Lorentzian Spectral Geometry for Globally Hyperbolic Surfaces
aa r X i v : . [ m a t h - ph ] O c t LORENTZIAN SPECTRAL GEOMETRY FORGLOBALLY HYPERBOLIC SURFACES
FELIX FINSTER AND OLAF M ¨ULLERNOVEMBER 2014
Abstract.
The fermionic signature operator is analyzed on globally hyperbolicLorentzian surfaces. The connection between the spectrum of the fermionic signa-ture operator and geometric properties of the surface is studied. The findings areillustrated by simple examples and counterexamples.
Contents
1. Introduction 21.1. The Minkowski Drum 21.2. Summary of Results for the Minkowski Drum 61.3. Lorentzian Surfaces in the Massless Case 71.4. Outlook: The Chiral Index and Causal Fermion Systems 92. The Massless Case 112.1. Simple Domains 112.2. Representation of S as an Integral Operator 152.3. Symmetry of the Spectrum 162.4. Computation of tr( S q ), Recovering the Volume 172.5. Length of Causal Curves and the Largest Eigenvalue 212.6. Length of Spacelike Curves and tr( S + ) 222.7. A Reconstruction Theorem 243. The Massive Case 243.1. Solution of the Cauchy Problem 243.2. Regularity of the Image of S S as an Integral Operator 353.5. Symmetry of the Spectrum 363.6. Computation of tr( S ) 374. Lorentzian Surfaces in the Massless Case 374.1. Conformal Embedding into Minkowski Space 374.2. Conformal Transformation of the Fermionic Signature Operator 404.3. Computation of tr( S ) and tr( S ): Volume and Curvature 424.4. A Reconstruction Theorem 44References 47 Introduction
It is a renowned mathematical problem if one can hear the shape of a drum, i.e.whether the spectrum of the Laplace operator on a domain in R with Dirichlet bound-ary conditions determines the shape of the domain (see [29, 24, 23]). More generally,the mathematical area of spectral geometry is devoted to studying the connection be-tween the geometry of a Riemannian manifold ( M , g ) and spectral properties of certaingeometric operators on M (see [22] for a survey). In the present paper, we proposea setting in which the objectives of spectral geometry can be extended to Lorentziansignature. In a more analytic language, our setting makes ideas and methods devel-oped for elliptic differential operators applicable to hyperbolic operators. Moreover, westudy the resulting “Lorentzian spectral geometry” in the simplest possible situations:for subsets of the Minkowski plane and for Lorentzian surfaces.In order to make the paper accessible to a broad readership, we now introducethe problem for subsets of the Minkowski plane without assuming a knowledge ofdifferential geometry or Dirac spinors (Section 1.1). Then we give a summary of theobtained results (Section 1.2) and outline our generalizations to Lorentzian surfaceswith curvature (Section 1.3). Finally, Section 1.4 puts our constructions and resultsinto a more general context.1.1. The Minkowski Drum.
Recall that in the classical drum problem one studiesthe eigenvalue problem for the Laplacian on a bounded domain Ω ⊂ R with Dirichletboundary values, − ∆ φ = λφ in Ω , φ | ∂ Ω = 0 . The naive approach to translate this problem to the Lorentzian setting is to replacethe Laplacian by the scalar wave operator. Denoting the variables in the plane byby ( t, x ) ∈ R , we obtain the boundary value problem (cid:0) ∂ t − ∂ x (cid:1) φ ( t, x ) = λ φ ( t, x ) ∀ ( t, x ) ∈ Ω ⊂ R , , φ | ∂ Ω = 0 . (1.1)This is not a good problem to study, as we now explain. As a consequence of the minussign, the wave operator can be factorized into a product of two first order operators, (cid:0) ∂ t − ∂ x (cid:1) = ( ∂ t + ∂ x )( ∂ t − ∂ x ) . (1.2)This changes the analytic behavior of the solutions completely. Namely, in the caseof the scalar wave equation ( ∂ t − ∂ x (cid:1) φ = 0, the factorization (1.2) implies that thegeneral solution can be written as φ ( t, x ) = φ L ( t + x ) + φ R ( t − x ) (1.3)with arbitrary real-valued functions φ L and φ R . Thus, thinking of x as a spatialvariable and t as time, the solution can be decomposed into components φ L and φ R which propagate to the left respectively right, both with the characteristic speed one(which can be thought of as the speed of light, which for convenience we set equalto one). As a consequence, a boundary value problem makes no sense. Namely, forDirichlet boundary conditions, there is only the trivial solution φ ≡
0. Prescribingnon-zero boundary values would give rise to consistency conditions for the boundaryvalues, which can only be satisfied for special boundary values. If the “eigenvalue” λ in (1.1) is non-zero, the structure of the equation becomes more complicated becausethe components φ L and φ R are coupled to each other. But again, boundary conditionsgive rise to consistency conditions, which cannot in general be satisfied. Unless in veryspecial cases, these consistency condition cannot be met even for a countable set of ORENTZIAN SPECTRAL GEOMETRY 3 values of λ ∈ C , making it impossible to distinguish a discrete set of “eigenvalues” ofthe wave operator.These mathematical problems reflect the fact that seeking for solutions of boundaryvalue problems for the scalar wave equation is not the correct question to ask. Instead,one should pose the problem in the way it usually arises in the applications, namelyas an initial-value problem . Thus, instead of imposing boundary values, we shouldprescribe initial values up to first order ( φ | N , ∂ t φ | N ) on a curve N and seek for solutionswhich satisfy the initial conditions. In order to avoid consistency conditions for theinitial data, the curve N should be spacelike. Moreover, this curve should be chosen insuch a way that the initial conditions determine a unique global solution in our “space-time”. Such a curve is referred to as a Cauchy surface , and the existence of a Cauchysurface is subsumed in the notion that space-time should be globally hyperbolic . Wepostpone the general definition of these notions to Section 4, and now simply explainwhat these notions mean for open subsets of the plane.First of all, the
Minkowski plane R , is the plane R endowed with the inner product g : R , × R , → R , g (cid:0) ( t, x ) , ( t ′ , x ′ ) (cid:1) = tt ′ − xx ′ . If the minus sign in the last equation were replaced by a plus sign, the inner product g would go over to the usual Euclidean scalar product on R . This minus sign accountsfor the Lorentzian signature. The inner product g is referred to as the Minkowskimetric . As already mentioned above, we regard the coordinates t and x of R , as timeand space, respectively. The speed of light is set to one. A regular smooth curve c ( s )in the Minkowski plane parametrized by s ∈ ( a, b ) ⊂ R with components ( c ( s ) , c ( s ) )is said to be causal if it describes a motion at most with the speed of light, i.e.causal curve: (cid:12)(cid:12) c ′ ( s ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) c ′ ( s ) (cid:12)(cid:12) for all s ∈ ( a, b ) . The physical principle of causality states that information can be transmitted onlyalong causal curves. Similarly, the curve is said to be spacelike if its speed is alwaysfaster than the speed of light,spacelike curve: (cid:12)(cid:12) c ′ ( s ) (cid:12)(cid:12) < (cid:12)(cid:12) c ′ ( s ) (cid:12)(cid:12) for all s ∈ ( a, b ) . Geometrically, for a spacelike curve the angle between the horizontal line and thetangent to the curve is less than 45 ◦ . The causal structure can also be expressed interms of the Minkowski metric. Namely, the curve c ( s ) is ( causal if g (cid:0) c ′ ( s ) , c ′ ( s ) (cid:1) ≥ g (cid:0) c ′ ( s ) , c ′ ( s ) (cid:1) < ) for all s ∈ ( a, b ) . The domain of dependence D of a spacelike curve c is the set of all points of R , suchthat every inextendible causal curve through the point intersects c . It can also becharacterized as the smallest rectangle enclosing c whose sides have an angle of 45 ◦ tothe horizontal line. More generally, we refer to a rectangle whose sides have an angleof 45 ◦ as a causal diamond .We consider a bounded open subset M ⊂ R , of the Minkowski plane. The as-sumption of global hyperbolicity of M implies that there is a space-like curve N suchthat its causal diamond D contains M (see Figure 1). We refer to M as a Minkowskidrum . It turns out that the curve N is a Cauchy surface of M . For the scalar waveequation, this can be verified by constructing the solution of the Cauchy problem forinitial data on N explicitly by determining the functions φ L and φ R in the general F. FINSTER AND O. M ¨ULLER NM ⊂ R , νD Figure 1.
