Detection of Hermitian connections in wave equations with cubic non-linearity

We consider the geometric non-linear inverse problem of recovering a Hermitian connection $A$ from the source-to-solution map of the cubic wave equation $\Box_{A}\phi+\kappa |\phi|^{2}\phi=f$, where $\kappa\neq 0$ and $\Box_{A}$ is the connection wave operator in the Minkowski space $\mathbb{R}^{1+3}$. The equation arises naturally when considering the Yang-Mills-Higgs equations with Mexican hat type potentials. Our proof exploits the microlocal analysis of nonlinear wave interactions, but instead of employing information contained in the geometry of the wave front sets as in previous literature, we study the principal symbols of waves generated by suitable interactions. Moreover, our approach relies on inversion of a novel non-abelian broken light ray transform.

This paper considers an inverse problem for a non-linear wave equation motivated by theoretical physics and differential geometry. The main problem we wish to address is the following: can the geometric structures governing the wave propagation be globally determined from local information, or more physically, can an observer do local measurements to determine the geometric structures in the maximal region where the waves can propagate and return back? There has been recent progress on this question when the geometric structure is space-time itself and the relevant PDEs are the Einstein equations [20].
Here we propose the study of a natural non-linear wave equation when the Lorentzian background is fixed and the goal is the reconstruction of a Hermitian connection. The main difference between the inverse problems for the Einstein equations and the equation considered here is that, in the former case, the geometric structure (the metric) to be reconstructed appears in the leading order terms, and in the latter case, it (the connection) appears in the lower order terms. This difference poses novel challenges, since a perturbation in the leading order affects the wave front sets of solutions whereas lower order perturbations do not.
The leading order terms can frequently be reconstructed via study of distances (or time separations/earliest arrival times), whereas lower order terms often require reductions to light ray transform questions. Nevertheless, our approach exploits the recent philosophy that non-linear interaction of waves creates new singularities and enriches the dynamics [20,23,28]. As we shall see this interaction leads to a broken non-abelian light ray transform on lightlike geodesics that has not been previously studied.
Our main long term goal is the study of inverse problems for the Yang-Mills-Higgs equation. The present paper is the first stepping stone in this direction and our objective here is to start exposing the main features that this problem will have by considering a simplified, but non-trivial model case. Since the bundle M × g is trivial, A is a connection, and it is called the Yang-Mills potential; Φ is the Higgs field. The Yang-Mills-Higgs equations are where F A := dA + A ∧ A is the curvature of A, D A Φ := dΦ + [A, Φ] is the associated covariant derivative, and V is the derivative of a smooth function V : [0, ∞) → R. The adoint D * A is taken with respect to g and hence A := D * A D A is the wave operator associated with g and A.
An extensively studied case is the Yang-Mills-Higgs equations with the Mexican hat type potential, where κ, b ∈ R, see e.g. [8,Eq. (10.5)] where the Lagrangian formulation of the problem is used. We will consider the potential (3) with κ = 0, and to simplify the notations, with b = 0. The case b = 0 is not substantially different. Our choice can be viewed as the simplest potential introducing a non-linearity. We refer also to [36] where Yang-Mills-Higgs equations, with the potential (3), are discussed in a purely mathematical context, (M, g) being a Riemannian manifold there.
As it is well known, equations (1)- (2) are invariant under the group of gauge transformations which in this case coincides with the set of maps u ∈ C ∞ (M ; G) and the action on pairs is When Φ = 0 we obtain the pure Yang-Mills equation D * A F A = 0.
1.2. Formulation of the inverse problem in the model case. Dealing with the equations (1)-(2) from the outset might be too ambitious, so here we propose a simplified model. We shall suppose that we have a trivial bundle E = M × C n and a Hermitian connection A on E giving rise to a covariant derivative d + A. In this case, the gauge group is U (n). We take V to be the Mexican hat type potential (3) with b = 0, discard equation (1) completely and focus on the analogue of equation (2), with M × g replaced by E. That is, we consider the equation (4) A φ + κ|φ| 2 φ = 0, where φ is a section of E, A = (d + A) * (d + A) and |φ| is the norm with respect to the standard Hermitian inner product of C n . We shall further simplify matters by assuming that M is R 1+3 and that g is the Minkowski metric.
We discuss the existence of L A in more detail in Section 2 below.
The goal of the observer is to determine the Yang-Mills potential A up to the natural obstructions, given the source-to-solution map L A . The causal structure of (M, g) encodes the finite speed of propagation for the wave equation (4). Given x, y ∈ M we say that x ≤ y if x = y or x can be joined to y by a future pointing causal curve, and denote the causal future of x ∈ M by J + (x) = {y ∈ M : x ≤ y}. The causal future J + (x) is the largest set that waves generated at x can reach. The causal past of a point z ∈ M is denoted by J − (z) = {y ∈ M : y ≤ z}. If waves generated at x are recorded at z, the finite speed of propagation dictates that no information on the potential A outside the causal diamond The model problem is to determine A given L A in the largest domain possible, that is, in up to the natural gauge, where u ∈ C ∞ (D; U (n)) and u| = id. The sets and D are visualized in Figure 2.
Observe that if we have two connections A and B on M such that there exists a smooth map u : M → U (n) with the property that B = u −1 du + u −1 Au and u| = id, then B = u −1 A u and |uφ| = |φ|. Moreover, as f has compact support in it holds that uf = f . Therefore φ solves (8) for B if and only if uφ solves (8) for A, and it follows that L A = L B . This shows that the gauge (10) is indeed natural.
Our main theorem asserts that the model problem has a unique solution, or in more physical terms, the measurements performed on , as encoded by L A , determine the gauge equivalence class of the Yang-Mills potential A, in the largest possible causal diamond D. As D is strictly larger than , we can view the determination of the equivalence class of A as a form of remote sensing. We emphasize that the gauge equivalence classes of Yang-Mills potentials, not the potentials themselves, correspond to physically distinct configurations. Theorem 1. Let A and B be two connections in R 1+3 such that L A = L B where the source-to-solution map L A is defined as above, and L B is defined analogously, with A replaced by B in (8). Suppose that κ = 0 in (8). Then there exists a smooth u : D → U (n) such that u| = id and B = u −1 du + u −1 Au.
It is straightforward to see that L A = L B implies A = B on . The non-trivial content of the theorem is the gauge equivalence away from . To see that A and B coincide on , we fix y ∈ and choose φ ∈ C ∞ 0 ( ; E) such that φ(y) = 0. Then for small > 0 it holds that f := ( A φ + κ|φ| 2 φ) ∈ C. Since L A = L B we see that at y: cf. (14) below, and since dφ at y is arbitrary, there holds A = B at y.

