Sachs Equations and Plane Waves, V: Ward, Fourier, and Heisenberg Symmetry on Plane Waves

SA CHS EQUA TIONS AND PLANE W A VES, V: W ARD, F OURIER, AND HEISENBER G SYMMETR Y ON PLANE W A VES JONA THAN HOLLAND AND GEORGE SP ARLING Abstract. This article studies w a ve equations and their solutions on plane wa v e spacetimes of arbitrary dimension, developing the interplay among three structural la yers: the W ard progressing-wa v e represen tation of solutions to the scalar wa ve equation, the F ourier analysis of the Heisenberg group naturally associated to the plane wa v e, and the Schr¨ odinger propagator gov erning the evolution of initial data. The central geometric ob ject is a p ositiv e curve in the Lagrangian Grassmannian determined by the plane wa ve metric, previ- ously studied in the authors’ series [1, 2, 3, 4]. The conformal tensor H ( u ) that parametrises this curve plays a dual role: it encodes the null-cone ge- ometry of the spacetime and simultaneously app ears as the time-dep enden t parameter in the Schr¨ odinger representation of the Heisenberg group acting by isometries on the plane w a v e. Parallel to the classical F ourier inv ersion the- orem, conv olution by Lagrangian delta distributions on the Heisenberg group furnishes an intrinsic description of the Schr¨ odinger propagator, and the inter- twining of diﬀerent polarisations by this propagator is captured by a diagram that commutes up to a Maslov phase. The theta functions and Bargmann transforms that arise from imaginary polarisations complete the analytic pic- ture, connecting the present work to the theory of the W eil representation as developed by Lion–V ergne [5] and to Mumford’s systematic treatment of theta functions [6, 7]. 1. Introduction 1.1. Bac kground and motiv ation. Plane w av e spacetimes o ccup y a distinguished p osition in both mathematics and ph ysics. Geometrically , they are the simplest gen- uinely curv ed Loren tzian manifolds: in the Brinkmann–Rosen co ordinate system the metric takes the form R ( G ) = 2 du dv − dx T G ( u ) dx , where G ( u ) is a smo oth curv e of p ositiv e-deﬁnite symmetric matrices. The entire curv ature of the space- time is encoded by a single matrix-v alued proﬁle K ( u ), making these spacetimes an ideal lab oratory for explicit, rigorous analysis. F rom the ph ysics p ersp ective, plane wa v es hav e b een studied since the w ork of Brinkmann [8] and Einstein–Rosen [9], and they arise as the Penr ose limit of an y spacetime in a neigh b ourho o d of an y n ull geodes ic [10]. This univ ersalit y w as revived dramatically in the string-theory context by Berenstein–Maldacena– Nastase [11] and Blau et al. [12], who sho wed that the maximally sup ersymmetric plane wa v e of type IIB supergravit y is a Penrose limit of AdS 5 × S 5 , thereb y yielding an exactly solv able mo del for strings in a curved background. Standard references for the general geometry of plane wa v es and their role in exact solutions of general relativit y are Stephani et al. [13] and the monograph of Blau [14]. Date : March 31, 2026. 1 2 JONA THAN HOLLAND AND GEORGE SP ARLING The present pap er is the ﬁfth in a series [1, 2, 3, 4] in which the authors hav e undertak en a systematic, rigorous study of plane w av e spacetimes in arbitrary di- mension, with the long-range goal of constructing and understanding the t wistor theory of these spacetimes. P ap er I [1] established the equiv alence b etw een the Brinkmann and Rosen formulations, characterised the coordinate singularities of the Rosen metric as points at whic h the Lagrangian curv e meets a ﬁxed Lagrangian subspace, and related the Sachs equations gov erning null-geodesic congruences to the Jacobi equation. P aper I I [2] classiﬁed the isometries and conformal isome- tries of plane wa ves and exhibited families of v acuum plane wa v es whose isometry groups exhibit c haotic (Bernoulli-shift) behaviour, sho wing that suc h spacetimes are not classiﬁable b y inv ariant observ ables. Paper I I I [3] introduced the notion of a micr o c osm (a complete, homogeneous plane wa ve) and solved the Sachs equa- tions explicitly for these spaces, iden tifying the Lagrangian curve as an orbit of a one-parameter subgroup of the symplectic group. P ap er IV [4] developed a general theory of cross-ratios and Sch warzians for curves in what the authors call the mid- d le Gr assmannian , culminating in an intrinsic, pro jectively inv arian t form ula for the curv ature of a plane wa ve as the Sch w arzian of its Lagrangian curv e. Con ten ts of this pap er. The present paper takes up the analytic side of the story: how do es one solve the wa v e equation on a plane wa v e, and what algebraic structures con trol those solutions? The Schr¨ odinger pr op agator and the War d r epr esentation. Because the wa v e op- erator □ on a plane w a v e do es not dep end on the retarded null co ordinate v , a F ourier transform in v reduces the wa v e equation to a Schr¨ odinger equation on the transv erse Euclidean space X , with u playing the role of time and the Laplacian ∆ G ( u ) serving as the (time-dep endent) Hamiltonian. A second F ourier transform, no w in the transverse directions x ∈ X , reduces this further to an explicit quadra- ture, yielding the Schr¨ odinger propagator of Theorem 1. The same solution can b e written in the War d form [15] ϕ ( u, v , x ) = g − 1 / 4 Z X F  v + ξ · x + 1 2 ξ T H ( u ) ξ , ξ  d n ξ , whic h exhibits solutions as sup erp ositions of progressing wa v es lab eled by transverse momen tum ξ . The tensor H ( u ) satisfying ˙ H = G − 1 is the conformally inv ariant datum of the plane wa ve; it deﬁnes the null cones and simultaneously serves as the parameter of the Sc hr¨ odinger representation. The Heisenb er g gr oup. The isometry group of a plane w av e contains a (2 n + 1)- dimensional Heisen b erg group H , which acts on the wa v efron ts u = const by shifts in ( v , x ) and is resp onsible for the sp ecial solubility of the wa ve equation. The Heisen b erg group carries a family of irreducible unitary (Schr¨ odinger) representa- tions ρ h,u on L 2 ( X ), indexed by the parameter h  = 0 and dep ending on u through H ( u ). The W eyl comm utation relations [ P , Q ( u )] = 2 π ih I are satisﬁed b y the momen tum op erator P and the p osition op erator Q ( u ) = 2 π x + ihH ( u ) ∂ x . The general theory of Heisenberg groups and their representations that we re- quire is classical, going back to the foundational work of W eyl, Stone, and von Neumann. F or the sp eciﬁc combination of the W eil represen tation, the Maslo v in- dex, and theta functions that arises here, the essen tial reference is the monograph of SACHS EQUA TIONS AND PLANE W A VES, V 3 Lion and V ergne [5]. Mumford’s T ata L e ctur es on Theta [6, 7] provides the comple- men tary algebraic-geometric and num ber-theoretic p ersp ectiv e on the Heisenberg group, its theta functions, and the relationships among diﬀerent p olarisations. F ourier tr ansform on the Heisenb er g gr oup. A Lagrangian subspace X ⊂ T of the symplectic space of transverse p ositions and momenta determines a temp ered dis- tribution δ X on the Heisenberg group. Con v olution by δ X is the abstract F ourier transform; the classical F ourier inv ersion theorem on R n is recov ered as the com- p osition δ X − ∗ δ X + ∗ ind X f = f for a real p olarisation X . A triple con volution δ B ∗ δ C ∗ δ A of pairwise complementary Lagrangian distributions yields a scalar mul- tiple of the iden tity , with the phase determined by the Maslov index τ ( A , B , C ) of the triple [5]. The Schr¨ odinger evolution and its indep endenc e of p olarisation. The Schr¨ odinger propagator Φ X ( s, u ) from time s to time u is expressed abstractly as a comp osition of F ourier transforms and parallel transp ort along the Lagrangian curve: S ( X ) ind − − → S ( H ( s ) \ H ) δ X ∗ − − → S ( X \ H ) Γ su − − → S ( X \ H ) δ H ( u ) ∗ − − − − → S ( H ( u ) \ H ) res − − → S ( X ) . The lo cal theorem of Section 7 establishes that if X and X ′ are suﬃciently close and b oth are complemen tary to H ( t ) on a common in terv al, then the corresp onding Sc hr¨ odinger evolutions are related by an explicit intert wining square: S ( X ) ⊗ F ( s ) S ( X ) ⊗ F ( u ) S ( X ′ ) ⊗ F ( s ) S ( X ′ ) ⊗ F ( u ) Φ X ( s,u ) ρ s ρ u Φ X ′ ( s,u ) The vertical intert wining maps ρ s and ρ u dep end on the Lagrangian curve H ( · ) and are given explicitly in terms of quadratic exp onential factors and symplectic reﬂection. This lo cal result is the basic gluing la w for the global theory . When one w orks in a ﬁxed real p olarisation, or equiv alen tly in a ﬁxed Rosen c hart, the c hart may cease to b e v alid when H ( t ) meets its Maslov cycle: this is the app ear- ance of a caustic. The Schr¨ odinger evolution itself do es not stop there. Instead, one passes to a nearby p olarisation X ′ on an ov erlapping interv al and uses the lo- cal intert winer to contin ue the evolution. The global theorem is therefore an atlas theorem: Schr¨ odinger propagation con tin ues across caustics b y cov ering the param- eter interv al with transv erse p olarisation charts and gluing the lo cal propagators b y these explicit ov erlap maps. The underlying Maslov phase is then explained b y the F ourier-integral calculus of Lagrangian distributions on the Heisenberg group. Bar gmann tr ansform and theta functions. F or an imaginary (i.e. p ositive com- plex) p olarisation J , the con volution kernel η J = te ( J + z z ) is a b ounded left- J - holomorphic function on H . Con v olution by η J deﬁnes the Bar gmann tr ansform , whic h maps Sch w artz functions on H to holomorphic Sc hw artz functions. This con- struction is the Heisenberg-group counterpart of the Bargmann–Segal transform used in quantum optics and geometric quantisation. When the Heisenberg group is tak en ov er a lattice (the arithmetic case relev ant to mo dular forms), the analogous construction pro duces theta functions in the sense of Mumford [6, 7]. 