A Minkowski drum.solution (1.3). If the parameter λ in (1.1) is present or if other linear geometric equa-tions are considered, the unique solvability of the Cauchy problem follows for examplefrom the theory of linear symmetric hyperbolic systems [28, 40].Clearly, the choice of the Cauchy surface is not unique, because any other spacelikecurve with the same endpoints is also a Cauchy surface (see the other dashed lines inFigure 1). Since we are interested in the geometry of space-time, our constructionsshould not depend on the choice of the Cauchy surface N .The basic question is which Hilbert space and which operator thereon should bechosen as the basic objects of a Lorentzian spectral geometry. An answer to thisquestion is proposed in the recent paper [20], where the so-called fermionic signatureoperator is introduced for space-times of finite lifetime of general dimension. We hereexplain the idea and construction in the simple setting of the Minkowski drum. Amain ingredient is that, instead of the scalar wave equation, we work with the Diracequation. We now explain how to get from the scalar wave equation to the Diracequation and why the Dirac equation is preferable for our purposes. The first stepfor getting to the Dirac equation is to write separate equations for the left- and right-moving components in (1.3), (cid:18) ∂ t + ∂ x ∂ t − ∂ x (cid:19) (cid:18) φ L φ R (cid:19) = 0 . (1.4)Next, we replace φ L and φ R by complex-valued functions ψ L , ψ R and combine them tothe so-called Dirac spinor ψ = ( ψ L , ψ R ) ∈ C (for the formulation with vector bundlesover a manifold see Section 1.3). Moreover, we insert a parameter m ∈ R , the so-calledrest mass, on the diagonal. We thus obtain the Dirac equation (cid:18) im ∂ t + ∂ x ∂ t − ∂ x im (cid:19) ψ ( t, x ) = 0 . Multiplying the differential operator from the left by the same differential operatorwith im replaced by − im , we obtain (cid:18) − im ∂ t + ∂ x ∂ t − ∂ x − im (cid:19) (cid:18) im ∂ t + ∂ x ∂ t − ∂ x im (cid:19) = (cid:0) ∂ t − ∂ x + m (cid:1) C . This shows that every component of a solution of the Dirac equation is also a solutionof the wave equation in (1.1) with λ = − m . Conversely, to a solution φ of (1.1) we ORENTZIAN SPECTRAL GEOMETRY 5 can associate a solution ψ of the Dirac equation by setting ψ = (cid:18) − im ∂ t + ∂ x ∂ t − ∂ x − im (cid:19) φ . We point out that this association is not one-to-one, because the mapping φ ψ isin general not injective (as is obvious in the example φ ≡ Dirac operator D , D ψ = mψ with D = iγ ∂ t + iγ ∂ x , (1.5)where the Dirac matrices γ and γ are given by γ = (cid:18) (cid:19) , γ = (cid:18) − (cid:19) . (1.6)The Dirac matrices can also be characterized in a basis-independent way in terms ofthe anti-commutation relations γ i γ j + γ j γ i = 2 g ij C , where g ij = diag(1 , −
1) is again the Minkowski metric. Next, it is useful to introducean inner product ≺ . | . ≻ on the spinors such that the Dirac matrices are symmetricwith respect to it. Clearly, the Dirac matrices are not symmetric with respect tothe canonical scalar product on C , because γ is anti-Hermitian. But they becomesymmetric if we introduce the inner product ≺ . | . ≻ by ≺ ψ | φ ≻ = h ψ | (cid:18) (cid:19) φ i C . (1.7)We denote C with this inner product by ( V ≃ C , ≺ . | . ≻ ) and refer to it as the spinorspace . To any two solutions ψ, φ of the Dirac equation we can associate the vector field J k ( t, x ) = ≺ φ ( t, x ) | γ k ψ ( t, x ) ≻ . This vector field is divergence-free, as is verified by the following computation, ∂ k J k = ∂ k ≺ φ | γ k ψ ≻ = ≺ ∂ k φ | γ k ψ ≻ + ≺ φ | γ k ∂ k ψ ≻ = ≺ γ k ∂ k φ | ψ ≻ + ≺ φ | γ k ∂ k ψ ≻ = ≺ ( − im ) φ | ψ ≻ + ≺ φ | ( − im ) ψ ≻ = 0(here it is essential that m is real). Integrating this divergence over a region Ω of M and applying the Gauß divergence theorem, one concludes that the flux of the vectorfield J through the boundary ∂ Ω vanishes (the Gauß divergence theorem in Minkowskispace is the same as in Euclidean space, except that one must work with the normalwith respect to the Minkowski metric). Choosing Ω as the region between two Cauchy
F. FINSTER AND O. M ¨ULLER surfaces N and N , one sees that the following integral is independent of the choiceof the Cauchy surface, ( ψ | φ ) = 2 π ˆ N ≺ ψ | γ j φ ≻ ν j dµ N . (1.8)Here dµ N the volume form of the induced Riemannian metric on N , and ν is thefuture-directed normal on N (see Figure 1). Moreover, a direct computation showsthat the inner product ≺ . | γ j . ≻ ν j is positive definite, so that (1.8) defines a scalarproduct on the solutions of the Dirac equation. Forming the completion of the smoothsolutions, we obtain a Hilbert space ( H m , ( . | . )). We remark that in physics, the vectorfield ≺ ψ | γ j ψ ≻ is called Dirac current , and the fact that it is divergence-free is referredto as current conservation . Thus the conservation of the Dirac current is essentialfor introducing the Hilbert space H m of solutions, independent of the choice of theCauchy surface N . We also remark that the function ≺ ψ | γ j ψ ≻ ν j has the physicalinterpretation as the probability density for the Dirac particle described by the wavefunction ψ to be at a certain position on the space-like hypersurface N . In this context,current conservation corresponds to the fact that the probability of the particle tobe anywhere in space must be preserved in time. Therefore, current conservationis intimately connected with the probabilistic interpretation of the wave function inquantum mechanics.In order to encode the global behavior of the solutions in space-time in an operator,we introduce on H m the inner product <ψ | φ> := ˆ M ≺ ψ | φ ≻ dµ , (1.9)where dµ = dt dx is the Lebesgue measure. Using that M is a bounded set, it followsthat this inner product is bounded, i.e. there is a constant c > | <ψ | φ> | ≤ c k ψ k k φ k for all ψ, φ ∈ H m (1.10)(for details see [20, Section 3]). Thus for any φ ∈ H m , the anti-linear form < . | φ> : H m → C is continuous. By the Fr´echet-Riesz theorem (see for example [31, Sec-tion 6.3]), there is a unique vector u ∈ H m such that <ψ | φ> = ( ψ | u ) for all ψ ∈ H m . The mapping φ u is linear and bounded. We thus obtain a bounded linear opera-tor S ∈ L( H m ) such that <φ | ψ> = ( φ | S ψ ) ∀ φ, ψ ∈ H m . (1.11)Moreover, taking the complex conjugate of (1.11) and exchanging φ and ψ , one seesthat the operator S is symmetric. The operator S is referred to as the fermionicsignature operator .With the fermionic signature operator S on the Hilbert space ( H m , ( . | . )), we haveintroduced the objects of our spectral geometry. We are interested in the question ifthe geometry of ( M , g ) is encoded in the operator S and how this geometric informationcan be retrieved.1.2. Summary of Results for the Minkowski Drum.
In short, our analysis re-veals that the spectrum of the fermionic signature operator encodes many geometricproperties of M . However, it does not determine the geometry of M completely. ORENTZIAN SPECTRAL GEOMETRY 7
The massless case m = 0 is easier to analyze because, similar as explained in (1.3)for the scalar wave equation, the Dirac equation has a simple explicit solution. For thisreason, in Section 2 we begin with the massless case. We first consider so-called simpledomains for which the fermionic signature operator can be represented by an explicitfinite-dimensional matrix (see Definition 2.1 and Lemma 2.2). We construct one-parameter families of simple domains which are isospectral but not isometric , showingthat the spectrum of S does not determine the geometry completely (Example 2.4).Then we represent the fermionic signature operator for general Minkowski drums as anintegral operator and show that it is Hilbert-Schmidt (Proposition 2.5). Moreover, thespectrum of the fermionic signature operator is shown to be symmetric with respectto the origin (Proposition 2.7). We proceed by computing the trace of powers of S .The trace of S encodes the total volume of space-time (Proposition 2.10),tr (cid:0) S (cid:1) = µ ( M )4 π . (1.12)The traces of higher powers give additional geometric information, as is explained inthe example of tr( S ) in Proposition 2.11. Then we explore the connection betweenthe spectrum of S and the lengths of curves. We prove that length ℓ of any timelike curve is bounded from above by the largest eigenvalue λ of S by (Proposition 2.12) λ ≥ ℓ π . Moreover, it is shown that the length ℓ of any spacelike curve is bounded from aboveby the trace over the positive spectral subspace of S (Proposition 2.13),tr (cid:16) χ (0 , ∞ ) ( S ) S (cid:17) ≥ ℓ π . Finally, it is shown that the geometry of M is completely determined if S is givenas an integral operator acting on the initial data set of any Cauchy hypersurface(Theorem 2.14).In Section 3 we turn attention to the massive case. A general solution of the Cauchyproblem is constructed using the Green’s function which is given in terms of Besselfunctions (Lemma 3.1). Next, we analyze how the regularity of the image of S dependson the smoothness of the boundary (see Propositions 3.3 and 3.4). This also makesit possible to estimate the asymptotics of the eigenvalues near the origin in terms ofthe total variation of the boundary curve (Theorem 3.7). In Proposition 3.12 it isshown that the spectrum of S is again symmetric with respect to the origin, but witha different method than in the massless case. We proceed by computing the trace ofpowers of the fermionic signature operator. The dependence on the mass parametergives additional geometric information, as is explained in Proposition 3.13 for the traceof S .1.3. Lorentzian Surfaces in the Massless Case.
In generalization of the Minkowskidrum, in this paper we also consider Lorentzian surfaces with curvature. The pointof interest is to analyze how the curvature of the surface affects the spectrum of thefermionic signature operator. For technical simplicity, we restrict attention to themassless case. This case is mathematically appealing because we can make use of the conformal invariance of the massless Dirac equation, making the tools of conformalgeometry available.