1.3.
Comparison with previous literature. The previous results on inverse problems for non-linear wave equations, such as [20,23,28], are based on analysis of four singular, interacting waves. A new feature in the present paper is that we consider interactions of three waves only. This leads to a more economic proof, and is particularly well suited for the cubic non-linearity in (8). A more detailed comparison of interaction three versus four waves is given in the beginning of Section 3.1.
Let us briefly explain what we mean by the interactions of three waves. The idea is to choose a source of the form f = 1 f 1 + 2 f 2 + 3 f 3 where j > 0 are small and f j are conormal distributions. Then the cross-derivative ∂ 1 ∂ 2 ∂ 3 φ| =0 satisfies a linear wave equation with a right-hand side that corresponds to a certain product of ∂ j φ| =0 , j = 1, 2, 3. Here = ( 1 , 2 , 3 ). As also the functions ∂ j φ| =0 satisfy the linear wave equation, we can view the cross-derivative as a result of their interaction.
The wave front set of the above cross-derivative was studied in the case of the 1 + 2-dimensional Minkowski space by Rauch and Reed [32], see also [6,30] for later results of similar nature. What is new in the present paper, is that, contrary to [32] and the previous results on inverse problems for non-linear wave equations, e.g. [20,23,28], we employ more precise information on the singular structure of the cross derivative than just its wave front set. Namely, in a suitable microlocal sense, the cross-derivative has a principal symbol, and the proof of Theorem 1 uses information contained in the principal symbol in order to recover a novel broken non-abelian light ray transform of the connection A along lightlike geodesics. The proof of Theorem 1 is completed by solving the subsidiary geometric inverse problem of inverting this transform in the Minkowski space, see Proposition 2 below, a result which has independent interest.
In more physical terms, we can say that the interaction of the three waves ∂ j φ| =0 , j = 1, 2, 3, produce an artificial source, that can be viewed either as two moving point sources or as a filament in spacetime, and that emits a wave encoded by the crossderivative ∂ 1 ∂ 2 ∂ 3 φ| =0 . We show that, when the sources f j , j = 1, 2, 3, are chosen carefully, the singular wave front emitted by the artificial source returns to . This wave front is visualized in Figure 1. Stretching the physical analogy further, we can say the leading amplitude of this singular wave front is the information used in the proof.
Paradoxically, Theorem 1 is open for the linear case, κ = 0, but a positive solution is known if A and B are supposed to be time-independent [24]. In the time-dependent case there are results [34] available only in the abelian case of a line bundle, n = 1, and it is an open problem if recovery of A in the optimal causal diamond D is possible in this case. Let us also mention that the linear, abelian, time-independent case has been studied extensively, see e.g. [2,3,17], but these results do not carry over to the time-dependent case. The reason for this is that they (and also above mentioned [24]) are based on Tataru's unique continuation principle [35], that again is known to fail for equations with time-dependent coefficients [1].
We emphasize once more that the focus of the current paper is on the recovery of the lower order terms in a non-linear wave equation. This makes definite progress towards Open Problem 5 in [25], and is different from the previous results where only the determination of the leading order terms is considered. See for instance [23,28] where the determination of the metric tensor (or its conformal class) is studied for scalar-valued non-linear equations, or [20,27] where the determination of the metric tensor is studied for the Einstein equations coupled with different matter field equations. The difference between recovery of leading and lower order terms All the three pieces intersect in the two black points, moving along the vertical axis over the point where the black lines intersect. The points act as artificial sources that produce a new propagating wave, the red surface. The line segment traced by the two points can be viewed also as the projection of a one dimensional filament acting as an artifical source. The filament curves in spacetime since the two points move with a non-constant speed. Bottom right. As time progresses, the red propagating wave front grows. Eventually it will reach the points where the pieces of the spherical waves originate from.
is reflected in the key novelty of our approach, namely, in the study of principal symbols instead of wave front sets.
Such difference is apparent also in the existing theory of inverse problems for linear wave equations. The case of a linear wave equation with time independent coefficients, and with sources and observations in disjoint sets, illustrates this. In this case the theory is still under active development, and the best results available are very different for leading and lower order terms: the recovery of the metric [26] is based on distance functions, whereas the recovery of the lower-order terms [19] is based on focussing of waves. The latter also requires additional convexity assumptions, that are not present in the former case.

1.4.
A conjecture on higher order non-linearities. One outcome of the current paper is the following emergent principle for dealing with inverse problems for waves with polynomial non-linearities using an approach similar to ours. Assume for simplicity that we are in the line bundle case (i.e. n = 1) and consider an equation of the form where κ = 0. Then to recover A (up to gauge) from a source-to-solution map it is necessary to consider the J-fold linearization of (11), where The necessity is discussed further in Remarks 2 and 4 below. We conjecture that (12) is also a sufficient condition, but the present paper establishes this only in the case N = 3.
1.5. Outline of the paper. This paper is organized as follows. Section 2 contains preliminaries mostly having to do with the direct problem (8). Section 3 contains the microlocal analysis for the interaction of three waves and shows that we can recover the broken non-abelian light ray transform along lightlike geodesics from the knowledge of L A . Section 4 solves the geometric inverse problem of determining A up to gauge from the broken non-abelian light ray transform and completes the proof of Theorem 1. Appendix A recalls the theory of conormal and Intersecting Pair of Lagrangian (IPL) distributions; Appendix B contains certain technical details concerning symplectic transformations to a model pair of intersecting Lagrangians, and Maslov bundles; and Appendix C gives full description of the wave front set of the cross-derivative ∂ 1 ∂ 2 ∂ 3 φ| =0 , that is, the red surface in Figure 1.
We would like to dedicate this paper to the memory of our friend and colleague Slava Kurylev who was instrumental in initiating the present line of research on inverse problems for the Yang-Mills-Higgs equations.
Acknowledgements. LO thanks Allan Greenleaf, Alexander Strohmaier and Gunther Uhlmann for discussions on microlocal analysis.
ML was supported by Academy of Finland grants 320113 and 312119. LO was supported by EPSRC grants EP/P01593X/1 and EP/R002207/1 and XC and GPP were supported by EPSRC grant EP/R001898/1. GPP thanks the University of Washington for hospitality while this work was in progress and the Leverhulme trust for financial support.