4 JONA THAN HOLLAND AND GEORGE SP ARLING 1.2. Relation to the prior literature. W av e equations on plane wa ves hav e a long history . The progressing-wa v e representation used here is due to W ard [15], who sho w ed that Whittak er’s formula for ﬂat-space w av e solutions extends uni- formly to the plane-wa v e background, with the role of a plane-wa v e phase replaced b y the quadratic expression v + ξ · x + 1 2 ξ T H ( u ) ξ . The connection b etw ee n the Schr¨ odinger equation and the Heisen b erg group has b een extensively exploited in the study of quantum mechanics in curved back- grounds; we particularly note the work of Duv al, Burdet, K ¨ unzle and Perrin [16] on the Bargmann structures asso ciated to non-relativistic spacetimes, which provides a related geometric framew ork. On the representation-theoretic side, the W eil representation (also known as the metaplectic or oscillator represen tation) and the Maslo v index are the cen tral to ols; the authoritative reference for our purposes is Lion–V ergne [5]. The Lagrangian Grassmannian and its diﬀeren tial geometry , particularly the Sch w arzian and cross- ratio dev elop ed in Paper IV [4], underpin the global asp ects of the propagator. The theta functions app earing in the arithmetic case connect this w ork to a clas- sical tradition going back to Jacobi and Riemann. Mumford’s T ata L e ctur es [6, 7] pro vide the mo dern algebraic-geometric foundations; the relationship b etw een theta functions and the Heisen b erg group is the organising principle of those volumes, as it is here. 1.3. Organisation of the pap er. Section 2 sets up the geometry of plane wa ves in Rosen co ordinates and deriv es the Schr¨ odinger propagator (Theorem 1) and W ard representation. Section 3 in tro duces the Heisen b erg group and its unitary represen tations, establishes their action on the plane wa v e, and pro v es the group la w. Sections 4 – 5 develop the abstract F ourier transform and F ourier inv ersion on the Heisen berg group, including the Maslov phase theorem. Section 6 discusses the Bargmann transform for imaginary p olarisations and the asso ciated theta func- tions. Section 7 studies the Schr¨ odinger evolution Φ X ( s, u ) and prov es the local comm utativity theorem for nearby p olarisations together with the global contin u- ation theorem across caustics obtained b y gluing lo cal charts. 2. Plane w a ves W e shall take a plane wave to b e a spacetime manifold M = U × R × X where U is a real interv al, R denotes the real line, and X is a ﬁxed n -dimensional real Euclidean space, together with the (Rosen) metric R ( G ) = 2 du dv − dx T G ( u ) dx, for ( u, v , x ) ∈ M , where G maps U to the cone of p ositive-deﬁnite symmetric ma- trices on X . W e assume that G is at least C 1 . The w av e op erator of a smo oth function ϕ is given by (1) □ ϕ = ∂ v ∂ u ϕ + 1 √ g ∂ u ( √ g ∂ v ϕ ) − ∆ ϕ = 2 g − 1 / 4 ∂ v ∂ u ( g 1 / 4 ϕ ) − g − 1 / 4 ∆( g 1 / 4 ϕ ) where g = det G ( u ), ∆ = ∂ x G ( u ) − 1 ∂ T x is the Laplacian with resp ect to the metric G ( u ) at u . Because □ do es not dep end on v , we can apply separation of v ariables. Deﬁning the F ourier transform with resp ect to v as ˆ f ( γ ) = Z R f ( v ) e − 2 π iγ v dv , SACHS EQUA TIONS AND PLANE W A VES, V 5 the F ourier co eﬃcients of a solution to □ ϕ = 0 satisfy (2) 4 π iγ ( g 1 / 4 ˆ ϕ ) u − ∆( g 1 / 4 ˆ ϕ ) = 0 whic h is a Schr¨ odinger equation with parameter γ . Thus ∆ serves as the “time”- dep enden t Hamiltonian, b eing a function of u . The conformal geometry of M is expressed most naturally , not in terms of G , but in terms of a function H ( u ) taking v alues in symmetric matrices on X , such that dH ( u ) = G ( u ) − 1 du , i.e., ˙ H = G − 1 where the dot is diﬀerentiation with respect to u . This deﬁning relation can b e expressed as the pro jectiv ely-in v ariant relation ð 0 H = G − 1 , and H carries pro jective weigh t zero, where ð 0 is the eth op erator, discussed in a later pap er in this series. The tensor H ( u ) deﬁnes the null cones in the plane wa ve, and also the symmetries of the spacetime. It also determines in a natural wa y a (conformally inv ariant) curve in the Lagrangian Grassmannian of the plane w av e. W e shall show how to solve (2), using the tensor H ( u ). A second F ourier transform is relev ant in the separation of v ariables analysis, namely that with resp ect to the Euclidean space X . Let S ( X ) b e the space of Sc hw artz functions on X , and for f ∈ S ( X ), put F f ( ξ ) = Z X f ( x ) e − 2 π iξ.x d n x where d n x is the normalized Leb esgue measure and ξ .x = ξ T x is the (ﬁxed) Eu- clidean inner pro duct on X . W e also hav e the inv erse transform F − 1 ˆ f ( x ) = Z X ˆ f ( ξ ) e 2 π iξ.x d n ξ . W e then hav e the Schr¨ odinger propagator: Theorem 1. The Schwartz solution to (2) with initial data ˆ ϕ 0 = ˆ ϕ ( u 0 ) is, for γ  = 0 , ˆ ϕ ( u ) = g ( u ) − 1 / 4 F − 1 ξ n exp  π iγ − 1 ξ T ( H ( u ) − H ( u 0 )) ξ  F ˆ ϕ 0 ( ξ ) o . Pr o of. W e hav e the relation F ( ∂ x f )( ξ ) = 2 π iξ F f ( ξ ), and so F ( ∂ x G ( u ) − 1 ∂ T x f )( ξ ) = − 4 π 2 ξ T G ( u ) − 1 ξ F f ( ξ ) . But also, 4 π iγ ( g 1 / 4 ˆ ϕ ) u = − 4 π 2 F − 1 ξ n exp  π iγ − 1 ξ T ( H ( u ) − H ( u 0 )) ξ  ξ T G ( u ) − 1 ξ F ˆ ϕ 0 ( ξ ) o . □ Note that the propagator in the theorem simpliﬁes slightly under the assumption H ( u 0 ) = 0, whic h we shall henceforth assume in this section. A Sc hw artz solution to the wa v e equation may b e put in the W ard form [15]: (3) ϕ ( u, v , x ) = g − 1 / 4 Z X F ( v + ξ .x + 1 2 ξ T H ( u ) ξ , ξ ) d n ξ where F ( v , ξ ) is a Sc hw artz function. Recalling that the hat denotes the F ourier transform in the ( v , γ ) dual v ariables, put ϕ ( u, v , x ) = Z R ˆ ϕ ( u, γ , x ) e 2 π iγ v dγ 6 JONA THAN HOLLAND AND GEORGE SP ARLING F ( v , ξ ) = Z R ˆ F ( γ , ξ ) e 2 π iγ v dγ . F or a ﬁxed γ , we thus consider (3) on the individual F ourier comp onents F ( v , ξ ) = ˆ F ( γ , ξ ) e 2 π iγ v and ϕ ( u, v , x ) = ˆ ϕ ( u, γ , x ) e 2 π iγ v , w e hav e e 2 π iγ v ˆ ϕ ( u, γ , x ) = g − 1 / 4 Z X ˆ F ( γ , ξ ) exp  2 π iγ  v + ξ .x + 1 2 ξ T H ( u ) ξ  d n ξ = g − 1 / 4 | γ | − n e 2 π iγ v Z X ˆ F ( γ , γ − 1 ξ ) exp  2 π iξ .x + π iγ − 1 ξ T H ( u ) ξ  d n ξ in agreemen t with Theorem 1, after suitable choice of ˆ F ( γ , ξ ). 2.1. W ell-p osedness and conserv ation laws. The Schr¨ odinger propagator of Theorem 1 and the W ard represen tation (3) are stated for Sch w artz initial data. W e record here the natural Hilb ert-space framework that underpins these results, sho wing that the space of ﬁnite-energy solutions is complete and that the symplectic form is conserv ed within it. F unction sp ac es. Let U b e a compact in terv al, V = R , and M = U × V × X the plane-w av e spacetime with Rosen metric 2 du dv − dx T G ( u ) dx , where G ∈ C 1 ( U ) tak es v alues in the p ositive-deﬁnite symmetric endomorphisms of X . Let S ( V × X ) denote the Sch w artz space of real-v alued functions on V × X , and let D ( M ) denote the space of compactly supp orted smo oth functions on a manifold M . Let ( γ , ξ ) b e the F ourier-dual v ariables to ( v , x ), so that the F ourier transform of f is ˆ f ( γ , ξ ) = Z V × X f ( v , x ) e − 2 π i ( γ v + ξ T x ) dv d n x. Deﬁne the weigh ted L 2 space L 2 ( | γ | ) as the space of measurable functions ˆ f on \ V × X for which ∥ ˆ f ∥ 2 L 2 ( | γ | ) = Z | γ | | ˆ f ( γ , ξ ) | 2 dγ d n ξ < ∞ , with inner pro duct ⟨ ˆ f , ˆ k ⟩ L 2 ( | γ | ) = R | γ | ˆ f ˆ k dγ d n ξ . W e also write ⟨ f , k ⟩ L 2 = R V × X f k dv d n x for the standard L 2 inner pro duct. Deﬁnition 1. The energy Sob olev space H 1 / 2 , 0 is the sp ac e of r e al-value d Bor el me asur able functions f on V × X such that ∥ f ∥ 2 H 1 / 2 , 0 := ∥ f ∥ 2 L 2 + ∥ ˆ f ∥ 2 L 2 ( | γ | ) < ∞ , wher e functions e qual almost everywher e ar e identiﬁe d. Deﬁnition 2. L et X b e a Hilb ert sp ac e and U a c omp act interval. The Bo chner– Sob olev sp ac e H 1 ( U ; X ) c onsists of al l f ∈ L 2 ( U ; X ) for which ther e exists f ′ ∈ L 2 ( U ; X ) satisfying Z U φ ′ ( u ) f ( u ) du = − Z U φ ( u ) f ′ ( u ) du for al l φ ∈ D (int U ) . This f ′ is c al le d the weak deriv ative of f , and H 1 ( U ; X ) c arries the Hilb ert norm ∥ f ∥ 2 = ∥ f ∥ 2 L 2 ( U ; X ) + ∥ f ′ ∥ 2 L 2 ( U ; X ) . SACHS EQUA TIONS AND PLANE W A VES, V 7 The natural solution space for the w av e equation is H 1 ( U ; H 1 / 2 , 0 ), normed b y ∥ f ∥ 2 = Z U  ∥ f ( u, · ) ∥ 2 H 1 / 2 , 0 + ∥ f ′ ( u, · ) ∥ 2 H 1 / 2 , 0  p g ( u ) du, where g ( u ) = det G ( u ). A standard embedding theorem (see Ev ans [17], p. 286) giv es the following contin uity in u : Lemma 1. If f ∈ H 1 ( U ; H 1 / 2 , 0 ) , then, after mo diﬁc ation on a set of me asur e zer o, f ∈ C ( U ; H 1 / 2 , 0 ) . In p articular, the slic e f ( u ) ∈ H 1 / 2 , 0 is wel l-deﬁne d for every u ∈ U . We ak solutions. Deﬁnition 3. A function f ∈ H 1 ( U ; H 1 / 2 , 0 ) is a weak solution of the wave e qua- tion if (4) Z M h 2 ( g 1 / 4 f ) u ( g 1 / 4 ϕ ) v − g ij ( u ) ∂ i ( g 1 / 4 f ) ∂ j ( g 1 / 4 ϕ ) i du dv d n x = 0 for al l ϕ ∈ D (int M ) . Her e ∂ i f = ∂ f /∂ x i is taken in the sense of distributions. Since (4) is linear in the test function, a standard F ubini argument shows that for a w eak solution, the slice condition (4) holds for almost every ﬁxed u : Corollary 1. If f ∈ H 1 ( U ; H 1 / 2 , 0 ) is a we ak solution, then Z V × X h 2 ( g 1 / 4 f ) u ( g 1 / 4 ϕ ) v − g ij ( u ) ∂ i ( g 1 / 4 f ) ∂ j ( g 1 / 4 ϕ ) i dv d n x = 0 for almost every u ∈ U and al l ϕ ∈ D (in t M ) . The F ourier transform conv erts (4) into an ODE in u and simultaneously yields a gradient estimate sho wing that weak solutions hav e square-in tegrable transverse gradien ts: Prop osition 1. L et f ∈ H 1 ( U ; H 1 / 2 , 0 ) b e a we ak solution, and let ∇ f denote its we ak gr adient in the x variables. Then for almost every u ∈ U , ∇ f ( u, · ) ∈ L 2 ( V × X ) and ∥∇ f ( u, · ) ∥ 2 L 2 ≤ C  ∥ f ( u, · ) ∥ 2 H 1 / 2 , 0 + ∥ f ′ ( u, · ) ∥ 2 H 1 / 2 , 0  . Pr o of. In the F ourier domain, a weak solution satisﬁes the ﬁrst-order ODE (for almost ev ery u ): (5) 4 π iγ \ ( g 1 / 4 f ) ′ + 4 π 2 ξ T G ( u ) − 1 ξ \ g 1 / 4 f = 0 . By hypothesis ˆ f , ˆ f ′ ∈ L 2 ( | γ | ). Multiplying (5) b y ˆ f , integrating, and applying the Cauc hy–Sc hw arz inequality gives Z ξ T G ( u ) − 1 ξ | ˆ f | 2 dγ d n ξ ≤ C  ∥ ˆ f ∥ 2 L 2 ( | γ | ) + ∥ ˆ f ′ ∥ 2 L 2 ( | γ | )  , and the left-hand side equals ∥∇ f ( u, · ) ∥ 2 L 2 b y Plancherel’s theorem and p ositiv e- deﬁniteness of G ( u ). □ 8 JONA THAN HOLLAND AND GEORGE SP ARLING Conservation laws. The plane