F. FINSTER AND O. M ¨ULLER
The structures introduced in Section 1.1 for the Minkowski drum all generalize tothe setting with curvature: We let ( M , g ) be a two-dimensional globally hyperbolicLorentzian manifold. Globally hyperbolic means that there is a space-like curve N being a Cauchy surface (for the precise definition of global hyperbolicity and a Cauchysurface see Section 4). The Cauchy surface can be either compact or non-compact.In the first case, it is diffeomorphic to a sphere, whereas in the latter case, it isdiffeomorphic to an open interval. For simplicity, we here restrict attention to thelatter case. We let S M be the spinor bundle on M . The fibers S x M are isomorphicto C . They are endowed with an inner product of signature (1 , ≺ . | . ≻ x . The smooth sections of the spinor bundle are denoted by C ∞ ( M , S M ).The Lorentzian metric induces a Levi-Civita connection and a spin connection, whichwe both denote by ∇ . Every vector of the tangent space acts on the correspondingspinor space by Clifford multiplication. We denote the corresponding map from thetangent space to the linear operators on the spinor space by γ : T x M → L( S x M ).Clifford multiplication is related to the Lorentzian metric via the anti-commutationrelations γ ( u ) γ ( v ) + γ ( v ) γ ( u ) = 2 g ( u, v ) 11 S x ( M ) . We also write Clifford multiplication in components with the Dirac matrices γ j anduse the short notation with the Feynman dagger, γ ( u ) ≡ u j γ j ≡ /u . The connections,inner products and Clifford multiplication satisfy Leibniz rules and compatibility con-ditions; we refer to [2, 30] for details. Combining the spin connection with Cliffordmultiplication gives the geometric Dirac operator D = iγ j ∇ j . The massless Diracequation reads D ψ = 0 . We remark for clarity that in the case with curvature, the square of the Dirac operatorno longer coincides with the wave operator. Indeed, by the Schr¨odinger-Lichnerowicz-Weitzenb¨ock formula D = −∇ j ∇ j + R these operators differ by a multiple of scalarcurvature R .In the Cauchy problem, one seeks for a solution of the Dirac equation with initialdata ψ N prescribed on a given Cauchy surface N . Thus in the smooth setting, D ψ = 0 , ψ | N = ψ N ∈ C ∞ ( N , S M ) . This Cauchy problem has a unique solution ψ ∈ C ∞ ( M , S M ). This can be seeneither by considering energy estimates for symmetric hyperbolic systems (see for ex-ample [28]) or alternatively by constructing the Green’s kernel (see for example [1]).These methods also show that the Dirac equation is causal, meaning that the solutionof the Cauchy problem only depends on the initial data in the causal past or future.In particular, if ψ N has compact support, the solution ψ will also have compact sup-port on any other Cauchy hypersurface. This leads us to consider solutions ψ in theclass C ∞ sc ( M , S M ) of smooth sections with spatially compact support. On solutions inthis class, one again introduces the scalar product (1.8), where /ν denotes Clifford mul-tiplication by the future-directed normal ν (we always adopt the convention that theinner product ≺ . | /ν. ≻ x is positive definite). Using current conservation ∇ j ≺ ψ | φ ≻ = 0,the scalar product (1.8) is independent of the choice of the Cauchy surface (similaras explained in Section 1.1 for the Minkowski drum). Now the fermionic signatureoperator S is defined exactly as for the Minkowski drum by expressing the space-timeinner product (1.9) in terms of the scalar product in the form (1.11). ORENTZIAN SPECTRAL GEOMETRY 9
For globally hyperbolic Lorentzian surfaces of finite lifetime and finite volume havinga non-compact Cauchy surface, we show that the fermionic signature operator encodesthe volume and the curvature in the following way. First, the Hilbert-Schmidt normof S again encodes the volume (1.12), where µ now is the volume measure correspondingto the Lorentzian metric. The formula for tr( S ) involves integrals of curvature:tr (cid:0) S (cid:1) = 18 π ˆ M dµ ( ζ ) ˆ J ( ζ ) exp (cid:18) ˆ D ( ζ,ζ ′ ) R dµ (cid:19) dµ ( ζ ′ ) , where J ( ζ ) denotes all space-time points which can be connected to ζ by a causalcurve. Moreover, D ( ζ, ζ ′ ) is the causal diamond of the space-time points ζ and ζ ′ , i.e. D ( ζ, ζ ′ ) = (cid:0) J ∨ ( ζ ) ∩ J ∧ ( ζ ′ ) (cid:1) ∪ (cid:0) J ∨ ( ζ ′ ) ∩ J ∧ ( ζ ) (cid:1) , (1.13)where J ∨ ( ζ ) and J ∧ ( ζ ) denotes the points which can be connected to ζ via a future-and past-directed causal curve, respectively.Finally, we show that the geometry of M can be reconstructed if S is given as anintegral operator acting on the initial data set of any Cauchy hypersurface (Theo-rem 4.11).1.4. Outlook: The Chiral Index and Causal Fermion Systems.
We now put theideas and constructions given in this paper into a more general context, also indicatingpossible directions of future research.We first point out that for simplicity, we here restrict attention to two-dimensionalspace-times and mainly the massless Dirac equation. But most constructions couldbe generalized to globally hyperbolic Lorentzian manifolds of arbitrary dimension.The continuity of the space-time inner product (1.10) can be subsumed in the notionthat the space-time should be m -finite, which means qualitatively that the space-timemust have finite lifetime (for details see [20, Section 3.2]). However, many interestingspace-times like asymptotically flat Lorentzian manifolds or Lorentzian manifolds withasymptotic ends have infinite life-time , implying that the continuity condition (1.10)fails. In such space-times, one must use a different construction which relies on theso-called mass oscillation property introduced in [21]. In all these situations, the con-nection between the spectrum of the fermionic signature operator and the geometryof the Lorentzian manifold is largely unknown, leaving many interesting mathematicalquestions open.We next remark that it is possible to associate an index to the fermionic signatureoperator, which takes integer values. In [14] simple examples of space-times with anon-trivial index are constructed, and the stability of the index under homotopiesis studied. But it is unknown if and how this index is related to the geometry orthe topology of the space-time. In order to introduce this index, one needs as anadditional structure a chiral grading operator Γ which acts on the spinor spaces andhas for all u ∈ T x M the propertiesΓ ∗ = − Γ , Γ = 11 , Γ γ ( u ) = − γ ( u ) Γ , ∇ Γ = 0 , (1.14)where γ is Clifford multiplication and the star denotes the adjoint with respect to theinner product ≺ . | . ≻ x . More generally, the operator Γ is defined in any even space-time dimension by Clifford multiplication with the volume form. In physics, Γ is called“pseudoscalar operator” and is usually denoted by γ . The grading operator gives rise to the two idempotent operators χ L = 12 (cid:0) − Γ (cid:1) and χ R = 12 (cid:0)
11 + Γ (cid:1) , referred to as the chiral projections (on the left respectively right handed componentof the spinors). The chiral signature operators S L and S R are defined by inserting thechiral projections into (1.11), <φ | χ L/R ψ> = ( φ | S L/R ψ ) . The first relation in (1.14) implies that S ∗ L = S R . We thus define the chiral index as theNoether index of S L (sometimes called Fredholm index; for basics see for example [31, § S := dim ker S L − dim coker S L = dim ker S L − dim ker S R . The fermionic signature operator is a technical tool in the fermionic projector ap-proach to quantum field theory and is also used for constructing examples of causalfermion systems. We now outline these connections. The fermionic projector P is ob-tained by composing the causal fundamental solution k m with the projection operatoron the negative spectral subspace of S , P = − χ ( −∞ , ( S ) k m . (1.15)This distribution implements the physical concept of the “Dirac sea”. Next, particlesand anti-particles are introduced to the system by adding to (1.15) additional occupiedstates or by creating “holes”. We refer the interested reader to the constructions in [20,Section 3] and the survey article [13].It is a general idea behind the fermionic projector approach that the geometry ofspace-time as well as all the objects therein should be described purely in terms ofthe physical wave functions of the system. This idea is made mathematically precisein the notion of a causal fermion system as introduced in [17]. In order to get intothis framework, one chooses H particle as a subspace of the solution space of the Diracoperator. A typical example is to choose H particle = χ ( −∞ , ( S ) as the image of thefermionic projector (1.15). By introducing an ultraviolet regularization, one arrangesthat the functions in H particle are continuous (for details see [20, Section]). Then for anyspace-time point x , one can introduce the so-called local correlation operator F ( x ) ∈ L( H particle ) via the relations( ψ | F ( x ) φ ) = −≺ ψ ( x ) | φ ( x ) ≻ x for all ψ, φ ∈ H particle . Denoting the signature of the spin scalar product by ( n, n ), the local correlation op-erator is a symmetric operator in L( H particle ) of rank at most 2 n , which has at most n positive and at most n negative eigenvalues. Finally, we introduce the universalmeasure dρ = F ∗ dµ as the push-forward of the volume measure on M under the map-ping F (thus ρ (Ω) := µ (( F ) − (Ω))). Omitting the subscript “particle”, we thus obtaina causal fermion system as defined in [17, Section 1.2]: Definition 1.1.
Given a complex Hilbert space ( H , h . | . i H ) and a parameter n ∈ N (the “spin dimension” ), we let F ⊂ L( H ) be the set of all self-adjoint operators on H offinite rank, which (counting with multiplicities) have at most n positive and at most n negative eigenvalues. On F we are given a positive measure ρ (defined on a σ -algebra ORENTZIAN SPECTRAL GEOMETRY 11 M ⊂ R , N N D (0 ,
0) (0 , b ) Figure 2.