Preliminaries
In this section, to accommodate further work, we let (M, g) be an arbitrary, globally hyperbolic Lorentzian manifold of dimension 1 + m. Also E can be taken as an arbitrary Hermitian vector bundle over M . Recall that a Lorentzian manifold (M, g) is globally hyperbolic if there are no closed causal paths in M , and the causal diamond J + (x) ∩ J − (z) is compact for any pair of points x, z ∈ M , see [5]. A globally hyperbolic manifold (M, g) is isometric to a product manifold R × M 0 with the Lorentzian metric given by where c : R × M 0 → R + is smooth and g 0 is a Riemannian metric on M 0 depending smoothly on t, see [4]. Moreover, the vector field ∂ t gives time-orientation on M .
To simplify the discussion, we make the further assumption that all the geodesics of (M, g) are defined on the whole R.
2.1. Direct problem. We write occasionally ∇ = d + A for the covariant derivative associated to the connection A, and view it as a map Writing g = g ij dx i dx j in coordinates, we denote by |g| and g ij the determinant and inverse of g ij , respectively. Moreover, A = A j dx j is a 1-form, and each A j is a skew-Hermitian matrix. Let us now write the wave operator A = ∇ * ∇ in coordinates. Consider compactly supported sections φ and ψ = ψ j dx j of E and T * M ⊗ E, respectively. Then where ·, · E is the inner product on E, and dV g denotes the volume form on (M, g).
Integrating by parts and using the fact that A is skew-Hermitian, we have Consequently, A φ takes the form Remark 1. To prove Theorem 1, we will need the operator A exclusively in Minkowski space where we can explicitly write where and div are the usual wave operator and divergence, Let T > 0 and let us consider the following nonlinear Cauchy problem where H : (R × M 0 ) × E → E is a smooth map operating section-wise such that H(t, x, 0) = 0, and f is a section of E. We will now give sufficient conditions on f in order for (15) to have a unique solution.
As the leading term of A is simply the canonical wave operator on (M, g), acting on each component of φ, we can use the standard results for quasilinear hyperbolic equations to show existence and uniqueness of solutions to this Cauchy problem. See, for example, Theorem 6 of [18] and its proof (with the notations explained in detail in Appendix C in [21]) or Theorems I-III and the proof of Lemma 2.7 in [15]. By these results, we have that for an integer r > m/2+2 and any compact set K ⊂ (0, T )×M 0 , there is 0 > 0 such that for any f ∈ C r 0 (K) satisfying f C r (K) < 0 , the initial value problem (15) has a unique solution. Recall that m is the dimension of the underlying space M 0 . In particular, in the case of the Minkowski space R 1+3 , we may take r = 4, and see that source-to-solution map L A is well-defined by (7).

2.2.
Notations for microlocal analysis. For a conic Lagrangian submanifold Λ 0 ⊂ T * M \ 0 and a vector bundle E over M , we denote by I p (M ; Λ 0 ; E) the space of Lagrangian distributions of order p ∈ R associated to Λ 0 , and taking values in E. If Λ 1 ⊂ T * M \ 0 is another conic Lagrangian submanifold intersecting Λ 0 cleanly, we denote by I p (M ; Λ 0 , Λ 1 ; E) the space of Intersecting Pair of Lagrangian (IPL) distributions of order p ∈ R associated to (Λ 0 , Λ 1 ), and taking values in E.
We use occasionally the notation ·, · for the duality pairing between covectors and vectors, and If Λ 0 coincides with the conormal bundle Although removing the zero section from N * K, when considering it as a conic Lagrangian manifold, is somewhat awkward notationally, it is natural to consider N * K as a submanifold of T * M , since then the fibres N * x K ⊂ T * x M , x ∈ K, are linear subspaces. We recall the basic properties of conormal and IPL distributions in Appendix A below.
The wave front set of a distribution u ∈ D (M ) is denoted by WF(u), see [11,Def. 2.5.2]. It is a subset of T * M \0, and its projection on M is called the singular support singsupp(u) of u. The wave front set WF(u) is conical and closed in T * M \ 0, and it is occasionally convenient to use the notation If K is the Schwartz kernel of a pseudodifferential operator χ on M , then the projection of WF(K ) ⊂ (T * M \ 0) 2 on the first factor T * M \ 0 is called the essential support of χ. (As WF(K ) is contained in the conormal bundle of the diagonal {(x, y) ∈ M 2 : x = y}, the choice between the first and second factor makes no difference.) Following [11, p. 124] we write WF(χ) for this set.
We denote by Ω 1/2 the half-density bundle over M . When Λ 0 and Λ 1 \ Λ 0 coincide with conormal bundles, and E = E ⊗ Ω 1/2 , there is a coordinate invariant way to define the principal symbol σ[u] of u ∈ I(M ; Λ 0 ; E), respectively u ∈ I(M ; Λ 0 , Λ 1 ; E), as an equivalence class of sections of E ⊗ Ω 1/2 over Λ 0 , respectively Λ 1 \ Λ 0 . We will not emphasize the difference between the equivalence class σ[u] and a representative of it, and we will also use the same notation for the half-density bundles over M and Λ j , j = 0, 1. Let us remark that there is typically no natural way to relate these bundles. For example, while it is natural to use |g| 1/4 to trivialize Ω 1/2 over M , the Lorentzian metric g on M typically does not induce a natural trivialization of Ω 1/2 over Λ j .
For IPL distributions in I(M ; Λ 0 , Λ 1 ; E), there is also a refined notion of principal symbol, with components on both Λ 0 and Λ 1 . We will use the refined principal symbol only in Appendix A. The notation σ[χ] is used also for the principal symbol of a pseudodifferential operator χ on M . In this case, σ[χ] is represented by a section of T * M \ 0.

2.3.
Microlocal analysis of the wave operator. It is convenient to rescale (8), and consider the following non-linear wave operator where |·| = |·| E is the norm with respect to the inner product ·, · = ·, · E . In order to make use of the microlocal machinery developed in [9], we conjugate the operator Q 0 with the half density |g| 1/4 and consider the operator Q(u) For the sake of convenience, we will slightly abuse the notation, and write for products of half-densities as functions. Then Q(u) = P u + κ|u| 2 u/2.
Writing ı = √ −1, the full symbol of the operator P reads We write also σ[P ] = ξ, ξ g /2 where ξ, ξ g denotes the inner product with respect to g. Let us remark that the subprincipal symbol transforming as a connection is discussed in [16] in the more general context of pseudodifferential operators on vector bundles.
We denote by H P the Hamiltonian vector field associated to σ[P ], and by Σ(P ) the characteristic set of P . That is, The covectors ξ satisfying ξ, ξ g = 0 are called lightlike. We denote by Φ s , s ∈ R, the flow of H P , and define for a set B ⊂ Σ(P ) the future flowout of B by Let us recall the parametrix construction for the linear wave equation that originates from [9]. We will follow the purely symbolic construction from [31], the only difference being that u is vector valued in our case. For the convenience of the reader, we give a proof of the below theorem in Appendix A.
where L H P is the Lie derivative with respect to H P , σ[u] and σ[f ] are the principal symbols of u and f on Λ 1 and Λ 0 , respectively, and R is a map, defined by (72) in Appendix A below, that acts as a multiplication by a scalar on E. Here it is assumed that Λ 0 and Λ 1 \ Λ 0 coincide with conormal bundles.