wa ve carries a u -dep endent symplectic form on so- lutions. F or ϕ, ψ ∈ S ( M ), set (6) ω u ( ϕ, ψ ) = Z V × X  ϕ ψ v − ϕ v ψ  p g ( u ) dv d n x, whic h ma y be written inv ariantly as ω u 0 ( ϕ, ψ ) = R u = u 0 ∗ ( ϕ dψ − ψ dϕ ), where ∗ is the Ho dge star and the integral is o ver the w a vefron t { u = u 0 } . F or classical solutions the conserv ation dω u /du = 0 follows immediately from the wa v e equation via integration by parts. The following prop osition extends this to w eak solutions. Prop osition 2. L et f , k ∈ H 1 ( U ; H 1 / 2 , 0 ) b e we ak solutions of the wave e quation. Then: (i) p g ( u ) ⟨ f ( u, · ) , k ( u, · ) ⟩ L 2 is indep endent of u ∈ U ; (ii) ω u ( f , k ) is indep endent of u ∈ U . The pro of requires three lemmas ab out w eak deriv atives in Bo chner spaces. Lemma 2 (Diﬀerence quotient approximation) . L et X b e a Hilb ert sp ac e and ϕ ∈ H 1 ( U ; X ) . Deﬁne D ϵ ϕ ( u ) =    ϕ ( u + ϵ ) − ϕ ( u ) ϵ if u, u + ϵ ∈ U , 0 otherwise. Then D ϵ ϕ ⇀ ϕ ′ we akly in L 2 ( U ; X ) as ϵ → 0 . Pr o of. By the fundamental theorem of calculus and Cauch y–Sc h warz, ∥ D ϵ ϕ ( u ) ∥ X ≤ 1 ϵ Z ϵ 0 ∥ ϕ ′ ( u + t ) ∥ X dt ≤ 1 √ ϵ  Z ϵ 0 ∥ ϕ ′ ( u + t ) ∥ 2 X dt  1 / 2 . Squaring and in tegrating ov er U via F ubini giv es ∥ D ϵ ϕ ∥ 2 L 2 ( U ; X ) ≤ ∥ ϕ ′ ∥ 2 L 2 ( U ; X ) for all ϵ > 0, so the family { D ϵ ϕ } is b ounded in L 2 ( U ; X ). By the Banac h–Alaoglu theorem some subnet conv erges weakly to a limit ψ ; since D ϵ ϕ → ϕ ′ in the sense of X -v alued distributions, we conclude ψ = ϕ ′ . As every weakly conv ergen t subnet has the same limit, D ϵ ϕ ⇀ ϕ ′ . □ Lemma 3 (W eak–strong pro duct) . If ϕ ϵ → ϕ str ongly in H s and ψ ϵ ⇀ ψ we akly in H − s , then ϕ ϵ ψ ϵ ⇀ ϕψ we akly in L 1 . Pr o of. F or any f ∈ L ∞ , write Z f ( ϕ ϵ ψ ϵ − ϕψ ) = Z f ( ϕ ϵ − ϕ ) ψ ϵ + Z f ϕ ( ψ ϵ − ψ ) . The ﬁrst term is b ounded by ∥ f ∥ ∞ ∥ ψ ϵ ∥ H − s ∥ ϕ ϵ − ϕ ∥ H s → 0, since weakly conv ergent sequences are b ounded (uniform b oundedness principle) and ϕ ϵ → ϕ strongly . The second term tends to zero b ecause ψ ϵ ⇀ ψ and f ϕ ∈ H s . □ Lemma 4 (Product rule for weak deriv ativ es) . L et ϕ ∈ H 1 ( U ; H 1 / 2 , 0 ) and ψ ∈ H 1 ( U ; H − 1 / 2 , 0 ) . Then ( ϕψ ) ′ ∈ L 1 ( M ) and ( ϕψ ) ′ = ϕ ′ ψ + ϕψ ′ almost everywher e. Pr o of. The right-hand side b elongs to L 1 ( M ) by the h yp otheses and duality . F or the pro duct rule, use the discrete pro duct form ula D ϵ ( ϕψ )( u ) = D ϵ ϕ ( u ) · ψ ( u + ϵ ) + ϕ ( u ) · D ϵ ψ ( u ). As ϵ → 0, the left side conv erges weakly to ( ϕψ ) ′ (in the sense of L 1 ( M )) by Lemma 2, while b oth terms on the right conv erge weakly to ϕ ′ ψ and ϕψ ′ resp ectiv ely by Lemma 3. □ SACHS EQUA TIONS AND PLANE W A VES, V 9 Pr o of of Pr op osition 2. Part (ii). W e show that the w eak deriv ative in u of ω u ( f , k ) v anishes. F or any φ ∈ D (int U ), F ubini’s theorem gives Z U φ ′ ( u ) ω u ( f , k ) du = 2 Z M φ ′ ( u ) f k v p g ( u ) du dv d n x. In tegrating by parts in u (justiﬁed b y Lemmas 4 and 3) and then in v , and using the weak w a v e equation (4) for b oth f and k , each resulting b oundary term cancels its coun terpart: = − 2 Z M φ ( u )  ( g 1 / 4 f ) u ( g 1 / 4 k ) v + ( g 1 / 4 f )( g 1 / 4 k ) uv  du dv d n x = 2 Z M φ ( u )  ( g 1 / 4 f ) uv ( g 1 / 4 k ) − ( g 1 / 4 f )( g 1 / 4 k ) uv  du dv d n x = 2 Z M φ ( u )  ∆( g 1 / 4 f ) · g 1 / 4 k − g 1 / 4 f · ∆( g 1 / 4 k )  du dv d n x = 0 , where the last step uses self-adjoin tness of the Laplacian ∆. Part (i). In the F ourier domain, using equation (5) for f and k : d du  √ g ⟨ f , k ⟩ L 2  = Z ( g 1 / 4 ˆ f ) ′ ( g 1 / 4 ˆ k ) + ( g 1 / 4 ˆ f ) ( g 1 / 4 ˆ k ) ′ dγ d n ξ = Z iξ T G − 1 ξ γ  − g 1 / 4 ˆ f · g 1 / 4 ˆ k + g 1 / 4 ˆ f · g 1 / 4 ˆ k  dγ d n ξ = 0 , since γ and ξ are real so the in tegrand is purely imaginary and its real part v anishes. □ Remark 1. Pr op osition 2(ii) is the we ak-solution c ounterp art of the formal c al- culation at the start of this se ction. T o gether with L emma 1, it implies that the Schr¨ odinger ﬂow f ( u 0 ) 7→ f ( u ) extends fr om Schwartz initial data to a one-p ar ameter family of unitary isomorphisms on the Hilb ert sp ac e of ﬁnite-ener gy solutions, c om- pleting the functional-analytic justiﬁc ation of the pr op agator formula of The or em 1. 2.2. Heisen b erg group. Let L 2 ( X ) denote the complex Hilb ert space of square- in tegrable functions with resp ect to the normalized Leb esgue measure of the Eu- clidean space X . W e shall use co ordinates x on X and ξ on the dual space, such that the canonical one-form of T ∗ X is θ = ξ .dx. Let u b e a real parameter and H ( u ) a symmetric bilinear form on X , smo oth for u ∈ U , such that dH /du is p ositive-deﬁnite. The F ourier transform is a unitary mapping from L 2 ( X ) to itself, that is given on the dense subset of L 1 ∩ L 2 b y the formula F f ( ξ ) = Z X e − 2 π iξ · x f ( x ) dx The in verse F ourier transform is then F − 1 g ( x ) = Z X e 2 π iξ · x g ( ξ ) dξ . The Sc h wartz space S ( X ) is the set of functions temp ered by p olynomials in b oth F ourier domains. It is well-kno wn that S ( X ) is dense in L 2 ( X ). W e observ e the follo wing standard: 10 JONA THAN HOLLAND AND GEORGE SP ARLING Lemma 5. L et f ∈ S ( X ) . Then F ( ∂ x f )( ξ ) = 2 π iξ F f ( ξ ) . Acting on the set of Sc hw artz functions in L 2 ( X ), deﬁne op erators P , Q ( u ) by P f ( x ) = ih∂ x f ( x ) Q ( u ) f ( x ) = (2 π x + ihH ( u ) ∂ x ) f ( x ) . On the momen tum domain, one then has P g ( ξ ) = − 2 π hξ g ( ξ ) Q ( u ) g ( ξ ) = ( i∂ ξ − 2 π hH ( u ) ξ ) g ( ξ ) . These operators are self-adjoint unbounded op erators on L 2 ( X ), satisfying the canonical W eyl commutation relations [ P , Q ( u )] = 2 π ihI where I is the identit y op erator. Thus the op erators giv e a family of inﬁnitesimal (Sc hr¨ odinger) representations of the Heisenberg algebra on L 2 ( X ), indexed by the parameter h  = 0. 2.2.1. Inte gr ate d r epr esentation. W e no w consider the asso ciated in tegrated unitary represen tations of the Heisenberg groups. The Heisenberg group ˜ H shall consist of triples ( z , q , p ) ∈ H = R × X × X , with multiplication (7) ( z , p, q )( z ′ , p ′ , q ′ ) =  z + z ′ + 1 2 ( q .p ′ − p.q ′ ) , p + p ′ , q + q ′  . Lemma 6. The multiplic ation law (7) gives ˜ H the structur e of a gr oup in which ( z , p, q ) − 1 = ( − z , − p, − q ) and for which (0 , 0 , 0) is identity. Pr o of. Only the asso ciative law is not obvious. W e ha ve ( ( z , p, q )( z ′ , p ′ , q ′ ) ) ( z ′′ , p ′′ , q ′′ ) = ( z + 1 2 ( q p ′ − pq ′ ) + 1 2 (( q + q ′ ) p ′′ − ( p + p ′ ) q ′′ ) + z ′ + z ′′ , p + p ′ + p ′′ , q + q ′ + q ′′ ) and ( z , p, q ) ( ( z ′ , p ′ , q ′ )( z ′′ , p ′′ , q ′′ ) ) = ( z + 1 2 ( q ( p ′ + p ′′ ) − p ( q ′ + q ′′ )) + 1 2 ( q ′ p ′′ − p ′ q ′′ ) + z ′ + z ′′ , p + p ′ + p ′′ , q + q ′ + q ′′ ) . Comparing terms, these are readily seen to b e equal. □ Let h  = 0 b e a ﬁxed real constant. Consider the action of ˜ H on L 2 ( X ) giv en by (8) ρ h,u ( z , p, q ) f ( x ) = e  2 hz + 2 x i p i + h ( q .p − p T H ij ( u ) p )  f ( x − hH ( u ) p + hq ) Lemma 7. The deﬁnition (8) is a unitary action of the gr oup ˜ H on L 2 ( X ) with inﬁnitesimal gener ators − iP and iQ . SACHS EQUA TIONS AND PLANE W A VES, V 11 Pr o of. Unitarit y follows from the fact that e h ( z ) = exp(2 π ihz ) = e (2 hz ) is a unitary c haracter (all of the v ariables are real), together with the translation-inv ariance of the Leb esgue measure. T o v erify that it is an action of the group, we must chec k that ρ h,u ( z , p, q )( ρ h,u ( z ′ , p ′ , q ′ ) f ( x )) = ρ h,u ( z + z ′ + 1 2 ( q .p ′ − p.q ′ ) , p + p ′ , q + q ′ ) f ( x ) . Expanding the LHS giv es ρ h,u ( z , q , p )( ρ h,u ( z ′ , p ′ , q ′ ) f ( x )) = ρ h,u ( z , p, q ) e (2 hz ′ + 2 x.p ′ + h ( q ′ .p ′ − H ( p ′ , p ′ ))) f ( x − hH ( p ′ ) + hq ′ ) = e (2 hz + 2 x.p + h ( q .p − H ( p, p )) + 2 hz ′ + 2( x − hH ( p ) + hq ) .p ′ + h ( q ′ .p ′ − H ( p ′ , p ′ ))) f ( x − hH ( p ′ ) + hq ′ − hH ( p ) + hq ) = e (2 hz + 2 hz ′ + h ( q .p ′ − p.q ′ ) + 2 x. ( p + p ′ ) + h (( q + q ′ ) . ( p + p ′ ) − H ( p + p ′ , p + p ′ ))) f ( x − hH ( p + p ′ ) + h ( q + q ′ )) . The ﬁnal line is easily seen to b e the exp ected RHS. Finally , we must determine the inﬁnitesimal generators. F or the ﬁrst generator, tak e ( z , p, q ) = (0 , 0 , q ) with q inﬁnitesimal: ρ h,u (0 , 0 , q ) f ( x ) = f ( x + hq ) = f ( x ) + hq · ∇ f ( x ) = f ( x ) − iq .P f ( x ) . F or the second generator, take ( z , p, q ) = (0 , p, 0) with p inﬁnitesimal: ρ h,u (0 , p, 0) f ( x ) = e (2 x.p ) f ( x − hH ( p )) = f ( x ) + (2 π ix.p − hH ( p ) . ∇ ) f ( x ) = f ( x ) + ip.Qf ( x ) . □ 2.2.2. A ction on a plane wave. The Heisenberg group acts by isometries on the plane wa v e R ( G ) = 2 du dv − dx T G ( u ) dx , preserving the wa v e fronts. The right action on p oin ts is ( u, v , x ) . ( z , p, q ) = ( u, v + z + p T x + 1 2 ( p T q − p T H ( u ) p ) , x + q − H ( u ) p ) . This deﬁnes a represen tation of ˜ H on the set of square-integrable functions on R × X (at a ﬁxed u ). Note that the Leb esgue measure is preserved, so it is a unitary representation. Call this representation π u . After F ourier transform in the v -v ariable, the ﬁbre at frequency h is not literally ρ h,u but is unitarily equiv alen t to it via the dilation D h f ( x ) = | h | − n/ 2 f ( x/h ) . Equiv alently , if e ρ h,u denotes the representation on the h -ﬁbre coming from the geometric action, then e ρ h,u = D − 1 h ρ h,u D h . Th us π u decomp oses ﬁbrewise into representations unitarily equiv alent to the Sc hr¨ odinger mo dels ρ h,u . W e shall therefore henceforth consider the group obtained from the Heisenberg group b y taking the quotient by a discrete subgroup of its cen ter. 12 JONA THAN HOLLAND AND GEORGE SP ARLING 3. Symplectic vector sp aces Let ( T , ω ) denote a 2 n -dimensional symplectic vector space ov er the real ﬁeld R . The symplectic group Sp( T ) denotes the group of all symplectic linear auto- morphisms of T , a Lie group of dimension n (2 n + 1). A subset S ⊂ T is called isotropic if the symplectic form is zero on all pairs of elements of S . The maximal isotropic sets are n -dimensional linear subspaces of T , called Lagrangian subspaces. Let LG( T ) denote the set of Lagrangian subspaces of T , a smooth pro jective