Choosing the Cauchy surface at t = 0. of subsets of F ), the so-called universal measure . We refer to ( H , F , ρ ) as a causalfermion system . Causal fermion systems provide a general mathematical framework in which thereare many inherent analytic, geometric and topological structures. This concept makesit possible to generalize notions of differential geometry to the non-smooth setting.From the physical point of view, it is a proposal for quantum geometry and an approachto quantum gravity. We refer the interested mathematical reader to the researchpapers [16, 18] or the textbooks [15, 19]. In the setting of causal fermion systems,the physical equations are formulated in terms of the causal action principle (see [12]or [15, Section 1.1]).In the setting of causal fermion systems, the fermionic signature operator is simplydefined by the integral S = − ˆ F x dρ ( x )(this operator can be viewed as the restriction of the fermionic signature operatordefined by (1.11) to H particle ; for details see [14, Section 4]). In this context, theobjectives of our “Lorentzian spectral geometry” generalize to the question of howthe spectrum of S is related to the objects of the Lorentzian quantum geometry asintroduced in [16]. 2. The Massless Case
Simple Domains.
Let M ⊂ R , be an open, bounded, globally hyperbolicsubset of the Minkowski plane R , (see the left of Figure 2, where also one Cauchysurface N is shown). The maximal solution for initial values on a Cauchy surface N isdefined on a causal diamond D in the Minkowski plane. The following constructionswill depend only on the space of solutions in this causal diamond, but the choiceof the Cauchy surface will be irrelevant. In particular, the Cauchy surface does notnecessarily need to lie entirely in M . With this in mind, we simply choose N as thestraight line joining the left and right corners of the causal diamond. Moreover, bya Poincar´e transformation we can arrange that the left corner is the origin, whereasthe right corner has the coordinates (0 , b ) with a parameter b >
0. Then the scalarproduct (1.8) simplifies to( ψ | φ ) = 2 π ˆ b ≺ ψ (0 , x ) | γ φ (0 , x ) ≻ dx = 2 π ˆ b h ψ (0 , x ) , φ (0 , x ) i C dx (2.1)(where in the last step we used (1.7)). M ⊂ R , x x x K x x x k x l ∆ kl Figure 3.
A simple domain.For simplicity, we begin the analysis in the massless case. Then the Dirac equa-tion (1.4) has the general solution ψ ( t, x ) = (cid:18) ψ L ( t + x ) ψ R ( t − x ) (cid:19) (2.2)with complex-valued functions ψ L and ψ R . In view of (2.1), the left- and right-movingcomponents are orthogonal. The following assumption makes it possible to analyze S explicitly. Definition 2.1. M is a simple domain if there are finitely many points x < x < · · · < x K = b such that the boundary of M is contained in the lightlike curves through these points, ∂ M ⊂ (cid:8) x , . . . , x K (cid:9) + R (1 ,
1) + R (1 , − . The name “simple domain” is motivated by simple functions in measure theory whichtake only a finite number of values. Figure 3 shows an example.We now introduce a basis of H in which the operator S will turn out to have aparticularly simple form. To this end, for c ∈ { L, R } , k ∈ { , . . . , K } and n ∈ Z wedefine functions which are plane waves on the subintervals, ψ k,nc ( x ) = 1 p π ( x k − x k − ) χ ( x k − ,x k ] ( x ) e πixk − xk − nx (2.3)(where χ denotes the characteristic function). As in (2.2) we regard the functions ψ k,nR and ψ k,nL as spinors in the first and second component, respectively. Solving the Cauchyproblem, we obtain the corresponding Dirac solutions (cid:18) ψ k,nL ( t + x )0 (cid:19) and (cid:18) ψ k,nR ( t − x ) (cid:19) , which with a slight abuse of notation we again denote by ψ k,nc ∈ H (note that theseare solutions only in the weak sense, as they are not continuous). A short computationshows that these vectors are orthonormal,( ψ k,nc | ψ k ′ ,n ′ c ′ ) = δ c,c ′ δ k,k ′ δ n,n ′ . Using that the plane waves e πixk − xk − nx form a Fourier basis of L (( x k − , x k ]), onealso sees that the vectors ( ψ k,nc ) form a basis of H . Hence ( ψ k,nc ) is an orthonor-mal basis of H . Moreover, a short computation shows that the space-time innerproduct <ψ k,nc | ψ k ′ ,n ′ c ′ > vanishes if n or n ′ are non-zero (because one integrates over a ORENTZIAN SPECTRAL GEOMETRY 13 full period of a plane wave) or if c = c ′ (because the inner product (1.7) involves anoff-diagonal matrix). The remaining inner products are computed by <ψ k,nR | ψ l,n ′ L > = <ψ l,n ′ L | ψ k,nR > = δ n δ n ′ π p ( x k − x k − )( x l − x l − ) µ (∆ kl ) = δ n δ n ′ π √ q µ (∆ kl ) , where µ is the Lebesgue measure on R , and∆ kl = (cid:16) ( x k − , x k ] + R (1 , (cid:17) ∩ (cid:16) ( x l − , x l ] + R (1 , − (cid:17) ∩ M (see Figure 3). We thus obtain the following result: Lemma 2.2.
On a simple domain M and for zero mass, the fermionic signatureoperator S has finite rank. More precisely, choosing the orthonormal basis (2.3) , ( ψ k,nc ) with c ∈ { L, R } , k ∈ { , . . . , K } , n ∈ Z , it has rank at most K . It vanishes on all the vectors ψ k,nc with n = 0 . On thesubspace spanned by the basis vectors ψ , L , . . . , ψ K, L , ψ , R , . . . , ψ K, R , it has the blockmatrix representation S = 12 π √ (cid:18) T ∗ T (cid:19) , where T is the matrix with components T kl = q µ (∆ kl ) . (2.4)From this matrix representation one can read off a few general properties of thespectrum of the fermionic signature operator: Corollary 2.3.
On a simple domain M and for zero mass, the following statementshold. (i) The spectrum of S is symmetric with respect to the origin and σ ( S ) = σ (cid:0) √ T ∗ T (cid:1) ∪ − σ (cid:0) √ T ∗ T (cid:1) . For an eigenvector u of √ T ∗ T corresponding to the non-zero eigenvalue λ , theeigenvectors of S corresponding to the eigenvalues ± λ have the form ψ = (cid:18) λu ± T u (cid:19) . (ii) The eigenvector corresponding to the largest eigenvalue of S is non-degenerate.Its components can be chosen to be non-negative. (iii) Tr (cid:0) S (cid:1) = µ ( M )4 π . Proof.
Follows from a direct computation. Part (ii) is a consequence of the Perron-Frobenius theorem for matrices with positive entries (see [39, Chapter 5]). (cid:3)
The spectrum of S does not determine the geometry completely, as the followingexample shows. x x x x x x x x Figure 4.
Simple domains corresponding to the matrices T (left)and ˜ T (right). Example 2.4. (Isospectral simple domains)
Consider the matrices T = a √ ab b √ bc c and ˜ T = d √ de √ df e √ ef f , where a, . . . , f are strictly positive parameters. The form of the off-diagonal matrix el-ements ensures that these matrices can be realized in the form (2.4) by simple domains.More precisely, in order to realize the matrix T , one chooses K = 3 and x − x = √ a , x − x = √ b , x − x = √ c . The simple domain is then chosen as all the squares in the future, except for thesquare on top (see the Figure 4 on the left). The matrix ˜ T is realized similarly by asimple domain with K = 3, but this time without removing the square on top (seeFigure 4 on the right). Obviously, the resulting simple domains are not isometric. Wewant to show that for suitable values of the parameters, the matrices T ∗ T and ˜ T ∗ ˜ T are isospectral. In view of Corollary 2.3, this implies that the fermionic signatureoperators corresponding to the two simple domains are isospectral.We choose the parameters d = e = 1 and f = δ with a small parameter δ >
0, sothat ˜ T = √ δ √ δ δ . As a necessary condition for T ∗ T and ˜ T ∗ ˜ T to be isospectral, the matrices T and ˜ T must have the same determinant (note that these determinants are obviously positive).We satisfy this condition by choosing c = δab . Computing and comparing the characteristic polynomials of T ∗ T and ˜ T ∗ ˜ T , one seesthat it remains to satisfy the conditions − a b − δ + aδ + bδ − δ + δ a + δ b + δ ab = 0 (2.5)3 − a − ab − b + 2 δ − δa + δ − δ a b = 0 . (2.6)Multiplying the second equation by a and adding the first equation, we obtain aquadratic equation for b of positive discriminant. Choosing the explicit solution whichfor δ = 0 and a = 1 gives b = 1 (so that we get back the matrix ˜ T ), and substituting this ORENTZIAN SPECTRAL GEOMETRY 15 solution into (2.5), we obtain one equation for the remaining unknown a . Expandingthis equation around δ = 0 and a = 1, we obtain the condition5 δ − a − + O (cid:0) δ (cid:1) + O (cid:0) δ ( a − (cid:1) + O (cid:0) ( a − (cid:1) = 0 . Thus for sufficiently small δ > a of the form a = 1 ± r δ O ( δ ) . We thus obtain a one-parameter family of isospectral pairs of matrices T ∗ T and ˜ T ∗ ˜ T . ♦ Representation of S as an Integral Operator. Let M ⊂ R , be a bounded,globally hyperbolic subset of the Minkowski plane. As explained at the beginning ofSection 2.1, we again choose the Cauchy surface (0 , b ) at time t = 0. Proposition 2.5.
The fermionic signature operator can be represented as an integraloperator ( S ψ )( x ) = ˆ b S ( x, y ) ψ ( y ) dy (2.7) with a bounded integral kernel, S ( ., . ) ∈ L ∞ (cid:0) (0 , b ) × (0 , b ) , L( V ) (cid:1) . (2.8) Proof.