2.4.
Flowout from a point in the Minkowski space. The following case will be of particular importance for us. We have also included a detailed discussion of Theorem 2 in the context of this example case in Appendix B.
Let c ∈ E 0 \ 0, that is, c is a non-zero vector in the fibre of E over the origin, and let χ be a pseudodifferential operator such that σ[χ] = 0 near ccl{(0, ξ 0 )}. We define is the future light-cone in the spacetime R 1+3 emanating from the origin. Letting u be the solution of (20), its restriction on R 1+3 \ 0 is a conormal distribution in , and ξ 0 is viewed also as an element of T γ(s) R 1+3 . The smaller the essential support WF(χ) is chosen around The pseudodifferential operator χ can be chosen for example as follows. Choose functions with |ξ| the Euclidean norm of ξ, we define the function Now χ 0 is positively homogeneous of degree q. Choose, furthermore, χ 4 ∈ C ∞ 0 (R 1+4 ) such that χ 4 = 1 near the origin. Then (1 − χ 4 (ξ))χ 0 (x, ξ) is smooth also near ξ = 0, and it is a symbol in the sense of [11, Def. 1.1.1]. Now we define a pseudodifferential operator by Ignoring 2π factors, the full symbol of χ is simply (1−χ 4 (ξ))χ 0 (x, ξ), and the principal symbol σ[χ] is the corresponding equivalence class modulo symbols of one degree lower order.
Let us check that indeed Λ 1 \ Λ 0 = N * K \ 0. The lightlike vectors in T 0 R 1+3 \ 0 are given by (λ, λθ) with λ ∈ R \ 0 and θ ∈ S 2 . In the Minkowski space, the tangentcotangent isomorphism corresponds to changing the sign of the first component. Therefore, We can also reparametrize For the convenience of the reader, we will still compute explicitly the conormal bundle of K. Toward that end, we choose local coordinates R 2 ⊃ B a → Θ(a) ∈ S 2 on S 2 and see that the tangent space of K at (t, θ 0 ) is given by the range of Writing 0 . Taking first δt = 1 and δθ = 0 we have ξ 0 = −ξ θ 0 . Letting then δθ vary we see that Here we can view θ 0 ∈ R 3 as a covector since the tangent-cotangent isomorphism in R 3 is the identity. Hence Observe that Λ 1 \ Λ 0 is embedded in the following smooth submanifold of T * M \ 0, that is the flowout to both past and futurê Note that while K is singular at t = 0, we see thatΛ 1 is not by considering the derivative analogous to (25), This matrix is injective since λ = 0.
We will next write the transport equation (21) for the principal symbol σ[u] as a parallel transport equation with respect to the covariant derivative ∇, and we begin by discussing Ω 1/2 over the flowout Λ 1 .

2.5.
Trivialization of the half-density bundle over the flowout. We want to trivialize Ω 1/2 in a way that preserves homogeneity properties, as possessed for example by χ 0 in (24). Let us point out that, even in the context of Example 1, there appears to be no canonical choice of a non-vanishing section of Ω 1/2 over the conormal bundle Λ 1 \ Λ 0 = N * K \ 0. For example, the Sasaki metric on T * R 1+3 , associated with the Minkowski metric, is degenerate when restricted on N * K \ 0.
The submanifold K in Example 1 is of codimension one in R 1+3 . This holds in general in the sense that, if Λ 0 and Λ 1 in Theorem 2 satisfy for a submanifold K ⊂ M , then K is of codimension one. This can be seen as follows. Observe first that Λ 1 ⊂ Σ(P ) simply because Λ 1 is the future flowout from Λ 0 ∩ Σ(P ). Therefore, for any x ∈ K, the fibre N * x K can contain only lightlike vectors with respect to g. On the other hand, if ξ 1 , ξ 2 ∈ T * x M are lightlike and linearly independent, then their linear span satisfies, see e.g.
In particular span(ξ 1 , ξ 2 ) contains vectors that are not lightlike, and therefore at most one of ξ j , j = 1, 2, can belong to N * x K. This shows that N * x K is of dimension one, or equivalently, K is of codimension one in M .
We will trivialize Ω 1/2 over Λ 1 by choosing a strictly positive half-density ω in C ∞ (Λ 1 ; Ω 1/2 ) that is positively homogeneous of degree 1/2. We begin by recalling the definition of positive homogeneity following [13, p. 13]. Let λ ∈ R \ 0 and define Then m λ restricts as a map on Λ 1 in a natural way, and we denote the restriction still by m λ . The half-density ω is said to be positively homogeneous of degree q ∈ R if m * λ ω = λ q ω for all λ > 0. If in local coordinates We emphasize that, as the conormal bundle of K coincides with the flowout Λ 1 in the sense of (27), the local coordinate x 0 is not the time coordinate t in (13). Here (x, ξ) = (x 0 , x ; ξ 0 , ξ ) are the induced coordinates on T * M . Considering the restriction Thus ω being positively homogeneous of degree 1/2 means that ω(x , λξ 0 ) = ω(x , ξ 0 ), moreover, ω being strictly positive means that ω(x , ξ 0 ) > 0. We will, in fact, choose a half-density ω that is also symmetric in the sense that a coordinate invariant formulation of which reads m * λ ω = |λ| 1/2 ω for all λ ∈ R \ 0. In general, a strictly positive half-density ω ∈ C ∞ (Λ 1 ; Ω 1/2 ) satisfying (30) can be constructed by choosing an auxiliary Riemannian metric on M , restricting the associated Sasaki metric h on T * M \ 0 to Λ 1 , and taking ω = |h| 1/4 . In particular, when (M, g) is the Minkowski space, it feels natural to choose the Euclidean metric on M .
2.6. Parallel transport equation for the principal symbol. Let us fix a strictly positive half-density ω over N * K satisfying (30). Here the closure is taken in T * M , and we remark that, in view of (27) . A computation in coordinates shows that L H P (aω) = (H P a)ω + aL H P ω. Introducing the notation div ω H P = ω −1 L H P ω, we write ω −1 L H P (aω) = H P a + adiv ω H P .
We want to further rewrite this as a conjugated differentiation along bicharacteristics, that is, along the flow curves of H P . Recall that Φ s , s ∈ R, denotes the flow of H P . Writing β(s) = Φ s (x 0 , ξ 0 ) for the bicharacteristic through (x 0 , ξ 0 ) ∈ N * K \ 0, we have (H P a) • β(s) = ∂ r a(Φ r (β(s)))| r=0 = ∂ s (a • β)(s). We further write α = a • β, and define Then . We denote by γ the projection of the bicharacteristic β to the base manifold M , and byγ * the tangent vector of γ as a covector, that is,γ * i = g ijγ j . It follows from (18) that γ is a geodesic of (M, g), and that β(s) = (γ(s),γ * (s)). As β(s) ∈ Σ(P ), the geodesic γ is lightlike. Moreover, using (17), where ·, · is again the duality pairing between covectors and vectors. The covariant derivative on the bundle E along the geodesic γ(s) is given by Therefore (21) along β is equivalent with e −ρ ∇γ(e ρ α) = 0. If we define the following symbol along β for conormal distributions in I(M ; N * K \ 0; E ⊗ Ω 1/2 ), then we can rewrite (21) along β as follows This is the parallel transport equation along γ with respect to the connection A. If x = γ(s 0 ) and y = γ(s 1 ) for some s 0 , s 1 ∈ R, then we write P A y←x : E x → E y for the parallel transport map from x to y along γ. That is, In general, the map P A y←x depends on the geodesic γ joining x and y, but not on the parametrization of γ. We do not emphasize the dependency on γ in our notation, since we are mainly interested in the Minkowski case, and in this case P A y←x depends only on the points x and y. To summarize, writing ξ =γ * (s 0 ) and η =γ * (s 1 ), it follows from (32) that 2.7. Positively homogeneous symbols. We will now consider how (34) changes under rescaling of ξ ∈ N * x K \ 0, assuming that σ[u] is positively homogeneous of degree q + 1/2 ∈ R at (x, ξ), that is, Proposition 1. Let Λ j , j = 0, 1, and u be as in Theorem 2, and suppose that (27) holds. Let (x, ξ) ∈ N * K \ 0 and (y, η) = Φ s 1 (x, ξ) for some s 1 ∈ R. Suppose that (35) holds at (x, ±ξ) and that (y, η) ∈ N * K \ 0. Then Recall that B = Λ 0 ∩ Σ(P ) = Λ 0 ∩ Λ 1 . As the symbol σ[u] on Λ 1 \ Λ 0 is smooth up to B, equation (36) holds also when (x, ±ξ) ∈ B. Then (y, η) ∈ N * K \ 0 implies x < y, that is, the causal relation x ≤ y holds and x = y. Writing again γ(s) for the projection of β(s) = Φ s (x, ξ) to M , we have Hence, if ξ is future pointing then s 1 > 0, and if ξ is past pointing then s 1 < 0. We emphasize that also past pointing singularities are propagated forward in time by the wave equation (20).
It remains to show that e ρ λ (s 1 /λ) = e ρ(s 1 ) . We denote the restriction of the flow Φ s on N * K \ 0 still by Φ s . In coordinates satisfying (29), the Lie derivative L H P ω is of the form where we used the fact that Φ 0 = id, and therefore |dΦ 0 /d(x , ξ 0 )| 1/2 = 1. We write for the Hamiltonian vector field H P on N * K \ 0. Then, see e.g. p. 418 of [29], Equation (18), together with the fact that ξ = 0 on N * K, implies that In particular, Moreover, as ω satisfies (30), The above six equations imply that L H P ω(x , λξ 0 ) = λL H P ω(x , ξ 0 ). This again implies that and therefore the change of variables s = λs gives div ω H P (γ(s ),γ * (s ))ds = ρ(s 1 ).