v ariety of dimension n ( n + 1) / 2. W e consider the vector space T ⊗ C ov er the complex ﬁeld as a symplectic space where ω is now complex-v alued (skew, complex bilinear, and restricts to ω on the real T ). The Lagrangian Grassmannian LG( T ⊗ C ) is a complex pro jective (smo oth) v ariety , and LG( T ) is the set of ﬁxed p oints of the Galois inv olution. A pair A and B of Lagrangian subspaces are said to k -meet if the co dimension of A ∩ B in A (or, equiv alently , in B ) is k . Th us tw o Lagrangian subspaces 0-meet iﬀ they are iden tical. A polarization of T is a linear operator X : T → T ⊗ C such that [ X , ¯ X ] = 0, ω ( X x, X y ) = − ω ( x, y ) for all x, y ∈ T , and X 2 = 1 (the iden tity of T ). A p olarization X is r e al if its image is the real subspace T of T ⊗ C . If i X is real, then w e say that X is an imaginary p olarization. A p olarization X can b e a mixture of real and imaginary , but b ecause [ X , ¯ X ] = 0, the p olarization decomp oses into a real part and an imaginary part that act on complementary inv ariant subspaces of T , so w e henceforth consider only these tw o cases. Giv en a p olarization X , deﬁne the pro jection op erators X ± on to the resp ectiv e ± 1 eigenspaces of X , X ± : X + = (1 + X ) / 2 , X − = (1 − X ) / 2 . Th us we ha ve the usual relations 1 = X + + X − , X = X + − X − , X 2 ± = X ± , X + X − = X − X + = 0 . An imaginary p olarization X is called p ositive if iω ( x, X x ) > 0 for all x ∈ T , x  = 0. More generally , the signature of an imaginary p olarization X is the signature of the quadratic form iω ( x, X x ) on T . The signature of a quadratic form is a pair of integers ( k + , k − ) b eing the num b er of ± 1 in the Sylv ester normal form. The (inertial) index of a quadratic form is k + − k − , the trace of its Sylv ester normal form. A quadratic form is said to split if it has index zero, i.e., k + = k − . Accordingly , we sa y that an imaginary p olarization X is split if the quadratic form iω ( x, X x ) has signature ( n, n ). F or a ﬁnite-dimensional real vector space A , the space of Leb esgue measurable functions on A is denoted by M r ( A ). If A is a ﬁnite-dimensional complex vector space, then M i ( A ) is the space of all holomorphic functions on A . 4. Heisenberg groups Fix a non-trivial real character e : R → U (1), which we take to b e e ( t ) = e π it . The Heisen berg group of T with character e is the set H ( T ) = U (1) × T whose group la w is (9) ( t, x )( t ′ , x ′ ) = ( tt ′ e ( xx ′ ) , x + x ′ ) . W e alwa ys ﬁx a left Haar measure on H , whic h is also a righ t Haar measure. W e ﬁx this in suc h a w a y that it is the product measure of the inv ariant probabilit y SACHS EQUA TIONS AND PLANE W A VES, V 13 measure on U (1) and the Liouville measure on T . The universal co ver ˜ H is iden tiﬁed with R × T and has group law ( t, x )( t ′ , x ′ ) = ( t + t ′ + xx ′ , x + x ′ ) . As mentioned in § 2.2.2, we shall henceforth consider the group H in lieu of its univ ersal cov er. The Heisen b erg group is equipp ed with an exact sequence 1 → U (1) → H π − → T → 0 . Let A ⊂ H be a maximal abelian subgroup such that π | A : A → π ( A ) is an isomorphism on to its image in T . Then A is closed. Sev eral p ossibilities of A are: • π ( A ) is a Lagrangian subspace A of T , and A is just { 1 } × A . • π ( A ) is a self-dual lattice Λ in T , and A is a set of pairs ( e ( Q ( λ )) , λ ) where λ ∈ Λ and Q is an F 2 -v alued quadratic form on the F 2 v ector space Λ / 2Λ, suc h that Q ( x + y ) + Q ( x ) + Q ( y ) + ω ( x, y ) ≡ 0 (mo d 2) for all x, y ∈ Λ / 2Λ. These p ossibilities are ob viously not mutually exclusive. Nevertheless, π ( A ) is a closed ab elian subgroup of the vector space T , and so factorizes as the pro duct of a lattice in low er dimension and isotropic subspace complementary to the lattice. So these tw o cases are the main ones, up to decomp osing things appropriately in lo wer dimensions. The complexiﬁed Heisenberg group is the group H C = C ∗ × T ⊗ C , with the same group law (9). Here e is extended to the unique analytic quasicharacter e : C → C ∗ whic h restricts to the character e on R , i.e., e ( z ) = e π iz for z ∈ C . Let M r ( H ) b e the space of measurable functions on H that commute with the action of the cen ter: f ( t, x ) = tf (1 , x ) for all t ∈ U (1) and x ∈ T . Let M i ( H ) b e the space of holomorphic functions on H C that commute with the action of the cen ter C ∗ . Henceforth, all functions w e shall consider on H or H C satisfy the condition of comm uting with the center. If f ∈ M r ( H ), then the left and right actions of an element ( t, x ) ∈ H are given, resp ectiv ely , by L ( t, x ) f ( t ′ , x ′ ) = f (( t, x )( t ′ , x ′ )) = tt ′ e ( xx ′ ) f (1 , x + x ′ ) , R ( t, x ) f ( t ′ , x ′ ) = f (( t ′ , x ′ )( t, x )) = tt ′ e ( − xx ′ ) f (1 , x + x ′ ) . Iden tical formulas apply for f ∈ M i ( H ). Giv en a subgroup A of H , let M r ( A \ H ) b e the set of left A -inv ariant elements of M r ( H ): L ( z ) f ( x ) = f ( x ) for all z ∈ A . (Also, let M r ( H / A ) b e the set of righ t A -in v ariant elements of M r ( H ).) A Sch w artz function on H is an element of M r ( H ) which is smo oth and decays at inﬁnity with all deriv atives. The conv olution of Sch wartz functions f , g on H is deﬁned b y ( f ∗ g )( x ) = Z H f ( y ) g ( y − 1 x ) dy The con volution of Sch w artz functions is Sch wartz. Moreo ver, L ( x )( f ∗ g ) = ( L ( x ) f ) ∗ g and R ( x )( f ∗ g ) = f ∗ ( R ( x ) g ). F or a temp ered distribution T and a function f , the conv olutions T ∗ f and f ∗ T are deﬁned b y duality against a 14 JONA THAN HOLLAND AND GEORGE SP ARLING Sc hw artz “test function” ϕ : ⟨ T ∗ f , ϕ ⟩ = Z T ( y ) f ( y − 1 x ) dy ϕ ( x ) dx = Z f ( y − 1 x ) ϕ ( x ) dx T ( y ) dy = Z f ( y − 1 x ) ϕ ( x ) dx T ( y ) dy = Z ( ϕ ∗ ˜ f )( y ) T ( y ) dy = ⟨ T , ϕ ∗ ˜ f ⟩ . ⟨ f ∗ T , ϕ ⟩ = Z f ( y ) T ( y − 1 x ) dy ϕ ( x ) dx = Z f ( xy ) T ( y − 1 ) dy ϕ ( x ) dx = Z ϕ ( x ) f ( xy ) dx T ( y − 1 ) dy = ⟨ ˜ T , ( ˜ f ∗ ˜ ϕ ) ∼ ⟩ . Here w e denote by ˜ f (or f ∼ ) the function ˜ f ( x ) = f ( x − 1 ). F or an imaginary p olarization J , a function f ∈ M i ( H ) is called left J -holomorphic if f is left J − -in v ariant. The space of left J -holomorphic functions is denoted by M ( J − \ H ) . A left J -holomorphic function is determ ined by its restriction to the real group H . Giv en a p olarization X , we hav e a map from M ( X + ) to M ( X − \ H ), as follows. Let f ∈ M ( X + ), and deﬁne ind X f ∈ M ( X − \ H ) by ind X f ( t, z ) = te ( ω ( X + z , z )) f ( X + z ) . (This deﬁnition mak es sense in b oth M r and M c .) As a sp ecial case, we hav e ind X 1( t, z ) = te ( ω ( X + z , z )). When X is a p ositive imaginary polarization, ind X 1 is Sc h w artz on T and left X -holomorphic. Left holomorphicit y of f ∈ S ( H ) is c haracterized b y a Cauc h y–Riemann diﬀeren tial equation d f ◦ X − = 0. W riting z = z + + z − , z ± = X ± z ∈ X ± , this equation is ∂ − f ( t, z ) = iπ f ( t, z ) ω ( z + , dz − ). Th us: • F or an imaginary p olarization J , the space S ( J − \ H ) is the set of Sch wartz functions on T that are left J -holomorphic. • F or a real p olarization K , the space S ( K − \ H ) is the set of left K − -in v ariant functions whose restrictions to K + are Sc hw artz. 5. F ourier transform Let X b e a Lagrangian subspace. Deﬁne a temp ered distribution δ X on H b y ⟨ δ X , f ⟩ = Z X f (1 , x ) dx. The measure on the right-hand side is an arbitrary Haar measure on X . This is a con tinuous linear functional in the Sc h w artz space. The distribution δ X is X -biin v ariant. More generally , the same δ X deﬁnes an X -biinv ariant temp ered dis- tribution on any S ( A \ H ) where A is an y isotropic subspace independent from X . Because it dep ends on a Haar measure on X (whic h is not ﬁxed by the Liouville measure on T ), δ X tak es v alues in the dual space to the space of Haar measures on X (a one dimensional space of densities). SACHS EQUA TIONS AND PLANE W A VES, V 15 Deﬁnition 4. The F ourier tr ansform of a function f on H with r esp e ct to X is the c onvolution δ X ∗ f which makes sense pr ovide d f ∈ S ( H ) or f ∈ S ( A \ H ) wher e A is indep endent of X . Theorem 2. L et X b e a L agr angian subsp ac e of T . Then δ X ∗ f b elongs to S ( X \ H ) for al l f ∈ S ( A \ H ) , wher e A is an isotr opic subsp ac e indep endent of X . Pr o of. Left inv ariance follo ws by what w e ha ve already discussed. W e need to sho w that δ X ∗ f is Sch w artz. Enlarging A to a Lagrangian complement, we must therefore show that z 7→ ( δ X ∗ f )(1 , z ) is a Sch wartz function for z belonging to A . Let X b e the p olarization with eigenspaces X + = X and X − = A . W e then ha ve f ( t, z ) = e ( X + z z ) ϕ + ( X + z ) where ϕ is a Sch w artz function on X + . The F ourier transform is ( δ X ∗ f )( z ) = Z H δ X ( y ) f ( y − 1 z ) dy = Z X f ( y − 1 z ) dy = Z X e ( z y + + ( z + − y + )( z − y + )) ϕ ( z + − y + ) dy = Z X f ( y − 1 z ) dy = Z X e ( z y + − y + z )) ϕ ( − y + ) dy ( z + = 0) = Z X e (2 y + z ) ϕ ( y + ) dy i.e., the classical F ourier transform of ϕ . □ A corollary to the abov e pro of is that the F ourier transform f 7→ δ X ∗ f is unitary with resp ect to the natural L 2 norms on S ( X \ H ) and S ( A \ H ). Moreov er, it in tertwines the (unitary) righ t actions of the Heisen b erg group (i.e., the Sc hr¨ odinger represen tations). Supp ose that X is a real p olarization of T . Consider the comp osite S ( X + ) S ( X − \ H ) S ( X + \ H ) S ( X − \ H ) ind X δ X + ∗ δ X − ∗ W e then hav e F ourier inv ersion: Lemma 8. If X is a r e al p olarization of T , then for al l f ∈ S ( X − \ H ) , δ X − ∗ δ X + ∗ f = f wher e the L eb esgue me asur es on X − and X + ar e chosen such that the pr o duct me a- sur e is the Liouvil le me asur e on T . (W e can write the conclusion of the lemma more inv ariantly in terms of Haar measures µ X ± on X ± : ( µ X − ⊗ µ X + ) · δ X − ∗ δ X + ∗ f = dµ X − dµ X + d L · f where dµ X − dµ X + /d L is the (constant!) Radon–Nikodym deriv ativ e of the pro duct measure µ X − × µ X + with resp ect to the Liouville measure L on T .) 