Global wave parametrices on globally hyperbolic spacetimes
aa r X i v : . [ m a t h . A P ] J un Global wave parametriceson globally hyperbolic spacetimes
Matteo Capoferri Claudio Dappiaggi Nicolò Drago29 June 2020
Abstract
In a recent work the first named author, Levitin and Vassiliev have constructedthe wave propagator on a closed Riemannian manifold M as a single oscillatoryintegral global both in space and in time with a distinguished complex-valued phasefunction. In this paper, first we give a natural reinterpretation of the underlyingalgorithmic construction in the language of ultrastatic Lorentzian manifolds. Subse-quently we show that the construction carries over to the case of static backgroundsthanks to a suitable reduction to the ultrastatic scenario. Finally we prove thatthe overall procedure can be generalised to any globally hyperbolic spacetime withcompact Cauchy surfaces. As an application, we discuss how, from our procedure,one can recover the local Hadamard expansion which plays a key role in all appli-cations in quantum field theory on curved backgrounds. Keywords: wave propagator, global Fourier integral operators, globally hy-perbolic spacetimes.
MSC classes: primary 58J40; secondary 35L05, 53C50, 35Q40.
Contents
MC: Department of Mathematics, University College London, Gower Street, London WC1E 6BT,UK; [email protected].
Current address:
School of Mathematics, Cardiff University, Senghen-nydd Road, Cardiff CF24 4AG, UK.CD: Dipartimento di Fisica, Università degli Studi di Pavia & INFN, Sezione di Pavia, Via Bassi 6,I-27100 Pavia, Italia; [email protected]: Department of Mathematics, Julius Maximilian University of Würzburg, Emil-Fischer-Straße31, D-97074 Würzburg, Germany; [email protected] atteo Capoferri, Claudio Dappiaggi, and Nicolò Drago Page 24 Extension to general globally hyperbolic spacetimes 13
The study of hyperbolic partial differential equations on manifolds, particularly withreference to the D’Alembert wave operator, has been and continues to be, in variousforms, at the forefront of scientific research. This is certainly true for mathematics,where the study of wave propagation has led over the years to major breakthroughs inthe pure and in the applied side of the subject alike.In this context, of particular relevance is the so-called wave propagator which is definedas follows. Consider (Σ , h ) a connected closed smooth Riemannian manifold of dimension dim Σ ≥ and let ∆ be the Laplace–Beltrami operator associated with h . The latteris known to be a non-positive operator, thus the square root √− ∆ is well-defined. Wecall wave propagator the Fourier Integral Operator (FIO) U ( t ) , t ∈ R , the distributionalsolution of the operator (pseudodifferential) initial value problem (cid:18) − i ∂∂t + √− ∆ (cid:19) U ( t ) = 0 , U (0) = Id . (1.1)It is well-known that the knowledge of U ( t ) is sufficient to solve the Cauchy problemassociated with the D’Alembert wave operator ∂ ∂t − ∆ . For this reason the study ofthe wave propagator has attracted a lot of attention, especially by means of microlocal-analytic techniques, see, e.g. , [15, 26].In this paper we are interested instead in a different approach, which aims at a globalconstruction of the wave propagator. This is based on the works of Laptev, SafarovVassiliev [20] and of Safarov and Vassiliev [25] in which it has been shown that thepropagator for a wide class of hyperbolic equations can be written as a single FIO,global both in space and in time, with complex-valued – as opposed to real-valued –phase function. This approach has the advantage of circumventing obstructions due tocaustics. Such viewpoint has recently been adopted in [6], where the authors proposeda global invariant definition of the full symbol of the wave propagator, together withan algorithm for the explicit calculation of its homogeneous components. It is worthobserving that the compactness assumption on Σ could be relaxed with quite some effort.In this paper we build upon [6], requiring that all underlying Riemannian manifolds areclosed, in order to avoid unnecessary technical difficulties, focusing instead on the mainideas and constructions.Our investigation originates from two key observations. On the one hand, the studyof hyperbolic partial differential equations sees one of its main applications in the descrip-tion of physical phenomena modelled over Lorentzian manifolds. These are the naturalplayground of several theories, such as general relativity and quantum field theory oncurved backgrounds, whose mathematical properties have been for decades the subjectof thorough investigations. On the other hand, the reinterpretation of the framework Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 3 depicted in [6] in the language of Lorentzian geometry is rather natural, particularly withreference to a specific class of backgrounds known as ultrastatic spacetimes .In this paper we will move from the observations above, asking ourselves to whichextent it is possible to adapt and generalise the results of [6] to the Lorentzian setting. Indoing so, we will focus on the construction of the wave propagator on a distinguished classof spacetimes, known as globally hyperbolic , see, e.g. , [1]. These are of notable importancesince they represent the natural collection of Lorentzian manifolds on which the Cauchyproblem for hyperbolic partial differential equations of the like of the D’Alembert waveequation is well-posed. It is important to stress that, by far, this is not the first paper inwhich FIOs are used in a Lorentzian framework to analyse distinguished parametrices, see, e.g. , [14, 27]. Yet, our method differs significantly from those used in earlier publications,especially with regards to the invariant approach relying on the use of a distinguishedcomplex-valued phase function.The main goal of this paper is to construct explicitly, in terms of Fourier integraloperators, solutions to the wave equation on a Lorentzian spacetime ( M, g ) with compactCauchy surface. More precisely, we will construct, modulo an operator with infinitelysmooth Schwartz kernel, the solution operator mapping initial data on a given Cauchysurface to full solutions in M , in a global and invariant fashion – effectively, a Lorentziananalogue of the wave group. As already mentioned, our strategy consists in extending andadapting results from [6], which yield an explicit – i.e. up to solving ordinary differentialequations – and invariant formula for the integral kernel of the solution operator (precisedefinitions will be provided later on). As our technique relies heavily on microlocalanalysis, all our constructions capture the singular structure of the mathematical objectsinvolved, modulo infinitely smooth contributions.In writing this paper we have in mind the mathematical results on the one side andtheir potential applications on the other. For this reason, we refrain, at times, fromproviding arguments and proofs in the most general possible case, in the attempt tomake the paper accessible also to a mathematical physics readership not fully familiarwith the technical theory of Fourier integral operators.Our main results are as follows.1. Working on an ultrastatic spacetime R × Σ with compact Cauchy surface, we extendresults from [6] to construct the operator e − i √− ∆ h + V t as a single oscillatory integral global in space and in time, see Subsection 3.2 andTheorem 3.3. Here ∆ h is the Laplace–Beltrami operator on Σ and V a time-independent smooth potential.2. Using the extended version of the results from [6], we provide an explicit algorithmfor the construction of the wave propagator on static spacetimes, by reductionto the ultrastatic case, see Section 3. Observe that, for this class of spacetimes,the knowledge of the wave propagator is sufficient to construct also the advancedand retarded fundamental solutions for the D’Alembert wave operator. These aretwo distinguished inverses characterised by their support properties. They playan important role in applications, especially in quantum field theory, since theycapture the finite speed of propagation encoded within the wave equation. Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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3. Working on a globally hyperbolic spacetime M with compact Cauchy surface andfixing a Cauchy surface Σ s := { s }× Σ , we construct the operator U ( t, s ) : C ∞ (Σ s ) → M whose integral kernel u (i) satisfies the wave equation (in a distributional sense)modulo C ∞ , (ii) has a wavefront set of Hadamard type, as a single oscillatory inte-gral, global in space and in time, with distinguished complex-valued phase function.See Theorem 4.7.4. We illustrate how our procedure can be applied to the construction of Hadamardstates and we demonstrate how to recover from our global integral kernel the (local)Hadamard expansion commonly used in algebraic quantum field theory.We remark that Gérard and Wrochna have also addressed the problem of constructingHadamard states on globally hyperbolic spacetimes in [12]. Their construction sharessome common features with our approach, though their work is based mainly on pseudo-differential techniques.The paper is structured as follows.In Section 2 we define the geometric objects we will be using throughout the paperand fix notation and conventions.In Section 3 we construct the wave propagator on static spacetimes. This is done inseveral steps. First, by a conformal transformation we reduce the problem at hand toan auxiliary problem on an ultrastatic background; secondly, working on an ultrastaticspacetime, we extend the results from [6] to encompass the case of wave operators witharbitrary time-independent infinitely smooth potential and solve the auxiliary problem;finally, we go back to the static case by conjugating by suitable multiplication operators.In Section 4 we construct, using Fourier integral operator techniques, a Lorentziananalogue of the wave group. The key idea is to identify a distinguished complex-valuedphase function encoding information about the propagation of singularities and theLorentzian geometry and then exploit the construction developed in [6] for the Rieman-nian case.Lastly, in Section 5 we discuss one of the potential applications of the method, namely,the global construction of Hadamard states, relevant in algebraic quantum field theoryon curved spacetime.The paper is complemented by an appendix, Appendix A, where we recall in anabridged manner basic facts about fundamental solutions of normally hyperbolic opera-tors. In this section we discuss the geometric data which will play a key role in this paper, andfix notation and conventions.We call spacetime a pair ( M, g ) where M is a connected, Hausdorff, second countable,orientable smooth manifold of dimension d ≥ and g is a smooth Lorentzian metric ofsignature (+ , − , ..., − ) .A spacetime is said to be globally hyperbolic if it possesses a Cauchy surface , that is aclosed achronal subset Σ ⊂ M whose domain of dependence D (Σ) is the whole spacetime M , see [28, §8.1]. Globally hyperbolic spacetimes are particularly important because Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 5 their geometry ensures the well-posedness of Cauchy problems for normally hyperbolicoperators ( e.g. , the wave operator).A remarkable and very convenient equivalent characterisation of global hyperbolicitywas provided by Bernal and Sanchez in [3, 4]. We report below their result as formulatedin [1, §1.3].