Microlocal analysis of the interaction of three waves
The core idea of the proof of Theorem 1 is to choose the source f in (8) as the weighted superposition of three singular sources, where f j , j = 1, 2, 3, are conormal distributions supported in satisfying Recall that is the set where the measurements are gathered, see the definition (7) of the source-to-solution map L A . Each f j will be similar to the source f in Example 1. We denote by φ = φ( ) the solution of (8) with the source f = f ( ), and write We will choose f 1 so that φ in is singular along a lightlike geodesic γ in intersecting . Then we choose f 2 and f 3 to be such perturbations of f 1 that φ out is singular along another lightlike geodesic γ out that intersects both γ in and . Let us fix the parametrization of γ in and γ out so that for some s in , s out ∈ R. Taking into account the fact that the wave equation (8) propagates singularities forward in time, we consider only the case x < y < z, see Figure 2, left.  (5), is the blue cylinder, and set D, see (9), is the diamond like region drawn with dashed curves. Left. Points x, z ∈ and y ∈ D as in (40). The geodesic segment from x to y is γ in and the segment from y to z is γ out . We write also γ in = γ y←x and γ out = γ z←y . Right. Points x j , j = 1, 2, 3, see (50), together with the geodesic segments γ y←x j . Here x 1 = x.
Assume from now on that (M, g) is the 1 + 3-dimensional Minkowski space, and denote the geodesic joining a pair (x, y) ∈ L by γ y←x . What follows is independent from the parametrization of γ y←x , but let us fix it by requiring that γ y←x (0) = x andγ y←x (0) = (1, θ) where θ ∈ S 2 . We define also the broken non-abelian light ray transform of the connection A by Now we can summarize the above discussion as follows: for each (x, y, z) ∈ S + ( ) we want to choose f j , j = 1, 2, 3, so that φ in is singular along γ y←x and that φ out is singular along γ z←y . Moreover, we will choose f j , j = 1, 2, 3, in such a way that, in a suitable microlocal sense, φ out is a conormal distribution. Then we will show using Proposition 1 that S A z←y←x is determined by the principal symbol of φ out | when f j , j = 1, 2, 3, are varied. As each f j is supported in , this will show that the source-to-solution map L A determines S A z←y←x for all (x, y, z) ∈ S + ( ).
3.1. The linear span of three lightlike vectors. We will begin with a lemma in linear algebra that will be a key component when proving that singularities propagating along any γ z←y , with the pair (y, z) in S out ( ) = {(y, z) : (x, y, z) ∈ S + ( ) for some x ∈ }, can be generated by choosing suitable f j , j = 1, 2, 3, supported in . We believe that this lemma can be also used to simplify the proofs of previous results, such as [22,23,28]. These results are based on using a superposition of four singular sources, which leads to more complicated computations.
To highlight the difference, we recall that all the previous results consider nonlinear wave equations on 1 + 3-dimensional Lorentzian manifolds, and use the fact that if N * K j \ 0, j = 1, 2, 3, 4, are flowouts analogous to N * K \ 0 in Example 1, then generically 4 j=1 K j is discrete. It follows that N * y ( 4 j=1 K j ) = T * y M for a point y ∈ 4 j=1 K j . Then, using notation analogous with the above, it is shown that φ out = ∂ 1 ∂ 2 ∂ 3 ∂ 4 φ| =0 can be made singular along any γ z←y with (y, z) ∈ S out ( ). Asγ z←y (0) ∈ N * y ( 4 j=1 K j ) trivially, there is no geometric obstruction for φ out being singular onγ z←y in view of the product calculus for conormal distributions, see Lemma 2 below.
On the other hand, we will use only three flowouts, and the intersection 3 j=1 K j will be of dimension one. Lemma 1 implies, however, that for any fixed y ∈ K 1 and any fixed lightlike η ∈ T * y M we can guarantee that (y, η) ∈ N * ( 3 j=1 K j ) by choosing K 2 and K 3 carefully.
Remark 2. Consider the solution φ( ) of (11) with vanishing initial conditions and the source f ( ) = 1 f 1 + 2 f 2 , with f j a conormal distribution supported in satisfying (38). Regardless of the degree of non-linearity N ≥ 2, it is not possible to make ∂ 1 ∂ 2 φ| =0 singular along arbitrary γ z←y with (y, z) ∈ S out ( ). Indeed, using (28) it can be shown that This explains the lower bound J ≥ 3 in (12).
The following lemma is formulated for the Minkowski space but, being of pointwise nature, it generalizes to an arbitrary Lorentzian manifold. Lemma 1. Let y be a point in R 1+3 , and let ξ 1 , η ∈ T y R 1+3 \ 0 be lightlike. In any neighbourhood of ξ 1 in T y R 1+3 , there exist two lightlike vectors ξ 2 , ξ 3 such that η is in span(ξ 1 , ξ 2 , ξ 3 ).
To simplify the notation, we assume that κ = −1 in (4). The general case is analogous. We introduce u j 1 ,...,j l = ∂ j 1 · · · ∂ j l u, v j 1 ,··· ,j l = u j 1 ,··· ,j l | =0 , and write Re for the real part of a complex number. With these notations, we differentiate (46), and obtain Together with the fact that u| =0 = 0, this implies P v j = |g| 1/4 f j . Differentiating the equation (46) twice yields and v jk = 0. Finally, one more differentiation gives the desired linear equation, that we call the three-fold linearization, The right-hand side of (48) is the sum of products of v 1 , v 2 , v 3 , and we call it the three-fold interaction of these three solutions to the linear wave equation (20).
Remark 4. Consider the solution φ( ) of (11) with vanishing initial conditions and the source f ( ) = N j=1 j f j , with f j a distribution supported in . Analogously to (47), we see that ∂ α φ| =0 = 0 for any multi-index α ∈ N 1+3 satisfying |α| < N where N is the degree of non-linearity in the equation (11). This explains the lower bound J ≥ N in (12).