The Cauchy problem with initial data at t = 0 has the explicit solution ψ ( t, x ) = (cid:18) ψ L (0 , x + t ) ψ R (0 , x − t ) (cid:19) . (2.9)Using this representation in (1.9), multiplying out and estimating each term gives (cid:12)(cid:12) <ψ | φ> (cid:12)(cid:12) ≤ ˆ M (cid:16) k ψ R (0 , x − t ) k k φ L (0 , x + t ) k + k ψ L (0 , x + t ) k k φ R (0 , x − t ) k (cid:17) dt dx . Estimating the integral by the integral over the whole causal diamond, we obtain ˆ M k ψ R (0 , x − t ) k k φ L (0 , x + t ) k dt dx ≤ ˆ D k ψ R (0 , x − t ) k k φ L (0 , x + t ) k dt dx = 12 (cid:18) ˆ b k ψ R (0 , x ) k dx (cid:19) (cid:18) ˆ b k φ L (0 , x ′ ) k dx ′ (cid:19) . We conclude that for all ψ, φ ∈ H , (cid:12)(cid:12) <ψ | φ> (cid:12)(cid:12) ≤ (cid:18) ˆ b k ψ (0 , x ) k dx (cid:19) (cid:18) ˆ b k φ (0 , x ′ ) k dx ′ (cid:19) . This means that the bilinear form <. | .> can be estimated in terms of the L -norm ofboth arguments on the Cauchy surface t = 0. In other words, <. | .> ∈ L (cid:0) (0 , b ) × (0 , b ) , dx dy (cid:1) ∗ . Since the dual space of L ( dx dy ) is the Banach space of L ∞ -functions acting by weakevaluation, we conclude that there is a kernel S ( ., . ) ∈ L ∞ ((0 , b ) × (0 , b ) , dx dy ) suchthat <ψ | φ> = 2 π ˆ b ˆ b h ψ ( x ) , S ( x, y ) φ ( y ) i C dx dy . Comparing with (1.11) and (2.1) gives the result. (cid:3)
We remark that the estimates used in the proof of this proposition will be generalizedand refined in Section 3.2.The fact that the kernel is pointwise bounded (2.8) and the domain (0 , b ) has finitevolume implies that the trace of any even power of S is finite and can be computedwith the standard formula: Corollary 2.6.
The fermionic signature operator S is Hilbert-Schmidt. Moreover, thetraces of even powers of S q , q ∈ N , are given by the integrals tr( S q ) = ˆ b dx . . . ˆ b dx q Tr (cid:0) S ( x , x ) · · · S ( x q , x ) (cid:1) , (2.10) where tr denotes the trace of an operator on the Hilbert space, and Tr is the trace of a (2 × -matrix.Proof. According to (2.8), we know that k S ( x, y ) k ≤ C for almost all x, y ∈ (0 , b )(where k · k denotes the Hilbert-Schmidt norm of a (2 × ˆ b dx ˆ b dy (cid:13)(cid:13) S ( x, y ) (cid:13)(cid:13) < C b . (2.11)We now choose on H = L ((0 , b ) , C ) the orthonormal basis ( ψ nc ) with n ∈ Z , c ∈{ L, R } given by plane waves ψ nc = 1 √ πb ψ c e πib nx where ψ L = (cid:18) (cid:19) , ψ R = (cid:18) (cid:19) . Then Parseval’s identity for double Fourier series shows that the series1(2 π ) X c,c ′ ∈{ L,R } X n,n ′ ∈ Z (cid:12)(cid:12) ( ψ nc | S ψ n ′ c ′ ) (cid:12)(cid:12) coincides with the integral in (2.11). We conclude that the operator S is Hilbert-Schmidt. Moreover, the trace of S can be computed by (2.10) specialized to thecase q = 1.By iterating (2.7) and using Fubini’s theorem, one obtains an integral representa-tion of the operator S q again with a pointwise bounded kernel. Repeating the aboveargument with S replaced by the operator S q , we conclude that also the operator S q isHilbert-Schmidt, and that its Hilbert-Schmidt norm can be computed by (2.10). Thisconcludes the proof. (cid:3) This corollary shows in particular that the operator S is compact. Thus S has apure point spectrum and finite-dimensional eigenspaces. Moreover, the eigenvaluescan accumulate only at the origin.2.3. Symmetry of the Spectrum.
In Corollary 2.3 (i) we saw that for a simpledomain, the spectrum is symmetric with respect to the origin. The next propositionshows why this is true even for general domains.
Proposition 2.7.
The spectrum of S is symmetric with respect to the origin. ORENTZIAN SPECTRAL GEOMETRY 17
Proof.
The matrix Γ := (cid:18) − (cid:19) (2.12)obviously anti-commutes with the Dirac matrices (1.6) and the Dirac operator (1.5),i.e. Γ D = −D Γ. Hence if ψ is a solution the massless Dirac equation, the same is truefor Γ ψ . In other words, Γ maps the solution space of the Dirac equation to itself,Γ : H → H . Using (1.7), one sees that Γ is anti-symmetric with respect to the spin scalar prod-uct ≺ ψ | Γ φ ≻ = −≺ Γ ψ | φ ≻ . Moreover, using (1.8) and (1.9), we find that for all ψ, φ ∈ H , <ψ | Γ φ> = ˆ M ≺ ψ | Γ φ ≻ x dµ ( x ) = − ˆ M ≺ Γ ψ | φ ≻ x dµ ( x ) = − < Γ ψ | φ> (2.13)( ψ | Γ φ ) = 2 π ˆ N ≺ ψ | /ν Γ φ ≻ x dµ N ( x ) = − π ˆ N ≺ ψ | Γ /νφ ≻ x dµ N ( x )= 2 π ˆ N ≺ Γ ψ | /νφ ≻ x dµ N ( x ) = (Γ ψ | φ ) . (2.14)In view of (1.11), the eigenvalue equation S ψ = λψ can be written as <φ | ψ> = λ ( φ | ψ ) for all φ ∈ H . Using the symmetries (2.13) and (2.14), one sees that if ψ is an eigenvector corre-sponding to the eigenvalue λ , then Γ ψ is an eigenvector corresponding to the eigen-value − λ . (cid:3) Computation of tr( S q ) , Recovering the Volume. Proposition 2.5 also im-plies that the operators S p are trace class for any p ∈ N . The symmetry of the spectrumshown Proposition 2.7 implies that the trace of an odd power of S vanishes. We nowwant to compute the trace of even powers of S . For computational purposes, it ismost convenient to work with the causal fundamental solution in light cone coordi-nates. In order to keep the setting as simple as possible, we here introduce the causalfundamental solution k simply as a device for expressing the solution of the Cauchyproblem. Lemma 2.8.
The solution ψ of the Cauchy problem D ψ = 0 , ψ | t =0 = ψ ∈ C ((0 , b )) has the representation ψ ( t, x ) = 2 π ˆ b k ( t, x − y ) γ ψ ( y ) dy , (2.15) where k ( t, x ) is the distribution k ( t, x ) = 14 π ( γ + γ ) δ ( t + x ) + 14 π ( γ − γ ) δ ( t − x ) . (2.16) Proof.
A direct computation using (1.6) shows that (2.15) indeed agrees with (2.9). (cid:3)
The distribution k ( t, x ) is referred to as the causal fundamental solution . At firstsight, the method of this lemma seems unnecessarily complicated, because (2.9) ismuch simpler than (2.15). The advantage of (2.15) is that this formula generalizesto the massive case (see Section 3.1) and even to globally hyperbolic space-times inarbitrary dimension (see for example [20, Lemma 2.1]).The integral in (2.15) can be regarded as a time evolution operator which maps thesolution at some initial time t = 0 to the solution at a final time t . Clearly, if one firsttakes the time evolution from time t to t and then the time evolution from t to t ,one gets the same as if one takes the time evolution directly from t to t . This fact isoften referred to as a group property of the time evolution, where the group operation isthe multiplication of the time evolution operators and the inverse of the time evolutionfrom t to t is the time evolution from t to t . This group property is reflected in aproperty of the causal fundamental solution. Namely, denoting a space-time point forsimplicity by ζ = ( t, x ), we have for any ζ, ˜ ζ ∈ M , k ( ζ − ζ ′ ) = 2 π ˆ b k ( ζ , ζ − x ) γ k ( − ˜ ζ , x − ˜ ζ ) dx (2.17)(this relation can also be verified by direct computation using (2.16)). Moreover, onesees directly from (2.16) that the kernel k ( t, x ) is symmetric in the sense that k ( t, x ) ∗ = k ( − t, − x ) (2.18)(where the star denotes the adjoint with respect to the inner product ≺ . | . ≻ x ).We next express the integral kernel of S in terms of the causal fundamental solution. Lemma 2.9.
The kernel S ( x, y ) of the fermionic signature operator in (2.7) can bewritten as S ( x, y ) = 2 π ˆ M k ( − t, x − z ) k ( t, z − y ) γ dt dz . Proof.