Theorem 2.1.
Let ( M, g ) be a time-oriented spacetime. Then the following statementsare equivalent:(i) ( M, g ) is globally hyperbolic(ii) ( M, g ) is isometric to R × Σ endowed with the line element ds = β dt − h t , (2.1) where t : R × Σ → R is the projection onto the first factor, β is a strictly positivesmooth function on R × Σ , and t h t is a one-parameter family of smooth Rieman-nian metrics. Furthermore, for all t ∈ R , { t } × Σ is a smooth, ( d − -dimensionalspacelike Cauchy surface. In this paper we will only consider globally hyperbolic spacetimes and we will always workwith the metric in the standard form (2.1). Furthermore, we will make the additionalassumption that Σ is compact, hence in particular closed. We shall adopt the conventionthat Greek (resp. Latin) indices are associated with quantities related to Σ (resp. to M ).Moreover, points of M are denoted with uppercase Latin letters, points of Σ are denotedwith lowercase Latin letters.For later convenience, we observe that, denoting by π : M → R and π : M → Σ the two natural projection maps, global hyperbolicity yields the isomorphisms of vectorbundles T M ≃ π ∗ T R ⊕ π ∗ T Σ , T ∗ M ≃ π ∗ T ∗ R ⊕ π ∗ T ∗ Σ , (2.2)where ⊕ indicates the Whitney sum between the pull-back bundles, see [16].For every point Y ∈ M , we denote by N ( Y ) the normal convex neighbourhood of Y consisting of all points X ∈ M such that there exists a unique geodesic γ connecting Y to X . Correspondingly we define the Synge’s world function as the scalar map σ ( X, Y ) := 12 ( λ − λ ) λ Z λ dλ g ( ˙ γ ( λ ) , ˙ γ ( λ )) , ∀ Y ∈ M, ∀ X ∈ N ( Y ) , (2.3)where ˙ γ ( λ ) is the tangent vector at γ ( λ ) to the geodesic γ connecting X to Y with theaffine parameter chosen in such a way that γ ( λ ) = Y and γ ( λ ) = X , see [19, §3]. It isimportant to stress that (2.3) corresponds also to half of the squared geodesic distance(in the Lorentzian sense) between X and Y .A globally hyperbolic spacetime is called static if it possesses an irrotational timelikeKilling vector field. This is equivalent to requiring that neither β nor h t in (2.1) dependon t . A globally hyperbolic spacetime is called ultrastatic if it is static and, in addition, β = 1 . Remark 2.2.
Observe that any ( d − -dimensional closed Riemannian manifold (Σ , h ) gives rise to a unique (up to isometries) ultrastatic globally hyperbolic d -dimensionalspacetime ( R × Σ , ds = dt − h ) , see [17]. Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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Let ∇ be the Levi-Civita connection associated with g and ρ g ( X ) := p | det g ab ( X ) | (2.4)be the Lorentzian density. Given a smooth real scalar field Φ : M → R , the Klein–Gordonequation is defined to be P Φ := ( (cid:3) g + ξ R + m )Φ = 0 , ξ ∈ R , and m ≥ , (2.5)where R is scalar curvature and (cid:3) g is the D’Alembert wave operator acting on scalarfunctions, defined by (cid:3) g := g ab ( X ) ∇ a ∇ b = [ ρ g ( X )] − ∂ a (cid:0) ρ g ( X ) g ab ( X ) ∂ b (cid:1) . (2.6)In our paper we will focus mainly on the wave equation, which is a special case of theKlein–Gordon equation obtained by setting ξ = m = 0 . In this section we have a twofold goal. First, we consider the d’Alembert wave operatoron a globally hyperbolic static spacetime with compact Cauchy surface and we show thatthe construction of the advanced and retarded fundamental solutions can be reduced tothat of the half-wave propagator for an associated partial differential operator. Secondly,we explicitly construct the latter, modulo an infinitely smoothing operator, by relyingon results and ideas from [6]. We start from the static scenario as it encompasses, inessence, all features of the most general case, hence providing an intuitive picture of theLorentzian extension, stripped of all technicalities here unnecessary.
Consider a globally hyperbolic static spacetime ( M, e g ) . On account of Theorem 2.1, M ≃ R × Σ and the metric tensor e g can be written as e g = β dt − e h, where both β and e h do not depend on t . Via a conformal rescaling we can construct anultrastatic metric g := β − e g = d t − h, where h = β − e h . The wave operators (cid:3) e g and (cid:3) g associated with e g and g , respectively,see (2.6), are related by β d (cid:3) e g β − d = (cid:3) g + V, (3.1)where d = dim M and V = β − − d (cid:16) ∆ h β − d (cid:17) . (3.2)Here and further on, ∆ h denotes the Laplace–Beltrami operator on Σ associated with h .Direct inspection of formula (3.1) reveals that constructing the advanced and retardedfundamental solutions e G ± associated to (cid:3) e g , cf. Theorem A.1, is equivalent to constructing G ± , their counterparts for the operator (cid:3) g + V = ∂ t − ∆ h + V. (3.3) Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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The two pairs of operators are related by the identity e G ± = β − d ◦ G ± ◦ β d . (3.4)The above argument shows that one can reduce the problem of constructing theadvanced and retarded fundamental solutions for the D’Alembert wave operator on astatic spacetime to the same problem for the operator (3.3) on an ultrastatic background,instead. Recently the first-named author, M. Levitin and D. Vassiliev have developed a geometricapproach to construct the wave propagator globally and invariantly on a closed Rieman-nian manifold [6], using Fourier integral operator techniques. We refer the reader to [6, 7]for a detailed review of the literature on global FIOs and their applications to hyperbolicpropagators.What presented in [6] is, effectively, the construction of advanced and retarded fun-damental solutions for the D’Alembert operator on an ultrastatic spacetime, up to anoperator with infinitely smooth Schwartz kernel. Dealing with an ultrastatic backgroundis much simpler than dealing with a general globally hyperbolic spacetime, in that in theformer case constructing the half-wave propagator e − it √− ∆ (3.5)is enough to the end of constructing the retarded-minus-advanced fundamental solution,see [6, Section 1].In the remainder of this Section, we will show how to adapt the results from [6] to thecase of static spacetimes, exploiting the reduction to an ultrastatic background presentedin Subsection 3.1. The extension to a wider class of globally hyperbolic spacetimes ispostponed to Section 4.Before doing so, let us recall in an abridged manner basic notions from symplecticgeometry. We refer the reader to [15, Vol. IV] for a comprehensive exposition.Let N be a smooth ( d − -dimensional manifold and let ω ∈ Ω ( N ) be the canonicalsymplectic form on the cotangent bundle T ∗ N . A diffeomorphism C : T ∗ N → T ∗ N iscalled a canonical transformation if it preserves ω under pullback. In addition we saythat C is positively homogeneous of degree k if C ( y, λ η ) = λ k C ( y, η ) for every λ > .The (twisted) graph Λ of C is a Lagrangian submanifold of T ∗ N × T ∗ N with respect tothe symplectic structure induced by ω . Similar definitions are given when the canonicaltransformation depends on additional parameters. This happens, remarkably, when C is the Hamiltonian flow generated by a Hamiltonian h ∈ C ∞ ( T ′ N ; R ) , where T ′ N := T ∗ N \ { } denotes the cotangent bundle with the zero section removed.For ( y, η ) ∈ T ′ N , we denote by ( x ∗ ( s ; y, η ) , ξ ∗ ( s ; y, η )) s ∈ R the solution to Hamilton’sequations ˙ x ∗ ( s ; y, η ) = h ξ (( x ∗ ( s ; y, η ) , ξ ∗ ( s ; y, η )) , ˙ ξ ∗ ( s ; y, η ) = − h x (( x ∗ ( s ; y, η ) , ξ ∗ ( s ; y, η )) , ( x ∗ (0; y, η ) , ξ ∗ (0; y, η )) = ( y, η ) . Here and further on the dot stands for differentiation with respect to the flow parameterwhereas the subscript ξ (resp. x ) denotes differentiation with respect to the ξ - (resp. x -)variable. It is easy to see that ( y, η ) ( x ∗ ( s ; y, η ) , ξ ∗ ( s ; y, η )) Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 8 is a one-parameter family of canonical transformations and, as such, it generates a one-parameter family of Lagrangian submanifolds Λ h ,s ⊆ T ′ N × T ′ N . If in addition the Hamil-tonian h is positively homogeneous of degree one in momentum, then the Lagrangiansubmanifold generated by the Hamiltonian flow is a conic submanifold of T ′ N × T ′ N . Definition 3.1.