3.3.
Wave interactions as products of conormal distributions. Consider a triple (x, y, z) ∈ S + ( ) and let us construct f j , j = 1, 2, 3, such that φ out , defined by (39), is singular on γ z←y . We write where s in > 0 satisfies γ y←x (s in ) = y.
The geodesic γ with the initial condition (x, ξ) is denoted by γ(·; x, ξ). Let V ⊂ T * y M be a small enough neighbourhood of ξ 1 so that for all ξ ∈ V it holds that γ(−s in ; y, ξ) ∈ and that ξ is future pointing. Let ξ 2 , ξ 3 ∈ V be two lightlike vectors as in Lemma 1 and write With a slight abuse of notation, we write also ξ j =γ(−s in ; y, ξ j ), see Figure 2, right, for the geometric setup.
Analogously to Example 1, we let c j ∈ E x j \ 0, and let χ j be a pseudodifferential operator with the following properties: (χ1) σ[χ j ] is positively homogeneous of degree q, symmetric with respect to 0 ∈ T * M , and real valued, (χ2) σ[χ j ] = 0 near (x j , ξ j ), (χ3) WF(χ j ) is contained in a small neighbourhood of ccl(x j , ±ξ j ). The degree q of χ j is chosen to be small enough so that f j ∈ C 4 (M ) where Here δ x j is the Dirac delta distribution at x j . Moreover, we choose χ j so that supp(χ j δ j ) ⊂ and that the support condition (38) is satisfied. Recalling that C ⊂ C 4 0 ( ; E) is the domain of the source-to-solution map L A , see (7), we have then that the linear combination 1 f 1 Recall that v j denotes the solution of Moreover, we denote the restriction of v j on R 1+3 \ {x j } still by v j , and write K j = x j + K where K is as in (23). That is, writing x j = (t j , x j ), Clearly y ∈ 3 j=1 γ y←x j (R + ) ⊂ 3 j=1 K j . It follows from Remark 3 that the covectors ξ j , j = 1, 2, 3, are linearly independent. This again implies that N * y (K 1 ∩ K 2 ∩ K 3 ) = span(ξ 1 , ξ 2 , ξ 3 ). When WF(χ j ), j = 1, 2, 3, are small enough, this also guarantees for distinct j, k, l that K j and K k , as well as K j and K k ∩ K l , are transversal in S := S 1 ∪ S 2 ∪ S 3 .
Let us now consider products of the form Re( v j , v k )v l , with distinct j, k, l, that appear on the right-hand side of (48). As s in > 0, it holds that y = x j for each j = 1, 2, 3, and v j are conormal distributions near y. Thus their products can be analysed using the product calculus for conormal distributions. We recall this calculus in the next lemma, that is a variant of [10, Lemma 1.1]. With obvious modifications it holds also when the conormal distributions u j , j = 1, 2, take values on E ⊗ Ω 1/2 and the product is defined in terms of ·, · E , and also when u 1 takes values on Ω 1/2 and u 2 on E ⊗ Ω 1/2 and the product is defined in terms of the scalar-vector product on E.
Lemma 2. Let K j ⊂ M , j = 1, 2, be transversal submanifolds, and let u j be a conormal distribution in I(M ; N * K j \ 0; Ω 1/2 ). For a fixed nowhere vanishing halfdensity µ ∈ C ∞ (M ; Ω 1/2 ), we define the product of u 1 and u 2 by Let χ be a pseudodifferential operator with WF(χ) disjoint from both N * K j , j = 1, 2. It follows that χ(u 1 u 2 ) ∈ I(M ; N * (K 1 ∩ K 2 ) \ 0; Ω 1/2 ), with the principal symbol, ignoring the 2π and ı factors, Let us remark that, due to the transversality of K 1 and K 2 , it holds that . The condition that both ξ (1) and ξ (2) are non-zero is equivalent with We return to our study of v j , j = 1, 2, 3. Due to Remark 3, we can choose a pseudodifferential operator χ such that χ = 1 near ccl(y, η), and that WF(χ) is contained in a small conical neighbourhood of ccl(y, η). Then applying Lemma 2 twice, we obtain where Λ 0 = N * (K 1 ∩K 2 ∩K 3 )\0 and j, k, l ∈ {1, 2, 3} are distinct. For the convenience of the reader, we give a detailed proof of this.
Let us denote the right-hand side of (48) by f out , that is, Furthermore, we write v out and w for the solutions of the following two wave equations where χ is as in (54). Then the solution v 123 of (48) satisfies v 123 = v out + w, and due to (54), it holds that χf out ∈ I(R 1+3 ; Λ 0 ; E ⊗ Ω 1/2 ). We will treat w as a remainder term.
Let us now consider the future flowout Λ 1 from Λ 0 ∩ Σ(P ), with Λ 0 as in (54). We will show in Appendix C that Λ 1 \ Λ 0 coincides with a conormal bundle, and thus we can apply Theorem 2 to the equation for v out in (57). Let us point out that this study of the structure of Λ 1 is not essential for the proof. We could alternatively treat v out as a Lagrangian distribution, but then its symbol should be viewed as a section of E ⊗ Ω 1/2 ⊗ M , with M the Maslov bundle over Λ 1 . However, we prefer to avoid technicalities related to general Lagrangian distributions in the present paper.