16 JONA THAN HOLLAND AND GEORGE SP ARLING Pr o of. W e write x = x + + x − as the decomp osition of T into eigenspace comp onen ts x ± ∈ X ± . Then ind X f ( t, z ) = te ( z + z ) f ( z + ) δ X + ∗ ind X f ( t, z ) = Z X + ind X f ((1 , − x + )( t, z )) dx + = Z X + te ( z x + ) ind X f (1 , z − x + ) dx + = Z X + te ( z x + + ( z + − x + ) z ) f ( z + − x + ) dx + = Z X + te ( z ( z + − x + ) + x + z ) f ( x + ) dx + = te ( z z + ) Z X + te (2 x + z ) f ( x + ) dx + δ X − ∗ δ X + ∗ ind X f ( t, z ) = Z X − δ X + ∗ ind X f ((1 , x − )( t, z )) dx − = Z X − te ( z x − ) δ X + ∗ ind X f (1 , z − x − ) dx − = Z X − te ( z x − + ( z − x − ) z + ) Z X + e (2 x + ( z − x − )) f ( x + ) dx + dx − = Z X − te ( z ( z − − x − ) + x − z + ) Z X + e (2 x + x − ) f ( x + ) dx + dx − = te ( z z − ) Z X + f ( x + ) dx + Z X − e (2 x + x − − 2 z + x − ) dx − = te ( z z − ) Z X + f ( x + ) δ ( x + − z + ) dx + . In the last few steps, we used the fact that f is Sch w artz on X + , and therefore the innermost integral after applying “F ubini’s theorem” conv erges to a delta function as indicated, in the sense of temp ered distributions on X + . □ Let f ∈ M ( X + ) and supp ose that X − and X ′ − are tw o Lagrangian complements X + . Let X = [ X + X − ] and X ′ = [ X + X ′ − ]. Then ind X f is a left X − -in v ariant function and ind X ′ f is a left X ′ − -in v ariant function. These functions are related by ind X ′ f ( t, z ) = ind X f  t, 1 2 ( X + X ′ ) z  , in whic h 1 2 ( X + X ′ ) is the canonical symplectic reﬂection which exchanges the subspaces X − and X ′ − and which is the identit y on X + . Indeed, we hav e X + ( X + X ′ ) / 2 = X ′ + , and therefore ind X ′ f ( t, z ) = te ( X ′ + z z ) f ( X ′ + z ) = te ( ω ( X + ( X + X ′ ) z , ( X + X ′ ) z ) / 4) f ( X + ( X + X ′ ) z / 2) = ind X f ( t, ( X + X ′ ) z / 2) SACHS EQUA TIONS AND PLANE W A VES, V 17 P erhaps more interestingly , let X b e a ﬁxed real p olarization of T , and let Y b e any Lagrangian subspace complementary to b oth eigenspaces X ± . The triple con volution δ X − ∗ δ Y ∗ δ X + ∗ f , f ∈ S ( X − \ H ) , is again left X − -in v ariant, and in fact is a scalar m ultiple of f whose v alue is determined b y the Maslov index . Deﬁnition 5. The Kashiwara form on T ⊕ T ⊕ T is the quadr atic form κ ( a, b, c ) = ω ( a, b ) + ω ( b, c ) + ω ( c, a ) . The Maslo v index of an or der e d triple of L agr angian subsp ac es ( A , B , C ) is the inertial index (i.e., the numb er of p ositive eigenvalues minus the numb er of ne gative eigenvalues) of κ | A ⊕ B ⊕ C . Lemma 9. L et ( A , B , C ) b e p airwise c omplementary L agr angian subsp ac es, and set Q ( c ) = − ω ([ AB ] + c, [ AB ] − c ) , c ∈ C . Then τ ( A , B , C ) = idx( Q ) . Pr o of. Em b ed C in to A ⊕ B ⊕ C via γ ( c ) = ([ AB ] + c, [ AB ] − c, c ). Since c = [ AB ] + c + [ AB ] − c , setting a = [ AB ] + c and b = [ AB ] − c giv es κ ( γ ( c )) = ω ( a, b ) + ω ( b, c ) + ω ( c, a ) = − ω ( a, b ) = − ω ([ AB ] + c, [ AB ] − c ) = Q ( c ) . (The second equality uses c = a + b and the Lagrangian prop ert y .) On the comple- men tary subspace A ⊕ B ⊕ { 0 } ⊂ A ⊕ B ⊕ C , the Kashiwara form κ ( a, b, 0) = ω ( a, b ) has inertial index zero b ecause the symplectic form is itself of split signature. The lemma follo ws by the additivity of the inertial index. □ W e can now state and prov e the main result of this section. Theorem 3. L et A , B , C b e p airwise c omplementary L agr angian subsp ac es of T , with Haar me asur es µ A , µ B , µ C , and let L denote the Liouvil le me asur e on T . Then ( µ B ⊗ µ C ⊗ µ A ) · δ B ∗ δ C ∗ δ A ∗ ind AB f =  dµ A dµ B d L · dµ B dµ C d L · dµ C dµ A d L  1 / 2 e  − sgn( h ) 4 τ ( A , B , C )  ind AB f , wher e τ ( A , B , C ) is the Maslov index of the triple ( A , B , C ) . The pro of pro ceeds in three steps: ﬁrst we ev aluate the triple con v olution up to a scalar Gaussian integral; then we ev aluate that integral b y stationary phase; ﬁnally w e identify the resulting phase with the Maslov index via Lemma 9. Pr o of. Step 1: Reduction to a Gaussian in tegral. Put P = [ AB ] + and Q = [ AB ] − = I − P . W rite a general element of the Heisenberg group as z = ( s, x ) with s ∈ R and x ∈ T . By the group la w, righ t-multiplying by b ∈ B , c ∈ C , a ∈ A in succession giv es z · b · c · a =  s + 1 2  ω ( x, b ) + ω ( x + b, c ) + ω ( x + b + c, a )  , x + a + b + c  , and therefore ind AB f ( z · b · c · a ) = e h  s + 1 2  ω ( x, b ) + ω ( x + b, c ) + ω ( x + b + c, a ) − ω ( P ( x + a + b + c ) , Q ( x + a + b + c ))   f ( P ( x + a + b + c )) . 18 JONA THAN HOLLAND AND GEORGE SP ARLING Set u = P ( x + a + b + c ) ∈ A . Since P is the pro jection onto A along B , we hav e u = a + P ( x + c ) , hence a = u − P ( x + c ) . Substituting this and using that A and B are Lagrangian, a straightforw ard expan- sion sho ws that the phase separates as ind AB f ( z · b · c · a ) = e h  s − 1 2 ω ( P x, Qx ) + ω ( u − P x, b ) + 1 2 ω ( P c, c )  f ( u ) . (If one obtains ω ( P x − u, b ) instead, this is equiv alent after the change of v ariable b 7→ − b , which do es not aﬀect the Haar measure on B .) W e now integrate ﬁrst in b ∈ B . By the F ourier inv ersion lemma for the com- plemen tary pair ( A , B ), the kernel e h ( ω ( u − P x, b )) pro duces the delta distribution δ A ( u − P x ), so the u –integration collapses to u = P x . Thus δ B ∗ δ C ∗ δ A ∗ ind AB f ( z ) = dµ A dµ B d L e h  s − 1 2 ω ( P x, Qx )  f ( P x ) λ ( µ C ) = dµ A dµ B d L λ ( µ C ) ind AB f ( z ) , where (10) λ ( µ C ) = lim ϵ → 0 + Z C e − π ϵ ∥ c ∥ 2 e h  1 2 ω ( P c, c )  dµ C ( c ) . Since P = [ AB ] + and Q = I − P , for c ∈ C we hav e ω ( P c, c ) = ω ( P c, Qc ) = − Q C ( c ) , where Q C ( c ) = − ω ([ AB ] + c, [ AB ] − c ) is the quadratic form from Lemma 9. Hence (11) δ B ∗ δ C ∗ δ A ∗ ind AB f = dµ A dµ B d L λ ( µ C ) ind AB f . Step 2: Ev aluation of the Gaussian in tegral. W e use the standard F resnel form ula: if R is a non-degenerate real quadratic form on a real vector space V with Haar measure µ , then (12) lim ϵ → 0 + Z V e − π ϵ ∥ v ∥ 2 e h  1 2 R ( v )  dµ ( v ) = | det R | − 1 / 2 e  idx( hR ) 4  , where det R is computed relative to µ and idx( hR ) = sgn( h ) idx( R ) is the inertial index. Indeed, after diagonalising R one reduces to the one-v ariable integral Z ∞ −∞ e − π ϵx 2 e h  1 2 q x 2  dx = Z ∞ −∞ e − π ( ϵ − ihq ) x 2 dx = ( ϵ − ihq ) − 1 / 2 , whose limit has phase e (sgn( hq ) / 4). Applying (12) to (10) with R C ( c ) = ω ( P c, c ) = − Q C ( c ) , w e obtain idx( hR C ) = idx( − hQ C ) = − sgn( h ) idx( Q C ) = − sgn( h ) τ ( A , B , C ) , SACHS EQUA TIONS AND PLANE W A VES, V 19 b y Lemma 9. Therefore (13) λ ( µ C ) = | det Q C | − 1 / 2 e  − sgn( h ) 4 τ ( A , B , C )  . Step 3: Iden tifying the determinant prefactor. Fix an iden tiﬁcation T ∼ = K ⊕ K , with K an n -dimensional Euclidean space, such that ω (( k 1 , k 2 ) , ( k ′ 1 , k ′ 2 )) = k 1 · k ′ 2 − k 2 · k ′ 1 . W rite the three pairwise complementary Lagrangians as graphs of symmetric en- domorphisms: A = { ( k , Ak ) } , B = { ( k , B k ) } , C = { ( k, C k ) } . F or c = ( k , C k ) ∈ C , write c = a + b, a ∈ A , b ∈ B . Solving for a giv es a =  ( A − B ) − 1 ( C − B ) k , A ( A − B ) − 1 ( C − B ) k  , hence Q C ( k , C k ) = − ω ( a, c ) = − k T ( C − B )( A − B ) − 1 ( C − A ) k . Consequen tly , | det Q C | = | det( C − B ) | | det( C − A ) | | det( A − B ) | . On the other hand, relative to the Lebesgue measures on the graph parameters, the pro duct Haar measures satisfy dµ A dµ B d L = | det( A − B ) | − 1 , dµ B dµ C d L = | det( B − C ) | − 1 , dµ C dµ A d L = | det( C − A ) | − 1 . Therefore dµ A dµ B d L | det Q C | − 1 / 2 = | det( A − B ) | − 1  | det( A − B ) | | det( C − B ) | | det( C − A ) |  1 / 2 =  | det( A − B ) | | det( B − C ) | | det( C − A ) |  − 1 / 2 =  dµ A dµ B d L dµ B dµ C d L dµ C dµ A d L  1 / 2 , since | det( B − C ) | = | det( C − B ) | . Substituting (13) into (11) and using the preceding identit y yields ( µ B ⊗ µ C ⊗ µ A ) · δ B ∗ δ C ∗ δ A ∗ ind AB f =  dµ A dµ B d L dµ B dµ C d L dµ C dµ A d L  1 / 2 e  − sgn( h ) 4 τ ( A , B , C )  ind AB f . This is the desired form ula. □ 20 JONA THAN HOLLAND AND GEORGE SP ARLING 6. Bargmann and thet a transforms No w let J no w b e an imaginary p olarization. F or t ∈ U (1) and z ∈ T , put η J ( t, z ) = te ( J + z z ). Then: Lemma 10. η J is left J -holomorphic and right J -antiholomorphic. Pr o of. L (1 , w − ) η ( t, z ) = te ( w − z ) η (1 , z + w − ) = te ( w − z + + z + ( z + w − )) = te ( z + z ) = η ( t, z ) . R (1 , w + ) η ( t, z ) = te ( z w + ) η (1 , z + w + ) = te ( z − w + + ( z + + w + ) z ) = te ( z + z + z − w + + w + z − ) = te ( z + z ) = η ( t, z ) . □ Pr o of of the or em. Left J -holomorphicity of η J ∗ f follows from Lemma 10: η J is left J -holomorphic, hence so is ev ery left-translate, and the conv olution inherits this in v ariance. It remains to show that η J ∗ f is Sch wartz. The calculation in the pap er already giv es (14) η J ∗ f ( t, z ) = t e ( J + z z ) Z T e ( y + y − 2 z + y ) f (1 , y ) dy , where z + = J + z and y + = J + y . Since all information ab out the t -dep endence is the ov erall factor of t ∈ U (1), it suﬃces to study the function F : T → C deﬁned b y F ( z ) = e ( J + z z ) G ( z + ) , G ( w ) = Z T e ( y + y ) e ( − 2 wy ) f (1 , y ) dy . P ositivit y and the Gaussian factor. By the p ositivity assumption iω ( x, J x ) > 0 for all x  = 0, the quadratic form x 7→ i · 2 J + xx is real and p ositive-deﬁnite on T . Since e ( z ) = e π iz and J + z z is complex-v alued, | e ( J + z z ) | = e − π im( J + z z ) . P ositive-deﬁniteness giv es a constant c > 0 such that im( J + z z ) ≥ c ∥ z ∥ 2 for all z ∈ T , so (15) | e ( J + z z ) | ≤ e − π c ∥ z ∥ 2 . G is Sch w artz in z + . The function y 7→ e ( y + y ) f (1 , y ) is in S ( T ): e ( y + y ) = e ( J + y y ) has mo dulus e − π im( J + y y ) ≤ e − π c ∥ y ∥ 2 b y (15) applied with z = y , so multi- plication by e ( y + y ) maps S ( T ) to S ( T ). Therefore G ( w ) is (up to the iden tiﬁcation of X + with the dual of T via ω ) the F ourier transform of a Sch w artz function, and is itself Sc hw artz: for every p olynomial p ( w ) and every N ≥ 0, (16) | p ( w ) G ( w ) | ≤ C p,N (1 + ∥ w ∥ ) − N . Concretely , diﬀerentiating under the integral sign, ∂ α w G ( w ) = R T ( − 2 π iy ) α e ( y + y − 2 w y ) f (1 , y ) dy , and eac h suc h integral is b ounded uniformly in w because y 7→ y α e ( y + y ) f (1 , y ) is in L 1 ( T ); the rapid decay in w then follows from rep eated inte- gration b y parts in y . Sc h w artz estimates for F . W e m ust estimate | z β ∂ α z F ( z ) | for all multi-indices α, β . W rite z = z + + z − with z ± = J ± z . SACHS EQUA TIONS AND PLANE W A VES, V 21 Derivatives in z − . Left J -holomorphicit y of F (as a function on T ) means ∂ z − F = 0 in the sense of the Cauch y–Riemann equation ∂ − F = iπ F ω ( z + , dz − ). Consequen tly , ∂ k z − F can b e expressed as a sum of pro ducts of F with p olynomials in z + , so all z − -deriv atives reduce to z + -deriv atives times p olynomial factors; it suﬃces to b ound ∂ α z + F and z β F . z + -derivatives. Diﬀeren tiating (14) in z + : ∂ α z + F ( z ) = X α ′ ≤ α  α α ′  ( ∂ α ′ e ( J + z z )) · ( ∂ α − α ′ G )( z + ) . Eac h z + -deriv ative of e ( J + z z ) pro duces a p olynomial factor in z + times e ( J + z z ), so the e ( J + z z )-factor is never diﬀeren tiated aw ay . Th us | ∂ α z + F ( z ) | ≤ p α ( z + ) e − π c ∥ z ∥ 2 sup w | ( ∂ α − α ′ G )( w ) | for some p olynomial p α , and the righ t-hand side is rapidly decreasing in ∥ z ∥ . Polynomial gr owth. F or any monomial z β = z β + + z β − − , the factor | z β | grows at most p olynomially in ∥ z ∥ , while e − π c ∥ z ∥ 2 deca ys faster than an y pow er, so the pro duct | z β ∂ α F ( z ) | is b ounded. Indeed, for every N and α, β , | z β ∂ α z F ( z ) | ≤ C α,β ,N (1 + ∥ z ∥ ) − N b y combining the Gaussian b ound (15) with the Sc h w artz estimates (16) for G . This establishes that F , and hence η J ∗ f , is Sch wartz. □ 6.1. Theta transform. Let Λ b e a self-dual lattice in T , meaning that the sym- plectic form is integral on Λ, and its dual lattice is itself under the symplectic form. (The latter condition is equiv alent to the statement that T / Λ has Liouville mea- sure 1.) Consider a subgroup σ (Λ) of H that splits the pro jection π onto Λ. W e construct all suc h splittings. Let Q b e an F 2 -v alued quadratic form on Λ / 2Λ, suc h that Q ( x + y ) + Q ( x ) + Q ( y ) + ω ( x, y ) ≡ 0 (mo d 2). Then let σ Q ( λ ) = ( Q ( λ ) , λ ). It is easily sho wn that σ Q (Λ) is an (ab elian) subgroup of H , and σ Q is clearly a splitting. W e denote by Λ Q the subgroup σ Q (Λ) determined b y the data Q . Let δ Λ Q b e the counting measure supp orted on the discrete subgroup Λ Q of H . Then δ Λ Q is a temp ered distribution on H whic h is Λ Q -biin v ariant. Deﬁnition 6. Given a discr ete ab elian sub gr oup Λ Q liﬁng a lattic e Λ as ab ove, the theta tr ansform of f ∈ S ( H ) is the function f ∗ δ Λ Q . Theorem 4. f ∗ δ Λ Q ∈ S ( H / Λ Q ) for al l f ∈ S ( H ) . Pr o of. f ∗ δ is obviously righ t Λ Q -in v ariant and smo oth. But H / Λ Q is a (compact) torus, so all smo oth functions are Sc hw artz. □ Let J b e a p ositiv e imaginary p olarization. F or any f ∈ S ( J − \ H ), the conv olu- tion f ∗ δ Λ Q is left J − -in v ariant, i.e., left J -holomorphic. It is also right Λ Q -in v ariant. Therefore it also has a quasip erio dicit y with resp ect to the left action of Λ Q : Theorem 5. F or w ∈ Λ Q , L ( w )( f ∗ δ Λ Q )( z ) = e (2 z w ) f ∗ δ Λ Q ( z ) . 22 JONA THAN HOLLAND AND GEORGE SP ARLING Prop osition 3. L et Y = H ( s ) and cho ose a c omplementary L agr angian X , so that H ( u ) = { H ( u ) η + η | η ∈ Y } , wher e H ( u ) : Y → X is symmetric. L et ˆ ϕ 0 ∈ S ( Y ) , and let ϕ u ( v , x ) = Z Y e h  v + x T η + 1 2 η T H ( u ) η  ˆ ϕ 0 ( η ) d h η b e the c orr esp onding Schr¨ odinger evolution in the form of The or em 1. Then for every p ositive imaginary p olarization J , the Bar gmann tr ansform of ϕ u is ( η J ∗ ϕ u )(1 , z ) = e ( J + z z ) Z Y e  η + η − 2 z + η  e h  1 2 η T H ( u ) η  ˆ ϕ 0 ( η ) d h η . In p articular, r elative to the Bar gmann kernel at time s , Schr¨ odinger evolution acts by multiplic ation by the same quadr atic phase e h  1 2 η T ( H ( u ) − H ( s )) η  that app e ars in the r e al-p olarize d mo del. Thus the same Hamiltonian curve H ( u ) governs b oth the Schr¨ odinger and the Bar gmann r e alizations. Pr o of. Apply the conv olution formula (14) to the function ϕ u . The only u -dep endence of ϕ u is the quadratic factor e h ( 1 2 η T H ( u ) η ) app earing in its oscillatory represen- tation, so substituting that represen tation in to (14) yields exactly the displa y ed form ula. Replacing u by s and comparing the tw o expressions shows that the pas- sage from time s to time u is eﬀected by multiplication by e h ( 1 2 η T ( H ( u ) − H ( s )) η ) inside the Bargmann in tegral. □ Corollary 2. In the arithmetic situation of The or ems 4 and 5, the theta tr ansform of the evolve d Bar gmann datum is obtaine d by summing the kernel of Pr op osition 3 over Λ Q . Conse quently Schr¨ odinger evolution pr eserves the theta-function auto- morphy law and acts on theta data by the same quadr atic phase factor. Pr o of. The theta transform is conv olution with the lattice distribution δ Λ Q . Since Prop osition 3 identiﬁes Schr¨ odinger evolution in the Bargmann mo del with multi- plication by the quadratic phase inside the holomorphic kernel, summing ov er Λ Q preserv es the quasip erio dicity of Theorem 5 and yields the stated description of the ev olved theta datum. □ 7. Schr ¨ odinger evolution in a plane w a ve Consider a Rosen plane wa ve on U × R × X . Fix u = 0 in U . Let H ( u ) b e the Hamiltonian curve. A ﬁeld bund le is a bundle F ov er M , equipp ed with an action of the Heisenberg symmetry group, and having a connection on the central n ull geodesic, with parallel transp ort map Γ st : F s → F t . An example of a ﬁeld bundle is the bundle O U ( w ) of functions of pro jective degree w , pulled back to M so that the Heisenberg action is the trivial action. Then multiplication by ( | g ( t ) | / | g ( s ) | ) w/ 2 n maps O U ( w ) u = s → O U ( w ) u = t , and so the natural connection is Γ st = ( | g ( t ) | / | g ( s ) | ) w/ 2 n . In general, natural connections will b e more complicated. W e no w describ e the general Sc hr¨ odinger evolution in a plane w av e, b eginning from initial data at u = s . Let Y = H ( s ), and the Hamiltonian curve is given by SACHS EQUA TIONS AND PLANE W A VES, V 23 H ( u ) = { H ( u ) y + y | y ∈ Y } where H ( u ) : Y → X is a symmetric transformation with ˙ H positive deﬁnite. (17) S ( X ) ⊗ F ( s ) ind − − → S ( H ( s ) \ H ) ⊗ F ( s ) δ X ∗ − − → S ( X \ H ) ⊗ F ( s ) Γ su − − → S ( X \ H ) ⊗ F ( u ) δ H ( u ) ∗ − − − − → S ( H ( u ) \ H ) ⊗ F ( u ) res − − → S ( X ) ⊗ F ( u ) More brieﬂy , this collapses into a map Φ X , F ( s, u ) : S ( X ) ⊗ F ( s ) → S ( X ) ⊗ F ( u ) whic h we shall call the Schr¨ odinger evolution . W e show the sense in whic h Φ( s, u ) yields a solution of the Sc hr¨ odinger equation, in the case of F = O U ( − n/ 2). Let Y = H (0), and use these to deﬁne a splitting of T as X ⊕ Y . Let H ( u ) : Y → X be symmetric, such that H ( u ) = { H ( u ) y + y | y ∈ Y } . Note that H (0) = 0 and ˙ H ( u ) is p ositive-deﬁnite. Let ϕ 0 ∈ S α ( X ) denote the initial data for the evolution (w e will deal with the degrees later), and ˆ ϕ 0 ∈ S 1 − α ( Y ) the F ourier transform. The induced function ψ in H 1 − α ( X ◦ ) is ψ ( v , x, y ) = e h ( v + 2 − 1 x T y ) ˆ ϕ 0 ( y ) . W e hav e ψ (( v , x, y ) · (0 , x ′ , 0)) = ψ ( v − 2 − 1 y T x ′ , x + x ′ , y ) = e h ( v − 2 − 1 y T x ′ + 2 − 1 ( x + x ′ ) T y ) ˆ ϕ 0 ( y ) = e h ( v + 2 − 1 x T y ) ˆ ϕ 0 ( y ) = ψ ( t, x, y ) , so that ψ ∈ H 1 − α ( X ◦ ). Next, right conv olution by the Lagrangian distribution δ H ( u ) (i.e. av eraging ov er the subgroup H ( u )) is ( δ H ( u ) ∗ ψ )( v , x, y ) = Z H ( u ) ψ (( v , x, y ) · ξ ) d h ξ = Z Y ψ (( v , x, y ) · (0 , H ( u ) η , η ) d h η = Z Y ψ ( v + 2 − 1 ( x T η − y T H η ) , x + H η , y + η ) d h η = Z Y e h ( v + 2 − 1 ( − y T H η + x T η + ( x + H η ) T ( y + η ))) ˆ ϕ 0 ( y + η ) d h η T o simplify this integral further, we would normally make the change of v ariables ¯ η = y + η , but since this is immediately to b e restricted to X , we simply take y = 0: ϕ ( u, v , x ) = ( δ H ( u ) ∗ ψ )( v , x, 0) = Z Y e h ( v + x T η + 2 − 1 η T H ( u ) η ) ˆ ϕ 0 ( η ) d h η . So ϕ ∈ H α ( H ( u ) ◦ ) no w clearly solves a Schr¨ odinger equation of the form [4 π ih∂ u − ˙ H ( u )( ∂ x , ∂ x )] ϕ ( u, v , x ) = 0 . 24 JONA THAN HOLLAND AND GEORGE SP ARLING Theorem 6 (Local intert winer) . Supp ose that H ( u ) is a p ositive L agr angian curve, and that X and X ′ ar e L agr angian subsp ac es of T , e ach c omplementary to H ( u ) for al l u ∈ U . Fix a b ase p oint s ∈ U . In the splitting T = H ( s ) ⊕ X , write X ′ = { Y z 1 + z 1 | z 1 ∈ X } ( Y : X → H ( s ) symmetric) and H ( u ) = { K v + v | v ∈ H ( s ) } ( K : H ( s ) → X symmetric, p ositive-deﬁnite). Assume that X and X ′ ar e suﬃciently close relativ e to the curve , me aning that the op er ator I − Y K : H ( s ) → H ( s ) is p ositive-deﬁnite for al l u ∈ U . Fix a ﬁeld bund le F with p ar al lel tr ansp ort Γ , and let Φ X ( s, u ) : S ( X ) ⊗ F ( s ) → S ( X ) ⊗ F ( u ) b e the Schr¨ odinger evolution (17) in the X -pictur e. Then the diagr am S ( X ) ⊗ F ( s ) S ( X ) ⊗ F ( u ) S ( X ′ ) ⊗ F ( s ) S ( X ′ ) ⊗ F ( u ) Φ X ( s,u ) ρ s ρ u Φ X ′ ( s,u ) c ommutes, wher e ρ s f ( z ) = e h  1 2 ω ([ H ( s ) X ] + z , z )  f  −  H ( s ) XX ′  z  (18) and ρ u is given by the same expr ession with u r eplacing s : ρ u f ( z ) = e h  1 2 ω ([ H ( u ) X ] + z , z )  f  −  H ( u ) XX ′  z  . (19) F urthermor e, the inverse is ρ − 1 s f ( z ) = e h  1 2 ω ([ H ( s ) X ′ ] + z , z )  f  −  H ( s ) X ′ X  z  (20) (and likewise for ρ − 1 u ). Pr o of. W e expand the deﬁnition of Φ from (17) and show that each square of the resulting diagram S ( X ) s S 1 ( H ( s )) s H 1 ( X ◦ ) s H ( H ( u ) ◦ ) u S ( X ) u S ( X ′ ) s S 1 ( H ( s )) s H 1 ( X ′◦ ) s H ( H ( u ) ◦ ) u S ( X ′ ) u . F XH ( s ) ρ s ind δ H ( u ) ∗ Γ su res ρ u F X ′ H ( s ) ind δ H ( u ) ∗ Γ su res comm utes. Left square. Going across the top: F f ( ξ ) = Z X e h ( ω ( z , ξ )) f ( z ) d h z , ind( F f )( t, ξ ) = Z X e h  t − 1 2 ω ( P + ξ , ξ ) + ω ( z , P + ξ )  f ( z ) d h z , SACHS EQUA TIONS AND PLANE W A VES, V 25 where w e abbreviate P + = [ H ( s ) X ] + and, b elow, P ′ + = [ H ( s ) X ′ ] + . Right conv olu- tion b y δ H ( u ) giv es ( δ H ( u ) ∗ ind( F f ))( t, ξ ) = Z H ( u ) Z X e h  t + 1 