We call phase function a function ϕ ∈ C ∞ ( R × N × T ′ N ; C ) which is • nondegenerate , i.e. det ϕ x α η β = 0 on C ϕ := { ( s, x ; y, η ) | ϕ η ( s, x ; y, η ) = 0 } ; (3.6) • positively homogeneous of degree one in momentum, i.e. ϕ ( s, x ; y, λη ) = λ ϕ ( s, x ; y, η ) ∀ λ > , ( s, x ; y, η ) ∈ R × N × T ′ N. Definition 3.2.
Given a Hamiltonian h positively homogeneous of degree , we say thata phase function ϕ = ϕ ( s, x, y, η ) is of class L h , and we write ϕ ∈ L h , if(i) ϕ | x = x ∗ ( s ; y,η ) = 0 ,(ii) ϕ x α | x = x ∗ ( s ; y,η ) = ξ ∗ α ( s ; y, η ) ,(iii) det ϕ x α η β (cid:12)(cid:12) x = x ∗ ( s ; y,η ) = 0 ,(iv) Im ϕ ≥ .Phase functions ϕ ∈ L h allow one to globally parameterise the Lagrangian submanifold Λ h ,s [20], namely, in local coordinates x and y and in a neighbourhood of a given pointof Λ h ,s , we have Λ h ,s ≃ { (cid:0) ( x, ϕ x ( s, x ; y, η )) , ( y, ϕ y ( s, x ; y, η )) (cid:1) | ( s, x ; y, η ) ∈ C ϕ } , (3.7)where C ϕ is given by (3.6).Specialising the above notation to the case when N = Σ is a closed Riemannian ( d − -manifold endowed with a Riemannian metric h , consider the ultrastatic spacetime ( M := R × Σ , dt − h ) and put E := ∆ h − V, (3.8)where V ∈ C ∞ (Σ; R ) is a time-independent smooth real potential.The operator E is a formally self-adjoint elliptic second order linear partial differentialoperator on C ∞ (Σ; C ) . As we are adding a zero order perturbation, the principal symbolof E E prin ( x, ξ ) := − h αβ ( x ) ξ α ξ β , ( x, ξ ) ∈ T ′ Σ , (3.9)coincides with that of ∆ h .It is well known that the singularities of the solutions to ( ∂ t − E ) f = 0 propagate alongnull geodesics. In this setting, these are nothing but the lift to M of the Hamiltonianflow on Σ associated with the Hamiltonian h ( x, ξ ) := p − E prin ( x, ξ ) , where t is equal tothe parameter s along the flow.Since Σ is compact, the spectrum of − E is discrete and accumulates to + ∞ , butunlike for the case of the Laplace–Beltrami operator, it is no longer guaranteed to be non-negative, as the presence of the potential V may bring about the appearance of negative Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 9 eigenvalues. Let us denote by ζ k the eigenvalues of − E and by v k the correspondingorthonormalised eigenfunctions, − E v k = ζ k v k , (3.10)labelling positive eigenvalues with positive index k and nonpositive eigenvalues with non-positive index k , in increasing order and with account of their multiplicities.Let Π − : L (Σ) → L (Σ) be the orthogonal projection onto the direct sum of the eigenspaces corresponding tonon-positive eigenvalues of − E and put Π + := Id − Π − . Clearly, Π − has finite rank, as E has at most a finite number of non-negative eigenvalues,all with finite multiplicity.Let us define U + ( t ) := X ζ k > e − it √ ζ k v k ( v k , · ) L (Σ) , (3.11)where √ ζ k is the positive square root of ζ k > . One can see that, for any choice of thesquare root √− E (with branch cut away from the spectrum) such that U + ( t ) = e − it √− E Π + , the operator U + ( t ) − e − it √− E = e − it √− E Π − is infinitely smoothing.This tells us that constructing U + ( t ) is equivalent to constructing e − it √− E up toan infinitely smoothing operator and, furthermore, that working modulo operators withinfinitely smooth kernel allows one to disregard the ambiguity in the definition of √− E on Π − L (Σ) and in the choice of the branch cut.Define L ± (Σ) := Π ± L (Σ) . The operator U + ( t ) satisfies, in a distributional sense, ( − i∂ t + √− E ) U + ( t ) = 0 , (3.12a) U + (0) | L (Σ) = Id L (Σ) . (3.12b)Our strategy consists in approximating the operator U ( t ) := U + ( t ) | L (Σ) ⊕ Id L − (Σ) : L (Σ) ⊕ L − (Σ) → L (Σ) . (3.13)by a single oscillatory integral, global in space and in time. The operator U ( t ) will(a) satisfy equations (3.12) on L (Σ) up to an infinitely smoothing operator and(b) coincide with U + ( t ) , hence with e − it √− E up to an infinitely smoothing operator. Theorem 3.3.
The Schwartz kernel u ( t, x, y ) = X ζ k > e − it √ ζ k v k ( x ) v k ( y ) + X ζ k ≤ v k ( x ) v k ( y ) (3.14) Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 10 of the operator (3.13) can be written, modulo an infinitely smooth function in all variables,as a single oscillatory integral u ( t, x, y ) mod C ∞ = 1(2 π ) d − Z T ∗ y Σ e i ϕ ( t,x ; y,η ) a ( t ; y, η ) χ h ( t, x ; y, η ) w ( t, x ; y, η ) dη, (3.15) global in space and in time, where(a) ϕ is a phase function of class L h , with h ( y, η ) = p h αβ ( y ) η α η β ;(b) a ∈ C ∞ ( R , S ( T ′ Σ)) , namely it is a polyhomogeneous symbol of order zero dependingsmoothly on t , a ∼ ∞ X j =0 a − j , a − j ( t ; y, λ η ) = λ − j a − j ( t ; y, η ) ∀ λ > (c) χ h is a smooth cut-off subordinated to h , namely, an infinitely smooth function on R × Σ × T ′ Σ satisfying(i) χ h ( t, x ; y, η ) = 0 on { ( t, x ; y, η ) | | h ( y, η ) | ≤ / } ;(ii) χ h ( t, x ; y, η ) = 1 on the intersection between { ( t, x ; y, η ) | | h ( y, η ) | ≥ } and anyconical neighbourhood of { ( t, x ∗ ( t ; y, η ); y, η ) } ;(iii) χ h ( t, x ; y, α η ) = χ h ( t, x ; y, η ) for α ≥ on { ( t, x ; y, η ) | | h ( y, η ) | ≥ } ;(d) the weight w is defined as w ( t, x ; y, η ) := [ ρ h ( x )] − / [ ρ h ( y )] − / (cid:2) det (cid:0) ϕ x α η β ( t, x ; y, η ) (cid:1)(cid:3) / , (3.16) where ρ h is the Riemannian density (cf. (2.4) ) and the branch of the complex root ischosen so that arg (cid:2) det (cid:0) ϕ x α η β ( t, x ; y, η ) (cid:1)(cid:3) / (cid:12)(cid:12)(cid:12) t =0 = 0 . Remark 3.4.
Let us point out that the weight (3.16) is a smooth scalar function in thevariables t , x and η and a smooth ( − -density in the variable y . The choice of powers ofthe Riemannian density serves the purpose of making the integral kernel (3.15) a scalarfunction in all variables. The weight w is a crucial element in our construction in that itensures the right covariance properties of (3.15) and its nondegeneracy – a consequenceof using complex-valued phase functions of class L h – is key to performing a constructionglobal in time, circumventing topological obstructions offered by caustics. Proof of Theorem 3.3.
Denote by F λ → t [ f ]( t ) = b f ( t ) = Z + ∞−∞ e − itλ f ( λ ) dλ the Fourier transform and by F − t → λ [ b f ]( t ) := f ( λ ) = 12 π Z + ∞−∞ e itλ b f ( t ) dt the inverse Fourier transform.Assume one has constructed u ∈ C ∞ ( R ; D ′ (Σ x × Σ y )) such that Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 11 ( ∂ t − E ( x ) ) u ( t, x, y ) = 0 mod C ∞ , where the superscript ( x ) indicates that E actsin the variable x ;2. u satisfies Z Σ u (0 , x, y ) ( · ) ρ ( y ) dy = Id mod C ∞ ;
3. for every ψ ∈ C ∞ ( R ) , F − t → λ [ ψ ( t ) u ( t, x, y )] = O ( | λ | −∞ ) as λ → −∞ . How to do so will be explained in Subsection 3.2.1. Then our theorem follows from anadaptation of results from [6].
Corollary 3.5.