Using (2.15) in (1.9) and applying (2.18), we obtain <ψ | φ> = 4 π ˆ M dt dz ˆ b dx ˆ b dy ≺ k ( t, z − x ) γ ψ (0 , x ) | k ( t, z − y ) γ φ (0 , y ) ≻ = 4 π ˆ M dt dz ˆ b dx ˆ b dy ≺ ψ (0 , x ) | γ k ( − t, x − z ) k ( t, z − y ) γ φ (0 , y ) ≻ . On the other hand, we know from (1.11) and (2.7) that <ψ | φ> = (cid:16) ψ (cid:12)(cid:12)(cid:12) ˆ b S ( x, y ) φ (0 , y ) dy (cid:17) . Comparing these formulas gives the result. (cid:3)
The light-cone coordinates ( u, v ) are defined by u = t + x and v = t − x . (2.19)Then du dv = 2 dt dx and t = u + v , x = u − v ∂ u = 12 ( ∂ t + ∂ x ) , ∂ v = 12 ( ∂ t − ∂ x ) ORENTZIAN SPECTRAL GEOMETRY 19 (to improve the readability we denote the indices u and v in roman style). Setting γ u = γ + γ and γ v = γ − γ , (2.20)we obtain the anti-commutation relations (cid:0) γ u (cid:1) = 0 = (cid:0) γ v (cid:1) , (cid:8) γ u , γ v (cid:9) = 4 . (2.21)The Dirac operator (1.5) and the causal fundamental solution (2.16) become D = iγ u ∂ u + iγ v ∂ v (2.22) k ( u, v ) = 14 π (cid:16) γ u δ ( u ) + γ v δ ( v ) (cid:17) . (2.23)Combining Lemma 2.9 with the integral representation of Proposition 2.5, we cancompute powers of the operator S . For example, (cid:0) S (cid:1) ( x, y ) = ˆ b S ( x, z ) S ( z, y ) dz = 4 π ˆ b dz ˆ M d ζ ˆ M d ˜ ζ k (cid:0) − ζ , x − ζ (cid:1) k (cid:0) ζ , ζ − z (cid:1) γ × k (cid:0) − ˜ ζ , z − ˜ ζ (cid:1) k (cid:0) ˜ ζ , ˜ ζ − y (cid:1) γ . Now we can carry out the z -integral using the group property (2.17). This gives (cid:0) S (cid:1) ( x, y ) = 2 π ˆ M d ζ ˆ M d ˜ ζ k (cid:0) − ζ , x − ζ (cid:1) k (cid:0) ζ − ˜ ζ (cid:1) k (cid:0) ˜ ζ , ˜ ζ − y (cid:1) γ . Iterating this method, we obtain (cid:0) S p (cid:1) ( x, y ) = 2 π ˆ M d ζ · · · ˆ M d ζ p × k (cid:0) − ζ , x − ζ (cid:1) k (cid:0) ζ − ζ (cid:1) · · · k (cid:0) ζ p − − ζ p (cid:1) k (cid:0) ζ p , ζ p − y (cid:1) γ . Taking the trace with the help of Corollary 2.6, one can again apply (3.2) to obtaintr (cid:0) S q (cid:1) = ˆ M d ζ · · · ˆ M d ζ q Tr (cid:16) k (cid:0) ζ − ζ (cid:1) · · · k (cid:0) ζ q − − ζ q (cid:1) k (cid:0) ζ q − ζ (cid:1)(cid:17) . (2.24)This formula is generally covariant, showing in particular that the trace of S p is indeedindependent of the choice of the Cauchy surface.The above formulas can be simplified considerably using the form of the causalfundamental solution in light-cone coordinates (2.23) and (2.21). Namely, k (cid:0) ζ − ζ (cid:1) · · · k (cid:0) ζ p − − ζ p (cid:1) k (cid:0) ζ p − ζ (cid:1) = 1(4 π ) p γ u δ ( u − u ) γ v δ ( v − v ) · · · + 1(4 π ) p γ v δ ( v − v ) γ u δ ( u − u ) · · · . Taking the trace and again using (2.21), we get zero if p is odd. If p is even, we obtainTr (cid:16) k (cid:0) ζ − ζ (cid:1) · · · k (cid:0) ζ p − − ζ p (cid:1) k (cid:0) ζ p − ζ (cid:1)(cid:17) = 1(2 π ) p δ ( u − u ) δ ( v − v ) · · · δ ( u p − − u p ) δ ( v p − v )+ 1(2 π ) p δ ( v − v ) δ ( u − u ) · · · δ ( v p − − v p ) δ ( u p − u ) . (2.25) ζ η ζ η ζ η ζ η ζ η Figure 5.
The geometric constraints given by Θ( ζ , . . . , ζ q ) for q = 2(left) and q = 3 (right).Setting p = 2, we see that the result of Corollary 2.3 (iii) also holds for generaldomains: Proposition 2.10.
Let M be a bounded, globally hyperbolic subset of Minkowski space.Then the trace of S encodes the space-time volume, tr (cid:0) S (cid:1) = µ ( M )4 π . Proof.
Using (2.25) in (2.24) in the case p = 2, we obtaintr (cid:0) S (cid:1) = ˆ M d ζ ˆ M d ζ π ) δ ( u − u ) δ ( v − v )= 1(2 π ) ˆ M d ζ ˆ M d ζ δ ( ζ − ζ ) = µ ( M )4 π , giving the result. (cid:3) For general q , one can compute the trace of S q with the following method. First,by renaming the variables one sees that the two summands in (2.25) give the samecontribution to the trace. Thustr (cid:0) S q (cid:1) = 2(2 π ) q ˆ M d ζ ˆ M d η · · · ˆ M d ζ q ˆ M d η q × δ ( u − ˜ u ) δ (˜ v − v ) · · · δ ( u q − ˜ u q ) δ (˜ v q − v ) , (2.26)where the points η j have the coordinates (˜ u j , ˜ v j ). Carrying out the integrals over η , . . . , η q , we obtaintr (cid:0) S q (cid:1) = 2(2 π ) q q ˆ M d ζ · · · ˆ M d ζ q Θ( ζ , . . . , ζ q ) , where the function Θ( ζ , . . . , ζ q ), which takes the values zero and one, gives geometricconstraints for the position of the points ζ , . . . , ζ q . More precisely, introducing thepoints η = ( ζ , ζ ) , η = ( ζ , ζ ) , . . . η q − = ( ζ q − , ζ q ) , η q = ( ζ q , ζ ) , the function Θ is defined byΘ( ζ , . . . , ζ q ) = (cid:26) η , . . . , η q ∈ M . The geometric constraints are illustrated in Figure 5. Via these constraints, the traceof S p depends on the geometry of the boundary curves of M . While these constraintsare rather complicated in general, in the case q = 2 they can be easily understoodgiving the following result. ORENTZIAN SPECTRAL GEOMETRY 21
Proposition 2.11.
Denoting by J ( ζ ) the set of all points which can be joined from ζ by a causal curve, tr( S ) = 18 π ˆ M µ (cid:0) M ∩ J ( ζ ) (cid:1) d ζ . (2.27) Proof.
For fixed ζ , the region where Θ( ζ , ζ ) = 1 coincides with J ( ζ ) (see the left ofFigure 5). This gives (2.27). (cid:3) Length of Causal Curves and the Largest Eigenvalue.
We now turn atten-tion to the length of causal curves. The length ℓ ( α ) of a causal curve α : (0 , → M is defined by ℓ ( α ) = ˆ q(cid:0) α ′ ( τ ) (cid:1) − (cid:0) α ′ ( τ ) (cid:1) dτ . (2.28) Proposition 2.12.
Let α be a causal curve. Then the length of this curve (as definedby (2.28) ) is bounded in terms of the largest eigenvalue λ of S by λ ≥ ℓ π . (2.29) This inequality is sharp if M is a causal diamond. Conversely, if equality holdsin (2.29) and M is connected, then M is a causal diamond.Proof. Let α be a timelike curve with end points ζ and ζ ′ . We let D be the causaldiamond whose upper and lower corners are ζ and ζ ′ , and again denote the left andright corners by η and η ′ . We may replace α by the straight line joining ζ and ζ ′ ,because this increases the length of α . By a Lorentz transformation we can arrangethat η = 0 and η ′ = (0 , ℓ ) with ℓ >
0. We choose on D the orthonormal functions ψ L = 1 √ πℓ (cid:18) (cid:19) , ψ R = 1 √ πℓ (cid:18) (cid:19) . Then λ ≥ (cid:12)(cid:12) ( ψ L | S ψ R ) (cid:12)(cid:12) = (cid:12)(cid:12) <ψ L | ψ R > (cid:12)(cid:12) = 12 πℓ ˆ D dt dx = 12 πℓ ℓ ℓ π , giving the result.If M is a causal diamond, the inequality (2.29) is sharp according to Lemma 2.2 incase K = 1. In order to prove the converse statement, assume that M is connected andthat equality holds in (2.29). Then λ = | ( ψ L | S ψ R ) | , and the Rayleigh-Ritz principleimplies that the wave functions ψ L ± ψ R are eigenvectors of S with eigenvalues ± λ .Next, we choose a Cauchy surface N which goes through the left and right corner ofthe above causal diamond D . Then any wave function ψ on N which is supportedoutside D ∩ N is obviously orthogonal to ψ L and ψ R . Using that the vectors ψ L ± ψ R are eigenvectors, it follows that0 = ( ψ | S ψ L/R ) = <ψ | ψ L/R > .
Choosing ψ as a piecewise constant function, one sees that the causal future and pastof N \ D does not intersect the causal future and past of D . Since M is connected,we conclude that M \ D is empty. This gives the result. (cid:3) Figure 6.
Approximating a spacelike curve by lightlike rectangles.2.6.
Length of Spacelike Curves and tr( S + ) . We now come to the analysis of thelength of spacelike curves. The length ℓ ( α ) of a spacelike curve α : (0 , → M isdefined by ℓ ( α ) = ˆ q(cid:0) α ′ ( τ ) (cid:1) − (cid:0) α ′ ( τ ) (cid:1) dτ . We first note that for globally hyperbolic subsets of Minkowski space, the supremum ofthe length of spacelike curves is always larger or equal to the supremum of the lengthsof causal curves, sup α causal ℓ ( α ) ≤ sup α spacelike ℓ ( α ) . (2.30)Namely, if α is a causal curve with end points ζ and ζ ′ , then the corresponding causaldiamond D ( ζ, ζ ′ ) is contained in the space-time M . But then the straight line joiningthe left and right corners of the causal diamond is a spacelike curve whose length is atleast as large as that of α . An example for a space-time where the maximal length ofspacelike curves is much larger than the length of causal curves is shown on the left ofFigure 2.The inequality (2.30) suggests that for estimating the length of spacelike curves, itis not sufficient to consider a single eigenvalue. Instead, one should form a suitablesum of eigenvalues. More precisely, we must consider the trace of the operator S + defined as the positive spectral part of S , S + = χ (0 , ∞ ) ( S ) S . Proposition 2.13.
