_- = 0$ on the future of $\Sigma_p^+$.
We can use these functions to define the following contracting homotopy of our complex into itself:
The homotopy maps are defined as follows:
The contracting identities
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 $P$ and $P^*$ 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:
(distributions on Green hyperbolic PDE solution space are the generalized PDE solutions)
Let $P, P^\ast \;\colon\; \Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast)$ be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ).
Then
1)
the canonical pairing (from prop. )
induces a continuous linear isomorphism
2)
a continuous linear functional on the solution space
is equivalently a distributional section (def. )
which is a generalized solution to the differential equation
and this is a continuous linear isomorphism given by pullback along the causal Green function $\mathrm{G}_P$ (def. ):
Similarly:
Observe that both $\ker_{scp}(P) \subset \Gamma_{\Sigma,scp}(E)$ and $im_{tcp}(P^*) \subset \Gamma'_{\Sigma,tcp}(\tilde{E}^*)$ are closed subspaces: the first by continuity of $P$ 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
and
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
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)
($P$-Symplectic and $P$-Peierls bracket)
Let
be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ) via a differential operator $K$ (1).
Then:
Consider $\ker_{scp}(P) \subset \Gamma_{\Sigma,scp}(E)$ and $\ker'(P^*) \subset \Gamma'_{\Sigma}(E)$, where $\ker'$ simply means that we are considering distributional solutions. For any Cauchy surface $\Sigma_p \overset{\iota_{\Sigma_p}}{\hookrightarrow} \Sigma$ the $P$-Symplectic bracket is the bilinear map
Note that, even though $\Phi_2\in \ker'(P^*)$ is a distribution on $\Sigma$, we are integrating it over a codimension-$1$ surface $\Sigma_p \hookrightarrow \Sigma$. In this case, the restriction to $\Sigma_p$ 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 $\Sigma_p \hookrightarrow \Sigma$ does not intersect $WF(\Phi_2)$. But, since the $\Phi_2$ is annihilated by $P^*$, by the theorem on the propagation of singularities (this prop. there), $WF(\Phi_2)$ contains only covectors that are characteristic with respect to $P^*$ (those on which the principal symbol of $P^*$ fails to be invertible). But by its definition, a Cauchy surface $\Sigma_p$ must be nowhere characteristic, meaning that its conormal bundle does not intersect $WF(\Phi_2)$.
Because the arguments $\Phi_1$ and $\Phi_2$ are annihilated respectively by $P$ and $P^*$, equation (1) ensures that the definition of $\{-,-\}_{\Sigma_p,K}$ is actually independent of the choice of $\Sigma_p$.
The $P$-Peierls bracket is the bilinear map
Since $\mathrm{G}_P$ annihilates $im_{cp}(P)$ and the image of $\mathrm{G}_P$ is annihilated by any distribution in $im'_{tcp}(P^*)$ (again, $im'$ refers to the fact that we are considering distributional sections), the $P$-Peierls bracket descends to a bilinear map on
(causal Green function transforms $P$-Peierls bracket to $P$-symplectic pairing)
Let
be a pair of Green hyperbolic differential operators (def. ) which are formally adjoint (def. ) via a differential operator $K$ (1).
Then the causal Green function intertwines the $P$-Symplectic bracket with the $P$-Peierls bracket (def. ) in that for every Cauchy surface $\Sigma_p \hookrightarrow \Sigma$ and all
we have
As currently defined, the pairings $\{-,-\}_{\Sigma_p,K}$ and $\{-,-\}_{\Sigma,G}$ are far from being anti-symmetric. In particular, the two arguments may come from very different spaces. In the self-adjoint case $P=P^*$, 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
This problem is analogous to studying (anti-)self-adjoint extensions of a symmetric unbounded operator on Hilbert space.
(Klein-Gordon operator is Green hyperbolic)
For $\Sigma$ a globally hyperbolic spacetimes then the Klein-Gordon operator $P = \Box - m^2$ (i.e. the wave operator for $m = 0$) is Green hyperbolic according to def. (e. g. Bär-Ginoux-Pfaeffle 07) and formally self-adjoint (this example). The corresponding $P$-Peierls bracket (def. ) is the original Peierls bracket.
(Dirac operator is Green hyperbolic)
The Dirac operator $D$ squaring to a Green hyperbolic operator is itself Green hyperbolic.
(Bär 14, corollary 3.15, example 3.16)
François Trèves, Topological Vector Spaces, Distributions and Kernels (Academic Press, New York, 1967)
Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (arXiv:0806.1036)
Ko Sanders, A note on spacelike and timelike compactness, Classical and Quantum Gravity 30, 115014 (2012) (doi, arXiv:1211.2469)
Christian Bär, Green-hyperbolic operators on globally hyperbolic spacetimes, Communications in Mathematical Physics 333, 1585-1615 (2014) (doi, arXiv:1310.0738)
Igor Khavkine, Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv:1402.1282)
Last revised on August 1, 2018 at 07:53:17. See the history of this page for a list of all contributions to it.