Let M be a static spacetime. Then, the integral kernel of the operators G ,t and G ,t defined in accordance with Proposition A.3 can be represented, modulo C ∞ ,as the sum of two oscillatory integrals of the form (3.15) , global in space and time.Proof. Let E be given by (3.8) for the choice of V (3.2). The operators G ,t and G ,t associated with the normally hyperbolic operator ∂ t − E on the ultrastatic spacetime R × Σ are given by G ,t = cos( √− E ( t − t )) , G ,t = sin( √− E ( t − t ))( √− E ) − + t Π , where Π is the projection onto the kernel of E , see [6, Section 1] and [11, Chap. 3].Hence, the statement follows from Theorem 3.3 combined with (3.4).Observe that, unlike in the construction of the half-wave propagator, there is noambiguity in defining G ,t and G ,t by means of spectral calculus even if the spectrum of − E has non-empty intersection with the negative real line. The reason lies in the factthat the functions cos( x ) and sin( x ) x are even. In the remainder of this subsection, we will explain how to construct the oscillatoryintegral in the RHS of (3.15), adapting [6, Section 5] to the case in hand.The starting main idea consists in fixing a distinguished phase function and settingout an algorithm for the construction of a in a global, invariant fashion. Definition 3.6 (Levi-Civita phase function [6, Definition 4.1]) . We define the
Levi-Civitaphase function to be ϕ ǫ ( t, x ; y, η ) := − h ξ ∗ ( t ; y, η ) , grad z [dist ( x, z )] (cid:12)(cid:12) z = x ∗ ( t ; y,η ) i + i ǫ h ( y, η ) dist ( x, x ∗ ( t ; y, η )) (3.17)when x lies in a Σ -geodesic neighbourhood of x ∗ ( t ; y, η ) and continued smoothly and arbi-trarily elsewhere, in such a way that Im( ϕ ) ≥ . Here dist Σ is the (Riemannian) geodesicdistance on Σ , h · , · i = h − ( · , · ) and ǫ > is a positive parameter pre-multiplying theimaginary part of ϕ . Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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The symbol a is determined as follows. Step 1 . Set χ ≡ — contributions from regions where χ = 1 are smooth, as one canestablish via a stationary phase argument — and act on the oscillatory integral (3.15)with the operator ∂ t − E ( x ) . This yields a new oscillatory integral I ϕ ( a ) := 1(2 π ) d − Z T ∗ y Σ e i ϕ ( t,x ; y,η ) a ( t, x ; y, η ) w ( t, x ; y, η ) dη (3.18)which has the same form of (3.15) though with a new amplitude a ( t, x ; y, η ) := e − i ϕ w ( ∂ t − E ) (cid:2) e i ϕ a w (cid:3) . (3.19)Observe that the amplitude a ∈ C ∞ ( R × Σ; S ( T ′ Σ)) depends on the variable x . Step 2 . As a next step, we exclude the dependence on x of the amplitude a , namely,we find b ∈ C ∞ ( R ; S ( T ′ Σ)) such that I ϕ ( a ) = I ϕ ( b ) mod C ∞ . This is achieved by theso-called reduction of the amplitude , which exploits suitably devised differential-evaluationoperators. For every α = 1 , ..., d − , put L α : C ∞ ( R × Σ; S j ph ( T ′ Σ)) → C ∞ ( R × Σ; ( S j ph T ′ Σ)) ,L α := (cid:2) ( ϕ xη ) − (cid:3) α β ∂∂x β , (3.20)where ϕ − xη is defined in accordance with [ ϕ − xη ] αβ ϕ x β η γ = δ αγ . In addition, for all k ∈ N , let S − k : C ∞ ( R × Σ; S j ph ( T ′ Σ)) → C ∞ ( R ; S j − k ph ( T ′ Σ)) bedefined as S f := f | x = x ∗ ( t ; y,η ) , (3.21a) S − k f := S i w − ∂∂η β w X ≤| α |≤ k − ( − ϕ η ) α α ! ( | α | + 1) L α L β k f , k > . (3.21b)Bold Greek letters in (3.21b) denote multi-indices in N n − , α = ( α , . . . , α n − ) , | α | = P n − j =1 α j and ( − ϕ η ) α := ( − | α | ( ϕ η ) α . . . ( ϕ η n − ) α n − . All derivatives act on whatever isto the right, unless otherwise specified. The operator (3.21b) is well defined, because thedifferential operators L α commute, cf. [6, Lemma A.2].Denoting by a ∼ P ∞ j =0 a − j the asymptotic polyhomogeneous expansion of a , weconstruct a polyhomogeneous symbol b whose homogeneous components are defined as b l := X − j − k = l S − k a − j , l = 2 , , , − , . . . . (3.22)This gives [6, Appendix A] π ) d − Z T ∗ y Σ e i ϕ a w dη = 1(2 π ) d − Z T ∗ y Σ e i ϕ b w dη mod C ∞ . (3.23) Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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We adopt the terminology from [6] and call the operator S ∼ P ∞ k =0 S − k amplitude-to-symbol operator . Clearly, the symbol b is x -independent. Step 3.
Impose ( ∂ t − E ( x ) ) u = I ϕ ( b ) = 0 mod C ∞ . In view of equations (3.19) and (3.22), this amounts to equating each asymptotic homo-geneous component of b to zero: b l ( t ; y, η ) = 0 , l = 2 , , , − , .... (3.24)This yields a hierarchy of transport equations – ordinary differential equations in thevariable t – whose unknowns are the (scalar) homogeneous components a − k of a .Initial conditions a − k (0; y, η ) for (3.24) are established by requiring that u (0 , x, y ) is,modulo a smooth function, the integral kernel of the identity operator.Note that the identity operator appears in our construction as an invariant pseudod-ifferential operator with integral kernel Z T ∗ y Σ e i φ ( x ; y,η ) s ( y, η ) χ h (0 , x ; y, η ) (cid:2) det φ xη (cid:3) / (cid:18) ρ h ( y ) ρ h ( x ) (cid:19) / dη mod C ∞ , (3.25)where φ ( x ; y, η ) := ϕ (0 , x ; y, η ) and s ∈ S ( T ′ Σ) , so that the initial conditions read a − k (0; y, η ) = s − k ( y, η ) . We refer the reader to [6, Section 6] for a detailed analysis of the operator (3.25).The above algorithm identifies uniquely and invariantly an element u ∈ C ∞ ( R ; D ′ (Σ x × Σ y )) satisfying conditions 1.–3. in the proof of Theorem 3.3. In this section we generalise the construction of the global propagator with FIO methodsto the case of a general globally hyperbolic spacetime ( M, g ) with compact Cauchy surface.For a general global hyperbolic spacetime, the strategy presented in Section 3 doesnot work any more, because space and time intertwine in an essential way, and onecannot identify a time-independent analogue of the operator E . In order to construct thewave propagator, one needs to abandon the operator (3.13) and adopt a more abstractapproach.The first crucial step towards our goal rests in the identification of a Lorentziananalogue of the Levi-Civita phase function, cf. Section 3.2.
Let ( M, g ) be a globally hyperbolic spacetime with compact Cauchy surface of dimension d ≥ , realised as R × Σ , cf. Theorem 2.1, and let Σ s ≃ { s } × Σ be an arbitrary but fixedCauchy surface. Notation 4.1.
Let Y = ( s, y ) ∈ M and let ι s : Σ s → M be the embedding of Σ s into M . Given η ∈ T ∗ y Σ we denote by η + the unique future pointing null covector in T ∗ Y M such that ι ∗ s η + = η . Furthermore, we put b η + := η + k η k Σ s , where k η k Σ s = q h αβs ( y ) η α η β and h s := ι ∗ Σ s g . Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 14Definition 4.2.
Let Y = ( s, y ) ∈ M . We call Levi-Civita flow the map ς ( e X ∗ ( ς ; s, y, η ) , e Ξ ∗ ( ς ; s, y, η )) , (4.1)where • e X ∗ ( · ; s, y, η ) : ς e X ∗ ( ς ; s, y, η ) is the unique null geodesic stemming from Y withinitial cotangent vector b η + , parameterised by proper time; • e Ξ ∗ ( ς ; s, y, η ) is the parallel transport along e X ∗ ( · ; s, y, η ) of η + , from Y to e X ∗ ( ς ; s, y, η ) .The Levi-Civita flow satisfies, by construction, ( e X ∗ ( s ; s, y, η ) , e Ξ ∗ ( s ; s, y, η )) = ( Y, η + ) and ( e X ∗ ( ς ; s, y, λ η ) , e Ξ ∗ ( ς ; s, y, λ η )) = ( e X ∗ ( ς ; s, y, η ) , λ e Ξ ∗ ( ς ; s, y, η )) , (4.2)for every λ > .We have the following standard result. Lemma 4.3.