Let α : [0 , → M be a spacelike curve. Then the length of thecurve is bounded from above by the trace of S + , tr( S + ) ≥ ℓ ( α )4 π . (2.31) Proof.
By a Poincar´e transformation we can arrange that the two end points of thecurve α lie on the x -axis. We approximate α by rectangles with lightlike sides (seeFigure 6). By choosing the rectangles sufficiently small (and possibly leaving out smallparts at the beginning or end of the curve), we can arrange that the rectangles all liein M . Next, we choose a simple domain such that the rectangles are all part of therectangles of the simple domain. Thus we can label the rectangles which approximatethe curve by ∆ k l , . . . , ∆ k N l N . The fact that the curve is spacelike implies that the indices k j are all different, andthe indices l j are also all different. Choosing again the orthonormal basis (2.3), thisimplies that the vectors ψ k , R , . . . , ψ k N , R and ψ l , L , . . . , ψ l N , L ORENTZIAN SPECTRAL GEOMETRY 23 are orthonormal. Moreover, N X j =1 <ψ k j , R | ψ l j , L > = 12 π √ N X j =1 q µ (∆ k j l j ) → ℓ ( α )4 π , (2.32)where the last convergence refers to the limit where the size of the rectangles tends tozero (note that area of a lightlike rectangle is half the square of the Minkowski lengthof its diagonals).It remains to show that the trace of S + bounds the left side of (2.32). Similar asshown in Lemma 2.2 for simple domains, in the massless case the fermionic signatureoperator is block off-diagonal, S = (cid:18) S R S L (cid:19) . Exactly as in Corollary 2.3, one concludes thattr( S + ) = tr (cid:16)q S ∗ L S L (cid:17) . (2.33)We denote the eigenvalues of S ∗ L S L (counting with multiplicities) by ν n , with theordering ν ≥ ν ≥ ν · · · . Let ( ψ n ) be a corresponding orthonormalized eigenvector basis. The computation( S L ψ n | S L ψ n ′ ) = ( ψ n | S ∗ L S L ψ n ′ ) = ν n δ n,n ′ shows that the ψ n are mapped to orthogonal vectors. Choosing φ n as orthonormalvectors which are collinear to S L ψ n , we conclude that ν n = ( ψ n | S ∗ L S L ψ n ) = ( S L ψ n | S L ψ n ) = (cid:12)(cid:12) ( φ n | S L ψ n ) (cid:12)(cid:12) . (2.34)Taking the square root and using (2.33), we obtaintr( S + ) = ∞ X n =1 (cid:12)(cid:12) ( φ n | S L ψ n ) (cid:12)(cid:12) . Now we can apply the min-max principle in the following way (for basics see [36, Sec-tion XIII.1]). The first n eigenvalues can be computed by noting that the orthonormalsets ψ , . . . , ψ n and φ , . . . , φ n maximize the expression in (2.34), in the sense that ν = sup φ ,ψ with k φ k = k ψ k =1 (cid:12)(cid:12) ( φ | S L ψ ) (cid:12)(cid:12) ν n = sup φn,ψn with k φn k = k ψn k =1 , φ n ⊥ φ ,...,φ n − , ψ n ⊥ ψ ,...,ψ n − (cid:12)(cid:12) ( φ n | S L ψ n ) (cid:12)(cid:12) . As a consequence, we can estimate the trace of S + from below by adding up theabsolute values of the expectation values for any pair of orthonormal systems ( φ n )and ( ψ n ). In particular, tr( S + ) ≥ N X i =1 (cid:12)(cid:12) ( ψ k j , R | S L ψ l j , L ) (cid:12)(cid:12) . Noting that the last inner products coincide precisely with the summands in (2.32),which are all non-negative, the result follows. (cid:3)
Exactly as for the inequality (2.29), the inequality (2.31) is sharp for a simple domainconsisting of one causal diamond (see Lemma 2.2 in the case K = 1). But it is not sharpgeneral, as can be seen in Example 2.4: The fermionic signature operators S and ˜ S are isospectral, so that the left side of (2.31) coincides. But the length of the longestspatial curve on the right of (2.31) is different for the simple domains correspondingto S and ˜ S . More precisely, these lengths are given by a + b + c = a + b + δ/ab respectively d + e + f = 2 + δ . Analyzing the equations for a and b for small δ , onefinds that these lengths differ by a term ∼ √ δ .2.7. A Reconstruction Theorem.
As shown in Example 2.4, the spectrum of S in general does not determine the geometry of M . This raises the question whichadditional structures must be given in order to encode the geometry completely. Wepropose that in order to describe the space-time geometry, one should not consider S as an operator on an abstract Hilbert space, but instead as an operator on a space ofspinorial functions on a given Cauchy surface. More precisely, we use the identification H = L ( N , S M ) ≃ L ( N , S N ) ⊕ L ( N , S N ) , where the two direct summands correspond to the left- and right-handed componentsof the spinors, respectively (note that the fiber of the intrinsic spinor bundle S N isisomorphic to C ). Knowing S on the hypersurface N , the geometry of the ambientspace-time is completely determined: Theorem 2.14.
Let S and ˜ S be the fermionic signature operators corresponding totwo bounded, globally hyperbolic sets M , ˜ M ⊂ R , . Moreover, let N and ˜ N be twoCauchy surfaces in M respectively ˜ M of the same length (with respect to the inducedRiemannian metrics). Identifying N and ˜ N via an isometry, and also identifying thecorresponding Hilbert spaces via this isometry, H = L ( N , S N ) ⊕ L ( N , S N ) ≃ L ( ˜ N , S ˜ N ) ⊕ L ( ˜ N , S ˜ N ) , we assume that S = ˜ S . Then there is a Poincar´e transformation Λ ∈ R , ⋊ SO(1 , which maps the space-timeregions and the corresponding Cauchy surfaces to each other, i.e. Λ M = ˜ M and Λ N = ˜ N . We note for clarity that SO(1 ,
1) are the isometries of R , of determinant one. Thesetransformations include reflections at the origin (which flip both the spatial orientationand the time orientation), but they do not include time reversals nor spatial inversions.To avoid repetitions, the proof will be given in Section 4.4 for general surfaces.3. The Massive Case
Solution of the Cauchy Problem.
We now generalize the integral representa-tion (2.15) as well as the group property (2.17) to the massive case.
Lemma 3.1.
A Dirac solution ψ ∈ H m can be expressed in terms of the initial dataat t = 0 by ψ ( t, x ) = 2 π ˆ b k m ( t, x − y ) γ ψ (0 , y ) dy , (3.1) ORENTZIAN SPECTRAL GEOMETRY 25 where k m ( t, x ) is the distribution k m ( t, x ) = 14 π ( γ + γ ) δ ( t + x ) + 14 π ( γ − γ ) δ ( t − x ) − im π J (cid:16) m p t − x (cid:17) ǫ ( t ) Θ( t − x ) − m π (cid:0) tγ − xγ (cid:1) J (cid:16) m √ t − x (cid:17) √ t − x ǫ ( t ) Θ( t − x ) . Moreover, k m ( ζ − ˜ ζ ) = 2 π ˆ b k m ( ζ , ζ − x ) γ k m ( − ˜ ζ , x − ˜ ζ ) dx . (3.2) Proof.