From the source-to-solution map to the broken light ray transform.
Having understood the propagation and interaction of the linear waves, we now prove that the source-to-solution map determines the broken non-abelian light ray transform.
Theorem 3. Let A and B be two connections in R 1+3 such that L A = L B , as in Theorem 1. Then S A z←y←x = S B z←y←x for all (x, y, z) ∈ S + ( ). We give a constructive proof of Theorem 3 in the form of a method that recovers S A z←y←x for any (x, y, z) ∈ S + ( ) given L A . Therefore we continue considering only a single connection A.
Recall that the source-to-solution map L A determines σ[v out ](z, ζ) via (59), and observe that both the factors α in and α out are independent from A. Therefore L A determines the parallel transport P A z←y c in . Letting where we used the fact that P A y←x is unitary. Thus L A determines S A z←y←x after varying c 1 ∈ E x \ 0, and we have shown Theorem 3.

Inversion of the broken light ray transform
From now on we assume that E is the trivial bundle M × C n . Recall the definition (5) of . More generally, we write ( ) = (0, 1) × B( ) where B( ) is the open ball of radius > 0, centred at the origin of R 3 . We use also the shorthand notation D( ( )) = {y ∈ M : there is (x, y, z) ∈ S + ( ( ))}, 0 < < 0 , where S + ( ( )) is defined by (41).
We recall that S A z←y←x is defined by (42), that is, S A z←y←x = P A z←y P A y←x , (x, y, z) ∈ S + ( ), and that the parallel transport map P A y←x is the fundamental solution to the ordinary differential equation (33). In this section we will prove the following: Proposition 2. Let A and B be two connections in R 1+3 such that for all 0 < < 0 there holds (65) S A z←y←x = S B z←y←x , for all (x, y, z) ∈ S + ( ( )). Then there exists a smooth u : D(Ω( 0 )) → U (n) such that Observe that the causal diamond, defined by (9), satisfies D = D( ( 0 )), and therefore Theorem 1 follows immediately by combining Theorem 3 and Proposition 2. Similarly to S out ( ) we define S in ( ) = {(x, y) : (x, y, z) ∈ S + ( ) for some z ∈ }.
Lemma 3. Let A and B be two connections in R 1+3 . We define u(y, x) = P A y←x P B x←y for (x, y) ∈ S in ( ). If (65) holds then u(y, x 1 ) = u(y, x 2 ) for all (x j , y) ∈ S in ( ), j = 1, 2.
Proof. Note that P A y←x is a linear isomorphism and (P A y←x ) −1 = P A x←y . In particular, (S A z←y←x ) −1 = S A x←y←z , and (65) implies that S A x←y←z = S B x←y←z for all (x, y, z) ∈ S + ( ). Consider now y, x 1 and x 2 as in the claim. Then there is z ∈ such that (x j , y, z) ∈ S + ( ) for j = 1, 2. We have , and therefore P A x 2 ←y P A y←x 1 = P B x 2 ←y P B y←x 1 . We apply P A y←x 2 on left and P B x 1 ←y on right, and obtain Proof of Proposition 2. For y ∈ D( ) there are x, z ∈ such that (x, y, z) ∈ S + ( ) and we define u(y, x) as in Lemma 3. It follows from Lemma 3 that u(y, x) = u(y) and u can be viewed as a function of y ∈ D( ). The parallel transport map takes values in U (n) and therefore u is a section of U (n) over D( ).
Observe that ∩ D( ) = ∅ by the definition of S + ( ), see (41). For this reason we shrink , that is, we define u as above but replace with ( ), 0 < < 0 . This allows us to define u on ( 0 ) \ µ([0, 1]), see (6) for the definition of the path µ. By the continuity of the parallel transport map, we can define u on the whole ( 0 ). Using the continuity again, we let x → y ∈ (0, 1) × ∂B( ) in u(y, x), and see that u(y) = id for any y ∈ (0, 1) × ∂B( ) and any 0 < < 0 . Using the continuity once more, we see that u = id in the whole = ( 0 ).
We write γ = γ y←x and consider the fundamental matrix solution of (33). That is, for fixed s ∈ R, the function U A (t, s) in t is the solution of Clearly P A γ = U A (s in , 0). We define U B analogously. By (67) it holds that Moreover, differentiating (67) in s, gives and writing W in terms of the fundamental solution U A gives We have u(γ(t), y) = U A (t, 0)U B (0, t), and using (67) for A and (68) for B, We obtain (66) after rearranging and multiplying both sides with u −1 , We have shown that (66) holds for any (x, y) ∈ S in ( ( )). As ( ) is open, it follows from Lemma 1, that the vectorsγ y←x span T y M as x varies in the set {x ∈ : (x, y) ∈ S in ( ( ))}. HenceÃ = B at y for each y ∈ D( ( )) and each 0 < < 0 .