The Levi-Civita flow can be parameterised using the global time coordinate t defined in accordance with Theorem 2.1 (ii).Proof. Let e X ∗ ( ς ; s, y, η ) = ( τ ∗ ( ς ; s, y, η ) , x ∗ ( ς ; s, y, η )) , where τ ∗ ( ς ; s, y, η ) := t ( e X ∗ ( ς ; s, y, η )) and x ∗ ( ς ; s, y, η ) := π Σ τ ∗ ( ς ; s,y,η ) ( e X ∗ ( ς ; s, y, η )) is the projection of e X ∗ ( ς ; s, y, η ) onto Σ τ ∗ ( ς ; s,y,η ) . Denoting by d e X ∗ d ς the tangent vector to e X ∗ ( · ; s, y, η ) , we have g ( d e X ∗ d ς , d e X ∗ d ς ) = g − ( b η + , b η + ) = 0 . Since η ∈ T ′ Σ , this implies that d τ ∗ d ς cannot vanish.In view of Lemma 4.3, in what follows we shall denote with ( X ∗ ( t ; s, y, η ) , Ξ ∗ ( t ; s, y, η )) the Levi-Civita flow parameterised by t .We are now in a position to define our Lorentzian phase function. Definition 4.4.
Let ǫ > be a positive parameter, let X = ( t, x ) , Y = ( s, y ) ∈ M and let ( X ∗ ( t ; s, y, η ) , Ξ ∗ ( t ; s, y, η )) be the Levi-Civita flow. We define the Lorentzian Levi-Civitaphase function to be the infinitely smooth function ϕ : M × R × T ′ Σ → C defined by ϕ ( τ, x ; s, y, η ) := −h Ξ ∗ ( τ ; s, y, η ) , grad Z σ ( X, Z ) | Z = X ∗ ( τ ; s,y,η ) i + i ǫ k η k Σ s σ ( X, X ∗ ( τ ; s, y, η )) , (4.3)if X lies in a geodesic normal neighbourhood of X ∗ ( τ ; s, y, η ) , and smoothly continuedelsewhere in such a way that the imaginary part is positive. Here σ is the Ruse–Syngeworld function, defined in accordance with (2.3). Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 15Remark 4.5.
Observe that the Lorentzian Levi-Civita phase function ϕ can be equiva-lently recast as ϕ ( τ, x ; s, y, η ) = − h ι ∗ Σ τ Ξ ∗ ( τ ; s, y, η ) , grad z (dist Σ τ ( x, z ) ) (cid:12)(cid:12) z = x ∗ ( τ ; s,y,η ) i + i ǫ k η k Σ s dist τ ( x, x ∗ ( τ ; s, y, η )) , (4.4)where dist Σ τ is the (Riemannian) geodesic distance on Σ τ := { τ } × Σ ≃ Σ . Whenever ( M, g ) is ultrastatic the latter equation coincides with equation (3.17) – in which case k η k Σ τ = h ( y, η ) for every τ .The following proposition tells us that Definition 4.4 identifies a function ϕ withessentially the same properties of a phase function of class L h – cf. Definition 3.1.
Proposition 4.6.
The Lorentzian Levi-Civita phase function (4.3) satisfies the followingproperties:(a) The function ϕ is positively homogeneous of degree one in momentum, i.e. ϕ ( τ, x ; s, y, λη ) = λ ϕ ( τ, x ; s, y, η ) , ∀ λ > . (b) Im ϕ ≥ and, moreover, ϕ ( τ, x ∗ ( τ ; s, y, η ); s, y, η ) = 0 , (4.5a) ϕ x α ( τ, x ∗ ( τ ; s, y, η ); s, y, η ) = ξ ∗ α ( τ ; s, y, η ) , (4.5b) det ϕ x α η β ( τ, x ∗ ( τ ; s, y, η ); s, y, η ) = 0 , (4.5c) where ξ ∗ := ι Σ s Ξ ∗ .(c) If we define Φ := { ( t, x ∗ ( t ; s, y, η ); s, y, η ) | t, s ∈ R , ( y, η ) ∈ T ∗ Σ } , C ϕ := { ( τ, x ; s, y, η ) ∈ M × R × T ′ Σ | ϕ η ( s, x ; y, η ) = 0 } , then Φ ⊆ C ϕ . Furthermore, there exists a neighbourhood W of Φ such that C ϕ ∩ ( W \ Φ) = ∅ .Proof. (a) The function ϕ is positively homogeneous in η of degree because X ∗ and Ξ ∗ are positively homogeneous of degree and , respectively – cf. equation (4.2).(b) The inequality Im ϕ ≥ follows at once from (4.4). So let us prove (4.5). Recallthe following well-known identities for the Ruse–Synge world function σ [19, Sec. 4.1]: σ ( X, X ) = 0 , (4.6a) d σ ( X, X ) = 0 , (4.6b) [ ∇ a ∇ ′ b σ ( X, X ′ )] | X = X ′ = [ ∇ a ∇ b σ ( X, X ′ )] | X = X ′ = − g ab ( X ) , (4.6c) [ ∇ a ∇ ′ b ∇ ′ c σ ( X, X ′ )] | X = X ′ = 0 . (4.6d)Here ∇ ′ indicates that the Levi-Civita connection ∇ on ( M, g ) acts on the second argu-ment of σ . Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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Since d σ ( X, X ) = 0 , (4.5a) is satisfied. Moreover, (4.6b) and (4.6c) ensure that (4.5b)holds. Indeed, ϕ x α = ∇ α ϕ = − g βν ∇ α ∇ ′ β σ Ξ ∗ ν + iǫ k η k Σ s ∇ α σ x = x ∗ ( t ; s,y,η ) = ξ ∗ α . Finally let us prove that (4.5c). In view of [25, Cor. 2.4.5], it is enough to check that
Im( ϕ x α x β ( τ, x ∗ ( τ ; s, y, η ) , s, y, η )) > as a bilinear form. A direct calculation shows that ǫ Im ϕ x α x β = k η k Σ s ∂ x β ∇ x α σ = k η k Σ s ∇ x β ∇ x α σ + k η k Σ s Γ aβα ∇ x a σ . The above equation, combined with (4.6a)– (4.6d), implies that
Im( ϕ x α x β ( τ, x ∗ ( τ ; s, y, η ) , s, y, η )) = ǫ k η k Σ s h αβ , which is positive definite.(c) Differentiating (4.5a) with respect to η β we get ∂ η β [ ϕ ( t, x ∗ ( t ; s, y, η ); s, y, η )]= ϕ η β ( t, x ∗ ( t ; s, y, η ); s, y, η ) + ϕ x α ( t, x ∗ ( t ; s, y, η ); s, y, η )( x ∗ ) αη β ( t ; s, y, η )= ϕ η β ( t, x ∗ ( t ; s, y, η ); s, y, η ) + ξ ∗ α ( t, x ∗ ( t ; s, y, η ); s, y, η )( x ∗ ) αη β ( t ; s, y, η ) , where we used equation (4.5b). The task at hand is to show that ξ ∗ α ( t, x ∗ ( t ; s, y, η ); s, y, η )( x ∗ ) αη β ( t ; s, y, η ) = 0 , from which Φ ⊆ C ϕ descends. For that, we observe that Ξ ∗ ( t ; s, y, η ) = k η k Σ s b Ξ ∗ ( t, s, y, η ) ,where b Ξ ∗ ( t ; s, y, η ) is the cotangent vector to the geodesic γ : t X ∗ ( t ; s, y, η ) , b Ξ ∗ a = g ab d( X ∗ ) b d t . Being the geodesic flow Hamiltonian, it follows that the canonical -form Θ = Ξ a d X a isconserved along the geodesic. Observing that Ξ ∗ differs from b Ξ ∗ by a constant factor, weconclude that ( η + ) a d Y a = Ξ ∗ a d( X ∗ ) a = Ξ ∗ a (cid:2) ( X ∗ ) aY b d Y b + ( X ∗ ) aη b d η b (cid:3) . The latter equation implies Ξ ∗ a ( X ∗ ) aY b = ( η + ) b as well as ∗ a ( X ∗ ) aη b = ξ ∗ α ( x ∗ ) αη b , wherewe have used the fact that X ∗ ( t ; s, y, η ) = ( t, x ∗ ( t, s, y, η )) – cf. Lemma 4.3.Finally, let us assume by contradiction that for all neighbourhoods W of Φ we have C ϕ ∩ ( W \ Φ) = ∅ . Then, there exists a sequence ( t n , x n ; s n , y n , η n ) ∈ C ϕ converging to ( t, x ∗ ; s, y, η ) ∈ Φ . A standard Taylor expansion argument shows that this implies det ϕ x α η β ( t, x ∗ ; s, y, η ) = 0 , which contradicts (4.5c). Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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The Lorentzian Levi-Civita phase function can be exploited to construct a suitableLorentzian analogue of the operator (3.13).Let Σ s be a fixed Cauchy surface. Let U ( t, s ) : C ∞ (Σ s ) → C ∞ (Σ t ) (4.7)be the linear operator uniquely defined (modulo an infinitely smoothing operator) by thefollowing properties. Property 1 . The Schwartz kernel u ( t, x, y ; s ) of U ( t, s ) satisfies (cid:3) ( t,x ) u ( t, x, y ; s ) = 0 , u ( s, x, y ; s ) = δ Σ s ( x, y ) , (4.8)where δ Σ s ( x, y ) denotes the integral kernel of the identity operator on C ∞ (Σ s ) . Property 2 . The Schwartz kernel u ( t, x, y ; s ) , seen as a distribution in D ′ ( M × M ) satisfies WF( u ) = { ( X, Y, k X , − k Y ) ∈ T ∗ ( M × M ) \ { } | ( X, k X ) ∼ ( Y, k Y ) and k X ⊲ } (4.9)where WF denotes the wavefront set, ∼ means that the point X and Y are connectedby a lightlike geodesic γ whose tangent vector at X is k X , and k Y is the paralleltransport of k X from X to Y along γ . The symbol ⊲ means that k X is future directed.The operator U ( t, s ) , effectively, maps initial data C ∞ (Σ s ) on the ‘initial’ Cauchy sur-face to spacelike compact wave solutions in C ∞ ( M ) . The technology developed in earliersections, allows us to construct U ( t, s ) explicitly and invariantly as a global oscillatoryintegral, up to an infinitely smoothing operator. Theorem 4.7.