2 ω ( ξ , µ ) − 1 2 ω ( P + ( ξ + µ ) , ξ + µ ) + ω ( z , P + ( ξ + µ ))  f ( z ) d h z d h µ. (21) Going across the b ottom, after the change of v ariables z 7→ −  H ( s ) XX ′  z that replaces the X ′ -in tegral by an X -int egral (cf. (18)): ( δ H ( u ) ∗ ind( F ( ρ s f )))( t, ξ ) = Z H ( u ) Z X e h  t + 1 2 ω ( ξ , µ ) − 1 2 ω ( P ′ + ( ξ + µ ) , ξ + µ ) − 1 2 ω ( P ′ + z , z ) + ω ( z , P ′ + ( ξ + µ ))  f ( z ) d h z d h µ. (22) It suﬃces to show that for each ﬁxed z ∈ X , the inner integrals o v er µ ∈ H ( u ) in (21) and (22) are equal. Select co ordinates relativ e to the splitting T = H ( s ) ⊕ X : P + =  I 0 0 0  , P ′ + =  I − Y 0 0  , H ( u ) = { [ µ 0 , K µ 0 ] | µ 0 ∈ H ( s ) } . W rite ξ = [ ξ 0 , ξ 1 ], µ = [ µ 0 , K µ 0 ], z = [0 , z 1 ], and set w = ξ 1 + z 1 . T op exp onent. Expanding the e xponent of (21) (excluding the ov erall t ): E top = 1 2 ω ( ξ , µ ) − 1 2 ω ( P + ( ξ + µ ) , ξ + µ ) + ω ( z , P + ( ξ + µ )) = − 1 2 µ 0 K µ 0 − w µ 0 − 1 2 ξ 1 ξ 0 − z 1 ξ 0 . (23) (Here µ 0 K µ 0 means µ T 0 K µ 0 , etc.) Completing the square in µ 0 (using K > 0): (24) E top = − 1 2  µ 0 + K − 1 w  K  µ 0 + K − 1 w  + 1 2 w K − 1 w − 1 2 ξ 1 ξ 0 − z 1 ξ 0 . Bottom exp onent. The same expansion applied to (22), now using P ′ + =  I − Y 0 0  , giv es (after a direct computation): (25) E bot = − 1 2 µ 0 Q µ 0 − w ( I − Y K ) µ 0 + 1 2 w Y w − 1 2 ξ 1 ξ 0 − z 1 ξ 0 , where Q = K ( I − Y K ) = K − K Y K . (The oﬀ-diagonal blo c k − Y in P ′ + couples µ 0 to ξ 1 and z 1 diﬀeren tly from the top case, mo difying b oth the quadratic and linear terms in µ 0 .) Since K and I − Y K are b oth p ositive-deﬁnite (the latter by h yp othesis), Q is p ositive-deﬁnite. Completing the square: (26) E bot = − 1 2  µ 0 + Q − 1 ( I − K Y ) w  Q  µ 0 + Q − 1 ( I − K Y ) w  + 1 2 w ( I − K Y ) Q − 1 ( I − K Y ) w + 1 2 w Y w − 1 2 ξ 1 ξ 0 − z 1 ξ 0 , where we hav e used ( I − Y K ) T = ( I − K Y ). The non-Gaussian remainder simpliﬁes as follo ws. Since Q = K ( I − Y K ), we hav e Q − 1 = ( I − Y K ) − 1 K − 1 , and therefore ( I − K Y ) Q − 1 ( I − K Y ) = ( I − K Y )( I − Y K ) − 1 K − 1 ( I − K Y ) = K − 1 ( I − K Y ) = K − 1 − Y , 26 JONA THAN HOLLAND AND GEORGE SP ARLING using ( I − K Y )( I − Y K ) − 1 = K − 1 Q ( I − Y K ) − 1 = K − 1 · K = I in the sandwiched pro duct. Hence (27) E bot = − 1 2  µ 0 + Q − 1 ( I − K Y ) w  Q  µ 0 + Q − 1 ( I − K Y ) w  + 1 2 w K − 1 w − 1 2 ξ 1 ξ 0 − z 1 ξ 0 . Comp arison. Comparing (24) and (27), the non-Gaussian (residual) part of b oth exp onen ts is the same: 1 2 w K − 1 w − 1 2 ξ 1 ξ 0 − z 1 ξ 0 . The Gaussian parts are absorb ed by translations of the Haar measure on H ( u ) ( µ 0 7→ µ 0 − K − 1 w in the top, µ 0 7→ µ 0 − Q − 1 ( I − K Y ) w in the b ottom), which are legitimate since the Haar measure is translation inv ariant. It remains to v erify that the Gaussian prefactors agree. Ev aluating the top Gaussian integral gives a factor prop ortional to (det K ) − 1 / 2 , whereas the b ottom giv es one prop ortional to (det Q ) − 1 / 2 = (det K ) − 1 / 2 (det( I − Y K )) − 1 / 2 . How ever, the in tegrals (21) and (22) use the self-dual Haar measure d h µ on H ( u ). In the top row, this measure is induced by the transversal X , while in the b ottom row it is induced by X ′ : the parameterization µ 0 7→ [ µ 0 , K µ 0 ] uses H ( s ) as the graph parameter, and the self-dual measures relative to X and X ′ diﬀer by the Radon– Nik o dym factor r dµ H ( s ) dµ X ′ d L . dµ H ( s ) dµ X d L = (det( I − Y K )) 1 / 2 , whic h exactly comp ensates the determinantal discrepancy . (Alternativ ely , both ro ws compute the same abstract con volution δ H ( u ) ∗ ψ on H ; the left- X -in v ariant and left- X ′ -in v ariant descriptions of a single Sch wartz function on H agree by construc- tion, so the prefactors must cancel, and the preceding Radon–Nikodym calculation pro vides the explicit mechanism.) Therefore, the left square comm utes. Righ t square. Let ϕ ∈ H ( H ( u ) ◦ ). F or any z ∈ T , z +  H ( u ) XX ′  z ∈ H ( u ) . Since ϕ is righ t- H ( u )-inv ariant, ϕ ( t, z ) = ϕ  ( t, z ) ·  0 , − z −  H ( u ) XX ′  z   = ϕ  t − 1 2 ω  z ,  H ( u ) XX ′  z  , −  H ( u ) XX ′  z  . Using the fact that ind XH ( u ) and res X are inv erse to one another, the ab o ve applied to z ∈ X ′ giv es ϕ (0 , z ) = ind( ϕ | X )  − 1 2 ω  z ,  H ( u ) XX ′  z  , −  H ( u ) XX ′  z  = e h  − 1 2 ω  z ,  H ( u ) XX ′  z  − 1 2 ω  [ XH ( u )] +  H ( u ) XX ′  z ,  H ( u ) XX ′  z  ϕ | X  −  H ( u ) XX ′  z  . SACHS EQUA TIONS AND PLANE W A VES, V 27 The argument of e h reduces (by the identit y − ω ( z , M z ) − ω ([ XH ( u )] + M z , M z ) = ω ([ H ( u ) X ] + z , z ), where M =  H ( u ) XX ′  , which follows from the deﬁnition of the mon- o drom y and the pro jection) to 1 2 ω ([ H ( u ) X ] + z , z ) , and so ϕ (0 , z ) = e h  1 2 ω ([ H ( u ) X ] + z , z )  ϕ | X  −  H ( u ) XX ′  z  as required. □ Remark 2 (Wh y the global statement must b e formulated by charts) . The lo c al intertwiner the or em pr ove d in The or em 6 gives an explicit formula for the change of r e al p olarization ρ X , X ′ t : S ( X ) ⊗ F ( t ) → S ( X ′ ) ⊗ F ( t ) pr ovide d X and X ′ ar e suﬃciently close and b oth ar e c omplementary to H ( t ) . What fails glob al ly is not the existenc e of Schr¨ odinger evolution, but the attempt to c om- pr ess al l p olarization changes into a single p ointwise formula valid for arbitr ary distant p airs ( X , X ′ ) : dir e ct c omp osition of the p ointwise op er ators lands at the wr ong p oint of the sour c e p olarization and ther efor e is not, in gener al, a sc alar multiple of the dir e ct map. This me ans that c austics should b e tr e ate d as chart singularities of a ﬁxe d p o- larization, not as singularities of the evolution itself. The c orr e ct glob al statement is ther efor e an atlas the or em: one pr op agates in any p olarization which r emains tr ansverse to H ( t ) on a subinterval, and one glues neighb ouring charts by the lo c al intertwiner the or em on overlaps. This is the analytic c ounterp art of the R osen-universe pictur e of [1] : ther e, R osen c o or dinate singularities o c cur when the L agr angian curve me ets a ﬁxe d L agr angian subsp ac e, wher e as her e the same event app e ars as the br e akdown of a single r e al- p olarization Schr¨ odinger chart, ne c essitating p assage to a new chart. Deﬁnition 7 (Admissible p olarization atlas) . L et [ s, u ] ⊂ U . A n admissible p o- larization atlas for the p ositive L agr angian curve H ( · ) on [ s, u ] c onsists of c omp act subintervals I j = [ a j , b j ] , j = 1 , . . . , N , and L agr angian p olarizations X j ⊂ T such that: (1) [ s, u ] = S N j =1 I j and I j ∩ I j +1  = ∅ for j = 1 , . . . , N − 1 ; (2) X j is c omplementary to H ( t ) for every t ∈ I j ; (3) after shrinking overlaps if ne c essary, for every t ∈ I j ∩ I j +1 the p air ( X j , X j +1 ) is suﬃciently close in the sense of The or em 6. Remark 3 (Existence of admissible atlases) . A dmissible atlases always exist. F or e ach t 0 ∈ [ s, u ] cho ose a L agr angian p olarization X c omplementary to H ( t 0 ) . Com- plementarity is an op en c ondition, so X r emains c omplementary to H ( t ) on a neigh- b ourho o d of t 0 . By c omp actness of [ s, u ] one obtains a ﬁnite c over by such neigh- b ourho o ds. Sinc e “suﬃciently close” is an op en c ondition ne ar the diagonal in LG ( T ) × LG ( T ) , the c over may b e r eﬁne d so that adjac ent charts ar e suﬃciently close on e ach overlap. 28 JONA THAN HOLLAND AND GEORGE SP ARLING Deﬁnition 8 (Global propagator attached to an atlas) . L et A = { ( I j , X j ) } N j =1 b e an admissible p olarization atlas on [ s, u ] , and cho ose p oints t j ∈ I j ∩ I j +1 , j = 1 , . . . , N − 1 . Write s 0 = s, s j = t j (1 ≤ j ≤ N − 1) , s N = u. The propagator asso ciated to A is the op er ator Φ A ( s, u ) := Φ X N ( s N − 1 , s N ) ρ X N − 1 , X N s N − 1 Φ X N − 1 ( s N − 2 , s N − 1 ) · · · · · · ρ X 1 , X 2 s 1 Φ X 1 ( s 0 , s 1 ) , wher e Φ X j is the lo c al Schr¨ odinger evolution in the X j -pictur e deﬁne d in (17) , and ρ X j , X j +1 t is the lo c al intertwiner of The or em 6 on the overlap I j ∩ I j +1 . Lemma 11 (Insertion of a lo cal chart) . L et [ a, d ] ⊂ U , and supp ose that X is c omplementary to H ( t ) for every t ∈ [ a, d ] . L et [ b, c ] ⊂ [ a, d ] , and supp ose that X ′ is c omplementary to H ( t ) for every t ∈ [ b, c ] , with ( X , X ′ ) suﬃciently close r elative to the curve on [ b, c ] in the sense of The or em 6. Then (28) Φ X ( c, d ) ρ X ′ , X c Φ X ′ ( b, c ) ρ X , X ′ b Φ X ( a, b ) = Φ X ( a, d ) . Henc e r eplacing the single chart ([ a, d ] , X ) by the thr e e charts ([ a, b ] , X ) , ([ b, c ] , X ′ ) , and ([ c, d ] , X ) do es not change the pr op agator. Pr o of. By Theorem 6 applied on the interv al [ b, c ], ρ X , X ′ c Φ X ( b, c ) = Φ X ′ ( b, c ) ρ X , X ′ b . Since ρ X ′ , X c = ( ρ X , X ′ c ) − 1 , this is equiv alent to ρ X ′ , X c Φ X ′ ( b, c ) ρ X , X ′ b = Φ X ( b, c ) . Substituting this in to the left-hand side of (28) gives Φ X ( c, d ) Φ X ( b, c ) Φ X ( a, b ) , whic h equals Φ X ( a, d ) by the semigroup prop erty . □ Lemma 12 (Existence of common suﬃciently-close reﬁnements) . L et A = { ( I j , X j ) } N j =1 , e A = { ( e I k , e X k ) } e N k =1 b e admissible p olarization atlases for H ( · ) on a c omp act interval [ s, u ] . Then ther e exists an admissible p olarization atlas C on [ s, u ] which r eﬁnes b oth A and e A . Mor e pr e cisely, after sub dividing [ s, u ] into ﬁnitely many c omp act intervals J ℓ = [ c ℓ − 1 , c ℓ ] sub or dinate to the two c overs, one may assign to e ach J ℓ a ﬁnite chain of p olarizations X ℓ, 0 , X ℓ, 1 , . . . , X ℓ,m ℓ such that (1) X ℓ, 0 is the p olarization c oming fr om A on J ℓ ; (2) X ℓ,m ℓ is the p olarization c oming fr om e A on J ℓ ; (3) every X ℓ,r is c omplementary to H ( t ) for al l t ∈ J ℓ ; (4) e ach c onse cutive p air ( X ℓ,r − 1 , X ℓ,r ) is suﬃciently close r elative to the curve on J ℓ . SACHS EQUA TIONS AND PLANE W A VES, V 29 Conc atenating these chains over ℓ yields the desir e d c ommon r eﬁnement. Pr o of. T ake the common sub division of [ s, u ] by all endp oin ts of the interv als in A and e A , and reﬁne further if necessary so that each resulting compact in terv al J ℓ = [ c ℓ − 1 , c ℓ ] lies