Appendix A. Conormal distributions and IPL distributions
We formulate the framework of E ⊗ Ω 1/2 valued conormal and IPL distributions for the application to the connection wave equation. The contents of the appendix is modelled on the pioneering works of Hörmander [11], Duistermaat-Hörmander [9] and Melrose-Uhlmann [31]. Additionally, we refer the reader to Hörmander's books [14,12] for the basics of distributions and half densities.
Let us begin with the notion of conormal distribution. Definition 1. Let X be an n-dimensional manifold and Y an (n − N )-dimensional closed submanifold. We say a section-valued distribution u ∈ D (X, E ⊗ Ω 1/2 ) is a member of the conormal distributions I m (X; N * Y ; E ⊗ Ω 1/2 ), if in local coordinates (x , x ) ∈ R N +(n−N ) on X, such that Y is defined by x = 0, the distribution u takes the following form where a ∈ S m+n/4 (R n × (R N \ 0); E ⊗ Ω 1/2 ).
From the viewpoint of Lagrangian distributions, u is associated with the conormal bundle N * Y \0 and WF(u) ⊂ N * Y \0. If ξ = (ξ , ξ ) denotes the induced coordinates on the cotangent space of X, N * Y is defined by {x = 0, ξ = 0}. The principal symbol σ[u] ∈ S m+n/4 (N * Y \ 0; E ⊗ Ω 1/2 ) is defined as σ[u](x , ξ ) = a(0, x , ξ ). We have the following exact sequence When the source term for a linear wave equation is a conormal distribution, the solution is not in the same class, and the wider class of IPL distributions was introduced in [31] to tackle this problem.
The wavefront set of A is contained in the unionΛ 0 ∪Λ 1 and away from the intersection ∂Λ 1 =Λ 0 ∩Λ 1 , the IPL distribution A is a conormal distribution in the following sense, see [31, pp.  Proposition 3. Suppose u ∈ I m (R n ;Λ 0 ,Λ 1 ; E ⊗ Ω 1/2 ) and χ is a zero-th order pseudodifferential operator. We have This microlocalization leads to the following local principal symbol maps, ignoring 2π and ı factors related to the normalization in (69), IPL distributions can be defined on any pair of Lagrangians (Λ 0 , Λ 1 ), Λ j ⊂ T * X \0, with a clean intersection. By a clean intersection of two Lagrangians, we mean given two Lagrangians Λ 0 and Λ 1 with Λ 0 ∩ Λ 1 = ∂Λ 1 . However, we will omit discussion of Lagrangian distributions, and make the additional assumption that for some submanifolds Y j ⊂ X. Definition 3. Suppose (Λ 0 , Λ 1 ), a pair of Lagrangians over a smooth n-manifold X, intersect cleanly at Λ 1 . The space I m (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) consists of distributions of the form where u 0 ∈ I m−1/2 (X; Λ 0 ; E ⊗ Ω 1/2 ); u 1 ∈ I m (X; Λ 1 \ ∂Λ 1 ; E ⊗ Ω 1/2 ); {F j } is a family of zero-th order Fourier integral operators associated with the inverse of the homogeneous symplectic transformation from V j to T * R n , where {V j } is a locally finite, countable covering of ∂Λ 1 ; and v j ∈ I m (R n ;Λ 0 ,Λ 1 ; E ⊗ Ω 1/2 ).
To symbolically construct the parametrix of the wave operator, the key thing is to understand the principal symbols of IPL distributions. As in the model case, away from the intersection of Λ 0 and Λ 1 , the corresponding IPL distributions are conormal distributions assuming (71).
Proposition 4. Suppose that (71) holds. Let u ∈ I m (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) and let χ be a properly supported zero-th order pseudodifferential operator. We have Choosing χ so that σ[χ](x, ξ) = 0 at (x, ξ) ∈ (N * Y 0 \ 0 ∪ N * Y 1 \ 0) \ ∂Λ 1 , we can define the principal symbols of u away from the intersection ∂Λ 1 , However, the map from the IPL distributions I m (X; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ) to the symbols can not be defined as the principal symbol map, since it is not surjective. Indeed, Proposition 3 implies that σ (1) [u] extends to a smooth section of E ⊗ Ω 1/2 up to ∂Λ 1 , whilst hσ (0) [u] extends to a smooth section over Λ 0 if h is a smooth function vanishing on ∂Λ 1 . Hence the principal symbol space should be a proper subspace of the product space of symbols.
Moreover, the fact that the Lagrangians Λ j , j = 0, 1, may not be conormal bundles near the intersection ∂Λ 1 causes additional complications. In the case of applications that we are interested in, Λ 1 fails to be a conormal bundle near ∂Λ 1 . The principal symbol of a general Lagrangian distribution is not a section of E ⊗ Ω 1/2 but a section of E ⊗ Ω 1/2 ⊗ L where L is the Maslov bundle. To avoid discussion of Maslov bundles over Λ j , j = 0, 1, we make the assumption that they are trivial. In all the cases that we are interested in, Λ 0 is a conormal bundle and Λ 1 is the future flowout of Λ 0 ∩ Σ(P ), with P the wave operator in the Minkowski space. We will show that the Maslov bundles over the flowouts of interest are trivial in Section B.1 below.
Apart from the Hermitian bundle factor, the map R was constructed by Melrose-Uhlmann [31, p.491-493]. But that factor is harmless. Indeed, after passing to the model case, we can simply define R by (70), that is, Ra (0) = ı(ξ 1 a (0) )| ∂Λ 1 . This definition entails, of course, that R does not depend on the choice of a homogeneous symplectic transformation that maps (Λ 0 , Λ 1 ) locally to the model case (Λ 0 ,Λ 1 ). Such coordinate invariance was shown in [31].
Having the principal symbol map (73), we are ready to give a proof of Theorem 2, following [31].
To do so, we use the symbol calculus. First of all, since P is elliptic on where we used the shorthand notation p = σ[P ] for the principal symbol of P . As an element of I k−2+1/2 (M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ), u (0) has the following principal symbol on ∂Λ 1 , This can be viewed as the initial condition of the bicharacteristic flow on Λ 1 emanating from Λ 0 . On the other hand, the symbol calculus on Λ 1 obeys the transport equation, where c = σ sub [P ] is the subprincipal symbol of P . Noting that WF(f ) does not intersect Λ 1 \ ∂Λ 1 , we have Combining (75) and (76), we have solved (74). Next, we iteratively solve the following equations where f (j) ∈ I k−j (M ; Λ 0 ; E ⊗ Ω 1/2 ) and e (j) ∈ I k−j−1/2 (M ; Λ 0 , Λ 1 ; E ⊗ Ω 1/2 ). This can be done by choosing u (j) obey We complete the proof by adding up the equations (76) and (77) for j = 1, · · · , N , and letting N → ∞.
Appendix B. Theorem 2 in the context of Example 1 Let Λ j , j = 0, 1, be as in Example 1. That is, Λ 0 = N * {0} \ 0 and, taking into account the microlocal cutoff χ, where V is a small neighbourhood of θ 0 in S 2 . The smaller WF(χ) is, the smaller we can choose V.

It holds that
where B is a symmetric matrix, the precise form of which is inconsequential. Writing shortly dF − for the above derivative and J for the symplectic form on T * R 1+3 as a matrix, we assert that (dF − ) T JdF − = J. We write η = ξ /|ξ | and Let us now write the initial condition (22) in this context. We denote by (x,ξ) = (x 0 ,x ;ξ 0 ,ξ ) the local coordinates on T * R 1+3 given by F + , that is, (x 0 ,x ;ξ 0 ,ξ ) = F + (x 0 , x ; ξ 0 , ξ ).
When the essential support WF(χ) is small around ccl(0, ±ξ 0 ), the future flowout Λ 1 is embedded in the following flowout to both past and future, a localized version of (26),Λ
Observe that ε = T (z) = z/ s 2 in + z 2 , and this is small when we localize near y = 0 as the notation suggests.
Let us show that Λ 1 \ Λ 0 coincides with a conormal bundle. It is enough to verify that its projection to the base space R 1+3 is a smooth manifold. For our purposes it is enough to consider Λ 1 \ Λ 0 only over the compact diamond D and therefore it is enough verify that the map F (t, θ, z) = (T (z) + t, tθ, z + tε) is injective and has injective differential for t > 0, θ ∈ S 1 and small |z|.
As t > 0 and ε is small, the second component of (83) implies θ 1 = θ 2 and t 1 1 − |T (z 1 )| 2 = t 2 1 − |T (z 2 )| 2 . Writing Z j = s 2 in + z 2 j , and using the latter equation reduces to t 1 /Z 1 = t 2 /Z 2 . On the other hand, the last component of (83) gives z 1 (1 + t 1 /Z 1 ) = z 2 (1 + t 2 /Z 2 ). As 1 + t 1 /Z 1 = 1 + t 2 /Z 2 > 0, we get z 1 = z 2 . This together with t 1 /Z 1 = t 2 /Z 2 implies that also t 1 = t 2 . We have shown that F is injective. Observe that, writing Z = Z j when z 1 = z, Therefore, letting R ⊃ B a → Θ(a) ∈ S 1 be local coordinates on S 1 , When z = 0, this reduces to which is invertible since t > 0, Z > 0 and Θ = 0. The same holds when |z| is small enough, and we have shown that Λ 1 \ Λ 0 coincides with a conormal bundle. Let us also remark that an argument similar to that in Section B.1 shows that the Maslov bundle over Λ 1 is trivial.