Let ( M, g ) ≃ ( R × Σ , βdt − h t ) be a globally hyperbolic spacetime withcompact Cauchy surface. Then the Schwartz kernel u ( t, x, y ; s ) of the operator U ( t, s ) canbe written, modulo an infinitely smooth function in all variables, as a single oscillatoryintegral u ( t, x, y ; s ) mod C ∞ = 1(2 π ) d − Z T ∗ y Σ e i ϕ ( t,x ; s,y,η ) a ( t ; s, y, η ) χ ( t, x ; s, y, η ) w ( t, x ; s, y, η ) dη, (4.10) global in space x and in time t , where(a) ϕ is the Lorentzian Levi-Civita phase function (4.4) ;(b) a is a polyhomogeneous symbol of order zero depending smoothly on t , a ∼ ∞ X j =0 a − j , a − j ( t ; s, y, λ η ) = λ − j a − j ( t ; s, y, η ) ∀ λ > (c) χ is an infinitely smooth function on M × R × T ′ Σ satisfying(i) χ ( t, x ; s, y, η ) = 0 on { ( t, x ; s, y, η ) | k η k Σ s ≤ / } ; Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 18 (ii) χ ( t, x ; s, y, η ) = 1 on the intersection between { ( t, x ; s, y, η ) | k η k Σ s ≥ } andany conical neighbourhood of { ( t, x ∗ ( t ; s, y, η ) , s, y, η ) } where ( t, x ∗ ( t ; s, y, η )) = X ∗ ( t ; s, y, η ) is the flow defined in equation (4.1) – cf. Lemma 4.3.(iii) χ ( t, x ; s, y, α η ) = χ ( t, x ; s, y, η ) for α ≥ on { ( t, x ; s, y, η ) | k η k Σ s ≥ } ;(d) the weight w is defined as w ( t, x ; s, y, η ) := [ ρ h t ( x )] − / [ ρ h s ( y )] − / (cid:2) det (cid:0) ϕ x α η β ( t, x ; s, y, η ) (cid:1)(cid:3) / , (4.11) where ρ h is the Riemannian density on the Cauchy surface (see (2.4) ) and the branchof the complex root is chosen so that arg (cid:2) det (cid:0) ϕ x α η β ( t, x ; s, y, η ) (cid:1)(cid:3) / (cid:12)(cid:12)(cid:12) t = s = 0 . Proof.
The proof is a straightforward adaptation of results from [6], combined with theproperties of the Lorentzian Levi-Civita phase function.Let us stress that, thanks to the adoption of the Levi-Civita phase function, (astraightforward adaptation of) the algorithm presented in Subsection 3.2.1 uniquely deter-mines the scalar, global symbol a of the operator U ( t, s ) . Furthermore, all homogeneouscomponents a − j , j ≥ , of a are also guaranteed to be scalar. In Section 4 we have seen how the results of [6] can be suitably generalised to an arbitraryglobally hyperbolic spacetime ( M, g ) with compact Cauchy surfaces and of arbitrarydimension d = dim M ≥ . In this section we shall discuss an application of our resultsby showing that starting from the wave propagator constructed in Theorem 4.7 one canobtain a representation of the integral kernel of a Hadamard state up to smooth errors.Hadamard states play a distinguished role in the algebraic formulation of quantumfield theory, see e.g. [5], and [18] for a recent review. Given a globally hyperbolicspacetime ( M, g ) and a normally hyperbolic second order linear partial differential op-erator P : C ∞ ( M ) → C ∞ ( M ) , a Hadamard two-point function is fully specified by abi-distribution ω ∈ D ′ ( M × M ) such that its integral kernel ω ( X, X ′ ) satisfies (cid:26) ( P ⊗ I ) ω ( X, X ′ ) = ( I ⊗ P ) ω ( X, X ′ ) ∈ C ∞ ( M × M ) ,ω ( X, X ′ ) − ω ( X ′ , X ) − i G ( X, X ′ ) ∈ C ∞ ( M × M ) , (5.1)and WF( ω ) = { ( X, Y, k X , − k Y ) ∈ T ∗ ( M × M ) \ { } | ( X, k X ) ∼ ( Y, k y ) and k X ⊲ } (5.2)where G = G + − G − is the retarded-minus-advanced fundamental solution introducedin Lemma A.2. The notation in (5.2) was introduced after (4.9). In a quantum fieldtheory Hadamard bi-distributions, subject to a further positivity requirement, play adistinguished role in that they identify a natural class of physically meaningful Gaussianquantum states.The characterisation of Hadamard bidistributions via (5.1) and (5.2) admits the fol-lowing alternative and equivalent description proven by Radzikowski in [22, 23]. Consider Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 19 ω ∈ D ′ ( M × M ) satisfying equation (5.1) and let ω ( X, X ′ ) indicate the associated in-tegral kernel. The bi-distribution satisfies (5.2) if, for every geodesically convex opensubset O ⊆ M and for every X, X ′ ∈ O , ω ( X, X ′ ) = lim ǫ → + ( H ǫ ( X, X ′ ) + W ( X, X ′ )) , (5.3)where the limit is understood in the weak sense. In the above expression W is an infinitelysmooth function in C ∞ ( O × O ) and H ǫ ( X, X ′ ) := β n U d ( X, X ′ ) σ d − ǫ ( X, X ′ ) + γ d V d ( X, X ′ ) log (cid:18) σ ǫ ( X, X ′ ) ℓ (cid:19) , (5.4)where • ℓ ∈ R is a reference length to make the logarithm dimensionless. • σ ǫ ( X, X ′ ) = σ ( X, X ′ ) + iǫ ( t ( X ) − t ( X ′ )) + ǫ . Here σ is the geodesic distancebetween X and X ′ while t : R × Σ → R is the global temporal function introducedin Theorem 2.1. • β d and γ d are normalisation constants such that β d = − Γ (cid:0) d (cid:1) π d , γ d = ( − d − d π d Γ( d ) , d even (5.5) β d = ( − d +12 π − d (cid:0) − d (cid:1) , γ d = 0 , d odd . (5.6) • U d ( X, X ′ ) , V d ( X, X ′ ) ∈ C ∞ ( O × O ) admit an asymptotic expansion as power seriesof the geodesic distance U d = ∞ X k =0 U d,k σ k , V d = ∞ X k =0 V d,k σ k . (5.7)All coefficients U d,k and V d,k are smooth and they are determined from the equa-tion of motion (5.1) via a hierarchy of transport equations with prescribed initialconditions, see [22, 23].With a slight abuse of notation one refers to H ≡ H ǫ as the Hadamard parametrix.From a rigorous standpoint, H identifies a distribution b H ∈ D ′ ( O × O ) via b H ( f, f ′ ) := lim ǫ → + H ǫ ( f, f ′ ) , ∀ f, f ′ ∈ C ∞ ( O ) . The above discussion can be summarised in the following theorem (see [22, 23] for theproof).
Theorem 5.1.
Let ( M, g ) be an d -dimensional globally hyperbolic spacetime and let P be a normally hyperbolic second order linear partial differential operator. For any ω ∈D ′ ( M × M ) satisfying (5.1) , the following two conditions are equivalent:1. the singular structure of ω is codified by (5.2) , Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 20
2. in every geodesically convex open neighbourhood
O ⊆ M the integral kernel of ω isof the form (5.3) .If one of these two conditions is met, ω is said to be of Hadamard form . It is worth observing that, given a bi-distribution ω ∈ D ′ ( M × M ) , a direct use of(5.3) to verify the Hadamard condition is often impractical, since it involves the explicitcontrol of the integral kernel of ω in every geodesic neighbourhood. This is why (5.3)plays a key role in many applications of Hadamard states, cf. [18], [10].In the following we shall prove that there exists an explicit connection between ourrepresentation of the wave propagator, Hadamard states and the associated Hadamardparametrix, (5.4). In this analysis we shall work under a few additional assumptions,namely • ( M, g ) is a d -dimensional ultrastatic globally hyperbolic spacetime with compactCauchy surface, • the normally hyperbolic second order linear partial differential operator P rulingthe dynamics is of the form ∂ t − ∆ where ∆ is the Laplace–Beltrami operator actingon scalars associated with the metric h , cf. equation (2.1).This choice is made only for computational and presentation simplicity. Mutatismutandis , our next results hold true replacing − ∆ with E = − ∆ + V , where V is atime-independent smooth potential.Under the above assumptions, we have the following. Theorem 5.2.