in some I j ( ℓ ) and some e I k ( ℓ ) . W rite X ℓ := X j ( ℓ ) , e X ℓ := e X k ( ℓ ) . Cho ose a p oint t ℓ ∈ J ℓ . Since b oth X ℓ and e X ℓ are complementary to H ( t ℓ ), they b elong to the aﬃne c hart U ℓ := { Z ∈ LG( T ) : Z is complementary to H ( t ℓ ) } . Cho ose aﬃne co ordinates on U ℓ , and join X ℓ to e X ℓ b y a line segment K ℓ ⊂ U ℓ . Because complementarit y is an op en condition and the compact set K ℓ × { t ℓ } lies in { ( Z , t ) ∈ LG( T ) × [ s, u ] : Z is complementary to H ( t ) } , after shrinking J ℓ (and hence reﬁning the sub division) if necessary , we may assume that ev ery p olarization in K ℓ is complemen tary to H ( t ) for all t ∈ J ℓ . Lik ewise, the suﬃcien tly-close condition is op en near the diagonal in U ℓ × U ℓ . Since the diagonal of the compact set K ℓ × K ℓ is compact and lies in that ope n set, there exists ε ℓ > 0 (for any ﬁxed metric on U ℓ ) such that whenever Z , Z ′ ∈ K ℓ satisfy d ( Z , Z ′ ) < ε ℓ , the pair ( Z , Z ′ ) is suﬃciently close relative to the curve on J ℓ . Sub divide the segmen t K ℓ in to ﬁnitely many p oin ts X ℓ, 0 = X ℓ , X ℓ, 1 , . . . , X ℓ,m ℓ = e X ℓ with successive distances < ε ℓ . Then each consecutive pair is suﬃciently close on J ℓ , and ev ery X ℓ,r is complemen tary to H ( t ) throughout J ℓ . No w concatenate these chains ov er ℓ . On the in terior of a ﬁxed J ℓ the admis- sibilit y condition holds by construction. A t a b oundary p oint c ℓ = J ℓ ∩ J ℓ +1 , any adjacen t pair of p olarizations in the concatenated atlas is complementary to H ( c ℓ ). Since the ov erlap is then the single point { c ℓ } , the suﬃciently-close condition is automatic there: taking the base p oint to b e c ℓ , one has K = 0 in Theorem 6, so I − Y K = I is p ositive-deﬁnite. Therefore the concatenated atlas is admissible. By construction it reﬁnes b oth A and e A . □ Theorem 7 (Global contin uation of Schr¨ odinger evolution across caustics) . L et H ( · ) b e a p ositive L agr angian curve and F a ﬁeld bund le with p ar al lel tr ansp ort Γ . L et A = { ( I j , X j ) } N j =1 b e an admissible p olarization atlas on [ s, u ] . Then: (1) The op er ator Φ A ( s, u ) is indep endent of the choic e of the overlap p oints t j ∈ I j ∩ I j +1 . (2) Φ A ( s, u ) is unchange d by the fol lowing elementary r eﬁnements of the atlas: (a) sub dividing some interval I j and ke eping the same p olarization X j on the smal ler pie c es; (b) inserting, on a subinterval [ b, c ] ⊂ I j , an additional chart ([ b, c ] , X ′ ) such that X ′ is c omplementary to H ( t ) on [ b, c ] and suﬃciently close to X j r elative to the curve ther e, with the c orr esp onding tr ansition maps at b and c . 30 JONA THAN HOLLAND AND GEORGE SP ARLING (3) Conse quently, any two admissible p olarization atlases on the c omp act in- terval [ s, u ] deﬁne the same op er ator. In p articular, this yields a c anonic al glob al c ontinuation of the lo c al Schr¨ odinger evolution acr oss any c austics of a ﬁxe d p olarization. Pr o of. The pro of is a formal consequence of Theorem 6 together with the semigroup prop ert y of the lo cal propagators. Step 1: indep endenc e of the overlap p oints. Fix j ∈ { 1 , . . . , N − 1 } and let t, t ′ ∈ I j ∩ I j +1 . Because ( X j , X j +1 ) is suﬃcien tly close throughout the ov erlap, Theorem 6 applied on the in terv al with endp oints t and t ′ giv es the commutativ e square S ( X j ) ⊗ F ( t ) S ( X j ) ⊗ F ( t ′ ) S ( X j +1 ) ⊗ F ( t ) S ( X j +1 ) ⊗ F ( t ′ ) Φ X j ( t,t ′ ) ρ X j , X j +1 t ρ X j , X j +1 t ′ Φ X j +1 ( t,t ′ ) Equiv alently , (29) ρ X j , X j +1 t ′ Φ X j ( t, t ′ ) = Φ X j +1 ( t, t ′ ) ρ X j , X j +1 t . No w compare the atlas propagator built using t with the one built using t ′ , k eeping all other ov erlap p oints ﬁxed. The only aﬀected factor is Φ X j +1 ( t, · ) ρ X j , X j +1 t Φ X j ( · , t ) v ersus Φ X j +1 ( t ′ , · ) ρ X j , X j +1 t ′ Φ X j ( · , t ′ ) . Using the semigroup prop ert y Φ X j ( a, c ) = Φ X j ( b, c ) Φ X j ( a, b ) , Φ X j +1 ( a, c ) = Φ X j +1 ( b, c ) Φ X j +1 ( a, b ) , and inserting (29), these t wo expressions agree. Hence Φ A ( s, u ) is indep endent of the c hoice of every ov erlap p oint. Step 2: elementary r eﬁnement by sub division. Supp ose one interv al I j = [ a j , b j ] is sub divided at some c ∈ ( a j , b j ), but the same p olarization X j is used on b oth subin terv als. Then no new transition map is inserted, and the corresp onding p ortion of the propagator c hanges from Φ X j ( a j , b j ) to Φ X j ( c, b j ) Φ X j ( a j , c ) , whic h is the same by the semigroup prop erty . Step 3: elementary r eﬁnement by chart insertion. Supp ose that on a subin terv al [ b, c ] ⊂ I j one inserts an additional c hart ([ b, c ] , X ′ ) as in (2b). The aﬀected segmen t of the atlas propagator is precisely the left-hand side of (28), with X = X j . By Lemma 11, this segmen t is equal to the original factor Φ X j ( a j , b j ). Hence this elemen tary insertion do es not change the atlas propagator. Step 4: existenc e of a c ommon r eﬁnement. Let e A b e another admissible atlas on [ s, u ]. By Lemma 12, there exists an admissible common reﬁnement C of A and e A . By rep eated application of Steps 2 and 3, Φ A ( s, u ) = Φ C ( s, u ) and Φ e A ( s, u ) = Φ C ( s, u ) . SACHS EQUA TIONS AND PLANE W A VES, V 31 Therefore Φ A ( s, u ) = Φ e A ( s, u ) . This pro ves (3) and hence the canonical global contin uation claim. □ Corollary 3 (Single-chart case) . If ther e exists a p olarization X c omplementary to H ( t ) for al l t ∈ [ s, u ] , then the glob al pr op agator of The or em 7 is simply the lo c al pr op agator Φ X ( s, u ) . Pr o of. T ake the admissible atlas with one chart: A = { ([ s, u ] , X ) } . Then, b y deﬁnition, Φ A ( s, u ) = Φ X ( s, u ). □ Remark 4 (Why the lo cal theorem matters) . The or em 7 shows that the lo c al in- tertwiner the or em is not a disp ensable te chnic al lemma: it is pr e cisely the transition la w ne e de d to glue lo c al Schr¨ odinger pictur es into a glob al evolution. When H ( t ) cr osses the Maslov cycle of a ﬁxe d p olarization X , the formula for Φ X do es not c e ase to describ e a genuine ﬁeld; r ather, the X -chart has br oken down. T o c ontinue the evolution one p asses to a ne arby p olarization X ′ on a neighb ouring interval, and the overlap is c ontr ol le d exactly by The or em 6. In this sense, c austics ar e not failur es of pr op agation but failur es of a single r e al-p olarization chart. The lo c al the or em is the analo gue of a change of c o or di- nates formula, and the glob al pr op agator is obtaine d by gluing these lo c al c o or dinate descriptions. Remark 5 (Where the Maslov phase b elongs) . The algebr aic sour c e of the meta- ple ctic/Maslov phase is the triple-c onvolution the or em of Se ction 4, not a glob al c o cycle for the p ointwise r eﬂe ction formula. What fails in the p ointwise appr o ach is exactly the dir e ct c omp osition law for arbitr ary distant p airs of p olarizations: the c omp ositions evaluate the sour c e function at diﬀer ent p oints and ther efor e c annot, in gener al, diﬀer by a sc alar. The Maslov phase should ther efor e b e understo o d as attache d to the F ourier-inte gr al/Heisenb er g r e alization of the tr ansition op er ators, or e quivalently to the c omp arison of diﬀer ent lo c al oscil latory descriptions of the same abstr act c ontinuation op er ator. References [1] Jonathan Holland and George Sparling. Sac hs equations and plane w a ves, i: Rosen universes, 2024. arXiv:2402.07036 [gr-qc]. [2] Jonathan Holland and George Sparling. Sachs equations and plane wa ves, ii: Isometries and conformal isometries, 2024. arXiv:2405.12748 [math-ph]. [3] Jonathan Holland and George Sparling. Sachs equations and plane wa ves, iii: Micro cosms, 2024. arXiv:2412.10990 [math-ph]. [4] Jonathan Holland and George Sparling. Sachs equations and plane wav es, iv: Pro jective diﬀerential geometry , 2025. arXiv:2503.12503 [math-ph]. [5] G. Lion and M. V ergne. The Weil R epr esentation, Maslov Index and Theta Series , volume 6 of Pr ogr ess in Mathematics . Birkh”auser, Boston, 1980. [6] David Mumford. T ata L e ctures on Theta, I , volume 28 of Pro gr ess in Mathematics . Birkh”auser, Boston, 1983. [7] David Mumford. T ata L e ctur es on Theta, III , volume 97 of Pr o gress in Mathematics . Birkh”auser, Boston, 1991. [8] H. W. Brinkmann. Einstein spaces which are mapped conformally on each other. Mathema- tische Annalen , 94:119–145, 1925. 32 JONA THAN HOLLAND AND GEORGE SP ARLING [9] A. Einstein and N. Rosen. On gravitational wa ves. Journal of the F r anklin Institute , 223:43– 54, 1937. [10] R. Penrose. Any space-time has a plane wa ve as a limit. In M. Cahen and M. Flato, edi- tors, Diﬀerential Ge ometry and Relativity , volume 3 of Mathematic al Physics and Applie d Mathematics , pages 271–275. Reidel, Dordrech t, 1976. [11] D. Berenstein, J. M. Maldacena, and H. Nastase. Strings in ﬂat space and pp wav es from N = 4 sup er yang–mills. Journal of High Energy Physics , 2002(4):013, 2002. [12] M. Blau, J. Figueroa-O’F arrill, C. Hull, and G. Papadopoulos. Penrose limits and maximal supersymmetry . Classic al and Quantum Gr avity , 19:L87–L95, 2002. [13] H. Stephani, D. Kramer, M. MacCallum, C. Ho enselaers, and E. Herlt. Exact Solutions of Einstein ’s Field Equations . Cambridge Monographs on Mathematical Physics. Cambridge Universit y Press, Cambridge, 2 edition, 2003. [14] M. Blau. Lecture notes on plane w av es and p enrose limits. Lecture notes, Universit ´ e de Neuchˆ atel, 2011. Av ailable at http://www.unine.ch/phys/string/Lecturenotes.html . [15] R. S. W ard. Progressing w aves in ﬂat spacetime and in plane-wa v e spacetimes. Classic al and Quantum Gravity , 4(3):775–778, 1987. [16] C. Duv al, G. Burdet, H. P . K”unzle, and M. Perrin. Bargmann structures and newton–cartan theory . Physic al Review D , 31:1841–1853, 1985. [17] L. C. Ev ans. Partial Diﬀer ential Equations , volume 19 of Gr aduate Studies in Mathematics . American Mathematical So ciet y , Providence, RI, 2 edition, 2010. University of Pittsburgh, Dep ar tment of Ma thema tics, 301 Thackera y Hall, Pitts- burgh, P A 15260

Sachs Equations and Plane Waves, V: Ward, Fourier, and Heisenberg Symmetry on Plane Waves

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