Consider the operator Ω( τ, s ) := 12 ( √− ∆) − / e − i √− ∆( τ − s ) , s, τ ∈ R , (5.8) where ( √− ∆) − / is the square root of the pseudoinverse of − ∆ , cf. [24, Chapter 2,Section 2]. Then the integral kernel ω of (5.8) is of Hadamard form as a distribution in D ′ ( M × M ) .Proof. To start with, observe that √− ∆ is an elliptic pseudodifferential operator of order − . Hence Ω( τ, s ) is a Fourier integral operator whose amplitude can be determined (non-invariantly) by [26, Thm. 18.2].It is easy to see that ω satisfies ( P ⊗ I ) ω = ( I ⊗ P ) ω ∈ C ∞ ( M × M ) . Put Λ( f , f ) := ω ( f , f ) − ω ( f , f ) , ∀ f , f ∈ D ( M ) . Of course, Λ defines a distribution in D ′ ( M × M ) . Denoting a generic point of M ≃ R × Σ by X = ( t, x ) as above, the integral kernel Λ( X, X ′ ) satisfies ( P ⊗ I )Λ( X, X ′ ) = ( I ⊗ P )Λ( X, X ′ ) ∈ C ∞ ( M × M ) (5.9) Λ( X, X ′ ) | t = t ′ ∈ C ∞ (Σ × Σ) , ∂ t Λ( X, X ′ ) | t = t ′ − δ Σ ( x, x ′ ) ∈ C ∞ (Σ × Σ) , (5.10)where δ Σ ( x, x ′ ) is the integral kernel of the identity operator acting on scalar functionson the Cauchy surface { t } × Σ . Hence − i Λ must coincide up to smooth terms with theretarded-minus-advanced fundamental solution G , cf. Lemma A.2.
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Finally we can determine the wavefront set of ω since, being ( √− ∆) − / an ellipticoperator, WF( ω ) = WF( u ) where u is the integral kernel of U ( τ − s ) . Now, on accountof [6], we know that WF( u ) is of the form (5.2). Thus, by Theorem 5.1 we can draw thesought-after conclusion. Remark 5.3.
Theorem 5.2 can in principle be recovered indirectly from results availablein the literature — see, for example, [27, Sections 5.1 and 9.1] and [2, Section 6.2].Theorem 5.2 ensures that the Fourier Integral Operator Ω in equation (5.8) identifiesa distribution ω of Hadamard form. Hence, on account of Theorem 5.1, we can concludethat in every geodesically convex open neighbourhood O ⊆ M the integral kernel of ω differs from the Hadamard one, cf. (5.4), by a smooth remainder.In view of the results of Section 3, we can write Ω( t, mod Ψ −∞ = Z Σ Z T ∗ y Σ e iϕ ( t,x ; y,η ) (2 π ) n − f ( t ; y, η ) χ ( t, x ; y, η ) w ( t, x ; y, η ) ( · ) ρ h ( y ) dy dη. (5.11)An explicit formula for the scalar symbol f can be obtained following the same procedureoutlined in Section 3, albeit with a different initial condition. In what follows we show thatthere exists a concrete relation between the coefficients in the homogeneous expansionof the amplitude f from (5.11) and those appearing in (5.7) for the expansion of theHadamard parametrix (5.4).Consider the integral kernel I f ( t, x ; y ) := 1(2 π ) n − Z T ∗ y Σ e iϕ ( t,x ; y,η ) f ( t ; y, η ) χ ( t, x ; y, η ) w ( t, x ; y, η ) dη, (5.12)with X = ( t, x ) ∈ M and y ∈ Σ . The task at hand is to perform a local expansion of thisintegral kernel in a geodesic neighbourhood of (0 , y ) in M . In what follows, we identifyall Cauchy surfaces { t } × Σ , t ∈ R , with Σ and we choose geodesic normal coordinatescentred at y . Furthermore, we adopt the same coordinates for x and y .Since we are only interested in obtaining a local expansion for x close to y and small t , when carrying out the algorithm to compute f we can choose ϕ to be ϕ ( t, x ; y, η ) = h exp − y ( x ) , η i − k η k t (5.13)in normal coordinates centred at y , where k · k is the Euclidean norm, h · , ·i the Euclideanpairing, and exp y the exponential map at y . Substituting (5.13) into (3.16) we obtain w ( t, x ; y, η ) = [ ρ h ( x )] − / . (5.14)As a consequence, the integral kernel (5.12) turns into I f ( t, x ; y ) = 1(2 π ) d − [ ρ h ( x )] Z T ∗ y Σ e i h exp − y ( x ) ,η i− i k η k t f ( t ; y, η ) χ ( t, x ; y, η ) dη (5.15)where x and t are now such that ( x, t ) lies in a convex geodesic neighbourhood of (0 , y ) .Furthermore, since ( M, g ) is ultrastatic, we can assume without loss of generality that t is positive. In addition we can drop the cut-off χ since it is equal to in a neighbourhoodof the region where the phase is stationary. Dropping it does not affect the final result. Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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Let ζ : [0 , + ∞ ) → R be a smooth cut-off such that ζ ( r ) = 0 for r ≤ / and ζ ( r ) = 1 for r ≥ . Then, for every N ∈ N we have I f mod S − N − = 1(2 π ) d − [ ρ h ( x )] Z T ∗ y Σ e i h exp − y ( x ) ,η i− i k η k t N X k =0 f − − k ( t ; y, η ) ζ ( k η k ) dη . (5.16)Switching to polar coordinates ( r, ω ) ∈ R × S d − in T ∗ y Σ we obtain I f = N π ) d − [ ρ h ( x )] Z ∞ Z S d − e i r ( h exp − y ( x ) ,ω i− t ) N X k =0 χ ( r ) r k ( t ; y, ω ) r d − dr dVol S d − = N π ) d − [ ρ h ( x )] N X k =0 Z ∞ e − irt r d − k − χ ( r ) Z S d − e i r h exp y ( x ) ,ω i f − − k ( t ; y, ω ) dr dVol S d − , (5.17)where = N is a shortcut notation for mod S − N − = . We observe that the function f ( ω ) := h exp y ( x ) , ω i is stationary at ω = ± ˆ x , where ˆ x := [dist Σ ( x, y )] − (exp y ( x )) ♭ . Furthermore, it holds
Hess f ( ± ˆ x ) = ∓ dist Σ ( x, y ) Id . This allows us to evaluate thespherical integral via the stationary phase formula for r → + ∞ (see, for example, [25,Theorem C.1]) as Z S d − e i r ( h exp y ( x ) ,ω i ) f − − k ( t ; y, ω ) dVol S d − = X κ = ± e i κ r dist Σ ( x,y ) e iκ (2 − d ) π (cid:18) πr dist Σ ( x, y ) (cid:19) d − K X j =0 r − j [ L j f − − k ]( t ; y, κ ˆ x ) + O ( r − K − ) (5.18)for every K ∈ N , where [ L j f − − k ]( t ; y, κ ˆ x ) = X µ,ν : ν − µ = j µ ≤ ν − ν µ ! ν ! h [Hess f ( κ ˆ x )] − D ω , D ω i ν ( f µ ,κ f − − k )( t ; y, ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = κ ˆ x , (5.19)with f ,κ ( ω ) = f ( ω ) − f ( κ ˆ x ) − h Hess f ( κ ˆ x )( ω − κ ˆ x ) , ( ω − κ ˆ x ) i . (5.20)The local asymptotic expansion is now obtained by substituting (5.18)–(5.20) into (5.17)and subsequently carrying out the integral in r . We will compute the first few termsexplicitly in the special case d = 4 , M = R × S . (5.21)The argument in the general case would proceed along the same lines, only it would bea bit more lengthy and technically involute. Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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When
Σ = S , it is easy to see that the symbol f in (5.11) depends only on themagnitude of η , not on its direction. Hence, setting ˆ f ( t ; y ) := f ( t ; y, ω ) , ω ∈ S , (5.22)it holds that, modulo S − N − , I f = N π ) [ ρ h ( x )] N X k =0 Z + ∞ e − irt r − k χ ( r ) Z S e i r h exp y ( x ) ,ω i ˆ f − − k ( t ; y ) dVol S dr = N [dist Σ ( x, y )] − π [ ρ h ( x )] N X k =0 ˆ f − k − ( t ; y ) ×× Z + ∞ K X j =0 X κ = ± κ e ir ( κ dist Σ ( x,y ) − t ) i r k + j χ ( r )[ L j (1)]( t ; y, κ ˆ x ) + O ( r − ( k + K +1) ) ! dr. (5.23)By analysing formula (5.20) for f ,κ and setting f − − k = 1 in (5.19), one obtains [ L j (1)]( t ; y, κ ˆ x ) = κ j [ L j (1)]( t ; y, ˆ x ) (5.24)so that (5.23) becomes I f = [dist Σ ( x, y )] − π [ ρ h ( x )] N X k =0 ˆ f − k − ( t ; y ) ×× Z + ∞ K X j =0 X κ = ± κ j e ir ( κ dist Σ ( x,y ) − t ) i r k + j χ ( r )[ L j (1)]( t ; y, ˆ x ) + O ( r − ( k + K +1) ) ! dr. (5.25)The last step in the computation consists in evaluating, in a distributional sense, integralsof the form Z + ∞ e ir b ϕ r n dr, n ∈ N , where b ϕ = κ dist Σ ( x, y ) − t .For n = 0 we have Z + ∞ e ir b ϕ dr = lim ε → + Z + ∞ e ir ( b ϕ + iε ) dr = lim ε → + i b ϕ + iε . (5.26)For n ∈ N \ { } we have Z + ∞ e ir b ϕ r n dr = lim ε → + Z + ∞ e ir ( b ϕ + iε ) ( − n +1 ( n − d n dr n log r dr = lim ε → + − i n ( b ϕ + iε ) n ( n − Z + ∞ e ir ( b ϕ + iε ) log r dr = lim ε → + − i n − ( b ϕ + iε ) n − ( n − γ + log( ε − i b ϕ )] . (5.27)When computing (5.27) we used the fact that, for ℜ z < , Z + ∞ e zr log r dr = γ + log( − z ) z . (5.28) Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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In view of (5.26), the term corresponding to j = k = 0 in (5.25) is given by π [ ρ h ( x )] f − ( t ; y, ˆ x )dist Σ ( x, y ) X κ = ± κκ dist Σ ( x, y ) − t = 14 π f − ( t ; y, ˆ x )[ ρ h ( x )] ( x, y ) − t = 18 π f − ( t ; y, ˆ x )[ ρ h ( x )] σ (( t, x ) , (0 , y )) (5.29)Since the principal symbol of the half-wave propagator is [6] and since h ( − ∆) − i prin ( y, η ) = 1 p h αβ ( y ) , η α η β [26, Theorem 18.1] implies that f − ( t ; y, ˆ x ) = . It follows that (5.29) corresponds to thefirst term in the expansion (5.4) π [ ρ h ( x )] − σ (( t, x ) , (0 , y )) . (5.30)Furthermore we recover the well-known fact that, given two generic points X = ( t, x ) , Y =( t ′ , y ) ∈ R × Σ , the function u appearing in (5.4) satisfies u ( t, y, t, y ) = 1 ) and that u ( X, Y ) coincides with the square root of the Van Vleck-Morette determinant, which, in normalcoordinates centred at y , reads [ ρ h ( x )] − [21]. Remark 5.4.
We would like to stress that, the calculation of the first term (5.30) isgeneral and works for an arbitrary four-dimensional ultrastatatic spacetime. In fact, wedid not use anywhere the assumption that the Cauchy surface is a sphere.The substitution of (5.30) and (5.27) into (5.25) yields I f ∼ π [ ρ h ( x )] − σ (( t, x ) , (0 , y ))+ [dist Σ ( x, y )] − π [ ρ h ( x )] ∞ X k,j =0 X κ = ± κ j ˆ f − k − ( t ; y )[ L j (1)]( t ; y, ˆ x ) ×× lim ε → + i j + k ( b ϕ + iε ) j + k − ( j + k − γ + log( ε − i b ϕ )] . (5.31)Let us write down explicitly the term with j + k = 1 . We have [dist Σ ( x, y )] − π [ ρ h ( x )] X κ = ± i (cid:16) κ ˆ f − + ˆ f − [ L − (1)] (cid:17) [ γ + log( i ( t − κ dist Σ ( x, y )))] == i [dist Σ ( x, y )] − π [ ρ h ( x )] (cid:20) ˆ f − log (cid:18) t − dist Σ ( x, y ) t + dist Σ ( x, y ) (cid:19) + ˆ f − [ L − (1)] (cid:0) γ + log(dist ( x, y ) − t ) (cid:1)(cid:21) = log(dist ( x, y ) − t ) i [dist Σ ( x, y )] − π [ ρ h ( x )] (cid:16) ˆ f − [ L − (1)] + ˆ f − (cid:17) + i [dist Σ ( x, y )] − π [ ρ h ( x )] h ˆ f − (cid:0) iπ + log(( t + dist Σ ( x, y )) ) (cid:1) + 2 γ ˆ f − [ L − (1)] i . (5.32)We observe that (5.32) contributes a log term and a smooth term to the expansion. It iseasy to see that subsequent contributions in (5.31) yield terms of the form v k σ k log σ andsmooth terms, where the functions v k are completely determined by the homogeneouscomponents of the symbol f and by the geometry. Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
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Acknowledgements
M.C. is grateful to Dima Vassiliev for enlightening conversations. The authors are grate-ful to Igor Khavkine and Alex Strohmaier for useful comments. The work of N.D. issupported by a fellowship of the Alexander von Humboldt foundation and he is gratefulto the University of Pavia and of Trento for the hospitality during the realisation of partof this work.
A Fundamental Facts on Fundamental Solutions
In this appendix we recall, for completeness, a few basic well-known properties of thefundamental solutions of the operator P defined in (2.5). In fact, these operators can beshown to exist for a larger class of backgrounds that those considered in this papers, forexample when relaxing the assumption of compactness of the Cauchy surface or allowingfor the presence of a timelike boundary, see, e.g. , [8, 9, 13]. Theorem A.1.
Let ( M, g ) be a globally hyperbolic spacetime and let P be the Klein–Gordon operator (2.5) . Then P possesses unique advanced ( − ) and retarded (+) fun-damental solutions G ± ∈ D ′ ( M × M ) such that, calling G ± : D ( M ) → C ∞ ( M ) theassociated maps via the Schwartz kernel theorem [15, Thm. 8.2.12], • P ◦ G ± = G ± ◦ P = Id | D ( M ) , • supp ( G ± ( f )) ⊆ J ∓ (supp( f )) , for all f ∈ D ( M ) , where J ± ( A ) stand for the causalfuture (+) and for the causal past ( − ) of any open subset A ⊆ M . We refer the reader to [1] for the proof. The distribution G := G + − G − ∈ D ′ ( M × M ) is called the retarded-minus-advanced fundamental solution .Using G ± , one can characterise the space of smooth solutions of (2.5) as follows. Lemma A.2.
Let ( M, g ) be a globally hyperbolic spacetime and let G ± : C ∞ ( M ) → C ∞ ( M ) be the maps associated via the Schwartz kernel theorem to the advanced andretarded fundamental solutions G ± of the Klein–Gordon operator P . Let S sc ( M ) = { Φ ∈ C ∞ sc ( M ) | P Φ = 0 } where the subscript sc stands for spacelike compact . Then there existsan isomorphism of topological vector spaces S sc ( M ) ≃ C ∞ ( M ) P [ C ∞ ( M )] , where the isomorphism is implemented by G . = G + − G − via the map C ∞ ( M ) P [ C ∞ ( M )] ∋ [ f ] ( f ) ∈ S sc ( M ) . In this paper, we are particularly interested in the map that associates to initial data ofthe Klein–Gordon equation P the corresponding solution. Proposition A.3.
Let ( M, g ) be a globally hyperbolic spacetime and let G be the retarded-minus-advanced fundamental solution associated the Klein–Gordon operator P . Let Σ t . = { t } × Σ be an arbitrary but fixed Cauchy surface and let n be its future pointing, normal Φ ∈ C ∞ sc ( M ) if it is smooth and the intersection between the support of Φ and any Cauchy surfaceis compact. Global wave parametrices on globally hyperbolic spacetimesatteo Capoferri, Claudio Dappiaggi, and
Nicolò Drago Page 26 vector of unit norm. Let ι t : Σ t → M be the standard inclusion map. Let G ,t : C ∞ (Σ t ) → C ∞ ( M ) and G ,t : C ∞ (Σ t ) → C ∞ ( M ) be the linear maps obtained via the Schwartzkernel theorem respectively from ( I ⊗ ι ∗ t,n ) G and ( I ⊗ ι ∗ t ) G where ι ∗ t is the pull-back mapand ι ∗ t,n := ι ∗ t ◦ ( n µ ∇ µ ) . Then, for every f , f ∈ C ∞ (Σ t ) , the Cauchy problem (cid:26) P Φ = 0 , Φ | Σ t = f , ( n µ ∂ µ Φ) | Σ t = f , admits as unique solution Φ = G ,t ( f ) + G ,t ( f ) . (A.1)The proof of the above proposition is a direct adaptation to our notations and settingof [1, Lemma 3.2.2]. References [1] C. Bär, N. Ginoux and F. Pfäffle,
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