We recall the general method for constructing the time evolution operator (fordetails see for example [20, Section 2] or [19, Section 4.2]). The unique solvability ofthe Cauchy problem gives rise to the existence of advanced and retarded Green’s func-tions s ∨ and s ∧ (see for example [1]). The causal fundamental solution k m is definedas the difference of the advanced and retarded Green’s functions; more precisely, k m := 12 πi (cid:0) s ∨ m − s ∧ m (cid:1) . (3.3)Then the Cauchy problem for the Dirac equation with initial data ψ N on a Cauchysurface N with future-directed normal ν has the solution (see [20, Lemma 2.1]) ψ ( ζ ) = 2 π ˆ N k m ( ζ ; ζ ′ ) ν j ( ζ ′ ) γ j ψ N ( ζ ′ ) dµ N ( ζ ′ ) . Choosing N as the Cauchy surface at t = 0, this formula simplifies to (3.1). (Indeed, tosee the correspondence, one should keep in mind that in Minkowski space, the causalGreens function depends only on the difference vector ζ − ζ ′ , so that in (3.1) we canuse the notation k m ( ζ ) ≡ k m ( ζ ; 0)).In Minkowski space, the causal fundamental solution is given as the integral overthe mass shell. More precisely, k m ( t, x ) = 1(2 π ) ˆ R ( ωγ − kγ + m ) δ ( ω − k − m ) ǫ ( ω ) e − iωt + ikx dω dk (3.4)(for detail see [19, Section 4.2] or [11, § k m ( t, x ) = 1(2 π ) ( iγ j ∂ j + m ) ˆ R δ ( ω − k − m ) ǫ ( ω ) e − iωt + ikx dω dk = 1(2 π ) ( iγ j ∂ j + m ) ˆ R \ [ − m,m ] ǫ ( ω ) e − iωt √ ω − m (cid:16) e i √ ω − m x + e − i √ ω − m x (cid:17) dω = 1(2 π ) ( iγ j ∂ j + m ) ˆ R \ [ − m,m ] ǫ ( ω ) e − iωt √ ω − m cos (cid:16)p ω − m x (cid:17) dω = − i π ( iγ j ∂ j + m ) ˆ ∞ m sin( ωt ) √ ω − m cos (cid:16)p ω − m x (cid:17) dω . In the case m = 0, we obtain k ( t, x ) = 12 π γ j ∂ j ˆ ∞ sin( ωt ) ω cos( ωx ) dω = 12 π ˆ ∞ (cid:0) γ cos( ωt ) cos( ωx ) + γ sin( ωt ) sin( ωx ) (cid:1) dω = 14 π ˆ ∞−∞ (cid:0) γ cos( ωt ) cos( ωx ) + γ sin( ωt ) sin( ωx ) (cid:1) dω = γ π (cid:16) δ ( t + x ) + δ ( t − x ) (cid:17) − γ π (cid:16) δ ( t + x ) − δ ( t − x ) (cid:17) = 14 π ( γ + γ ) δ ( t + x ) + 14 π ( γ − γ ) δ ( t − x ) , in agreement with (2.16). In the case m >
0, we obtain additional Lorentz invariantcontributions in timelike directions. In order to compute them, it is easiest to set x = 0and t >
0. Then the Fourier integral can be carried out in terms of Bessel functionsof the first kind (see [33, § k m ( t, x ) = − i π ( iγ ∂ t + m ) ˆ ∞ m sin( ωt ) √ ω − m dω = − i π ( iγ ∂ t + m ) J ( mt ) = − im π J ( mt ) − m π γ J ( mt ) , where in the last step we used [33, eq. (10.6.2)]. Rewriting these contributions inLorentz invariant form and using that k m ( t, x ) ∗ = k m ( − t, − x ) gives the desired for-mula for k m (here the star again denotes the adjoint with respect to the inner prod-uct ≺ . | . ≻ x ). This also concludes the proof of (3.1).Finally, the identity (3.2) follows immediately by using the group property of thetime evolution operator (similar as explained before (2.17)). (cid:3) The result of the previous lemma gives a pointwise estimate for the solution of theCauchy problem.
Lemma 3.2.
Let ψ be a solution of the Dirac equation (1.5) with m ≥ with initialvalues ψ (0 , . ) ∈ C ((0 , b ) , C ) . Then ψ satisfies for all ( t, x ) ∈ D the pointwise bound | ψ L/R ( t, x ) | ≤ | ψ L/R (0 , . ) | C + 2 √ mt | ψ (0 , . ) | C . Proof.
We first estimate the Bessel functions in Lemma 3.1 by | J ( x ) | ≤ x ) , | J ( x ) | ≤ x (1 + x ) . This gives | ψ L/R ( t, x ) | ≤ | ψ L/R (0 , . ) | C + | ψ (0 , . ) | C ˆ t − t m (1 + m ( t − x )) + m t (1 + m ( t − x )) ! dx . Carrying out the integral gives the result. (cid:3)
ORENTZIAN SPECTRAL GEOMETRY 27
Regularity of the Image of S . We now work out estimates which give informa-tion on the regularity of the functions in the image of S . We again consider the Diracequation ( D − m ) ψ = 0 in a bounded, globally hyperbolic space-time M ⊂ D ⊂ R , ,with the Cauchy surface N = { } × (0 , b ) (see the right of Figure 2). Moreover, welet ψ, φ ∈ H m ∩ C ∞ ( D ) be smooth solutions of the Dirac equation inside the causaldiamond D . We introduce the wave function θ = ( D + m ) γ ψ . (3.5)The calculation( D − m ) θ = ( D − m ) γ ψ = − ( (cid:3) + m ) γ ψ = − γ ( (cid:3) + m ) ψ = 0shows that θ is again a solution of the Dirac equation. Hence( θ | S φ ) = ˆ M ≺ ( D + m ) γ ψ | φ ≻ dµ = ˆ M (cid:0) ∂ j ≺ iγ j γ ψ | φ ≻ + ≺ γ ψ | ( D + m ) φ ≻ dµ (cid:1) dµ = − i (cid:18) ˆ ∂ M + − ˆ ∂ M − (cid:19) ≺ ψ | γ /νφ ≻ dµ ∂ M (3.6)+ 2 m ˆ M ≺ ψ | γ φ ≻ dµ , (3.7)where in the last step we applied the Gauss divergence theorem and used the Diracequation. Here ∂ M ± are the future and past boundaries of M , and ν is the future-directed normal.If the boundaries ∂ M ± of M are space-like, this estimate implies that S maps tothe H¨older continuous functions: Proposition 3.3.
Assume that the future and past boundaries of M are space-like.Then there is a constant c = c ( ∂ M ± ) such that for all ψ ∈ H m ∩ C ∞ ( D ) , (cid:13)(cid:13) S ( D + m ) γ ψ (cid:13)(cid:13) ≤ c + mb ) k ψ k . (3.8) Moreover, the operator S maps to the weakly differentiable and H¨older continuousfunctions, S : H m → W , ( N , C ) ֒ → C , ( N , C ) . The operator S : H m → H m is compact.Proof. First, the integral (3.7) can be estimated with the help of Fubini’s theorem andthe Schwarz inequality by ˆ M |≺ ψ | γ φ ≻| dµ ≤ b k ψ k k φ k . Moreover, since ∂ M ± are space-like curves, we can estimate the integrand in (3.6) interms of the probability density, (cid:12)(cid:12) ≺ ψ | γ /νφ ≻ (cid:12)(cid:12) ≤ c ( ∂ M ± ) q ≺ ψ | /νψ ≻ ≺ φ | /νφ ≻ . Estimating the resulting line integrals with the help of the Schwarz inequality, weobtain (3.8).
Using that ψ solves the Dirac equation, we can use the anti-commutation relationsto obtain( D + m ) γ ψ = ( D + m ) γ ψ − γ ( D − m ) ψ = (cid:2) D , γ (cid:3) ψ + 2 mγ ψ = 2 iγ γ ∂ x ψ + 2 mγ ψ = 2 Hψ , (3.9)where H is the Dirac Hamiltonian H = − iγ γ ∂ x + mγ . (3.10)Using this identity together with the fact that S is symmetric, we can rewrite (3.8) inthe “dual form” k H S φ k ≤ ( c + mb ) k φ k for all φ ∈ H m . This shows that S maps to the W , -functions. We now apply Morrey’s embeddinginto the H¨older continuous functions (see [9, Theorem 5.7.6]). The compactness of S follows from the Arzel`a-Ascoli theorem. (cid:3) We point out that the above proposition only applies if the boundaries ∂ M ± arespace-like. This assumption is crucial in view of the examples of simple domains (seeLemma 2.2), in which case the eigenfunctions were characteristic functions, which areclearly not H¨older continuous. We now prove a weaker statement without assumingthat the curves ∂ M ± are space-like. Thinking of the characteristic functions in simpledomains, one is led to considering the total variation. In fact, we now show that thevectors in the image of S always have bounded variation. As usual, we denote the totalvariation by TV [0 ,b ] and denote the functions of finite total variation by BV([0 , b ] , C )(for basic definitions see for example [10]). Proposition 3.4.
The fermionic signature operator maps H m to BV([0 , b ] , C ) and TV [0 ,b ] ( S φ ) ≤ c k φ k , where the constant c depends only on m and b . The operator S : H m → H m is compact. We begin with a preparatory lemma.
Lemma 3.5.
For any smooth solutions ψ, φ ∈ H m and θ according to (3.5) , thefollowing estimate holds: (cid:12)(cid:12) ( θ | S φ ) (cid:12)(cid:12) ≤ √ b (cid:0) √ mb (cid:1) k φ k | ψ | C . (3.11) Proof.
We want to estimate ( θ | S φ ) in terms of the Hilbert norm k φ k and the sup-norm | ψ | C . To this end, we estimate (3.7) by ˆ M (cid:12)(cid:12) ≺ ψ | γ φ ≻ (cid:12)(cid:12) dµ ≤ b k ψ k k φ k ≤ b | ψ | C k φ k . (3.12)In (3.6) we first apply the Schwarz inequality, ˆ ∂ M + (cid:12)(cid:12) ≺ ψ | γ /νφ ≻ (cid:12)(cid:12) dµ ∂ M ≤ k φ k (cid:18) ˆ ∂ M + ≺ γ ψ | /νγ ψ ≻ dµ ∂ M (cid:19) . We would like to relate the last line integral to a corresponding integral on the Cauchysurface t = 0. To this end, we first note that the line integral can be recovered ORENTZIAN SPECTRAL GEOMETRY 29 as the boundary integral when applying the Gauss divergence theorem to the vectorfield ≺ γ ψ | γ j γ φ ≻ . However, this vector field is not divergence-free, because ∂ j ≺ γ ψ | γ j γ φ ≻ = 2 Re ≺ γ ψ | γ j γ ∂ j φ ≻ = 4 Re ≺ γ ψ | γ γ ∂ t φ ≻ − ≺ γ ψ | γ γ j ∂ j φ ≻ = 4 Re ≺ ψ | γ ∂ t φ ≻ = 2 ∂ t ≺ ψ | γ φ ≻ . Hence ˆ ∂ M + ≺ γ ψ | /νγ ψ ≻ dµ ∂ M − ˆ b ≺ ψ | γ ψ ≻ (0 , x ) dx = ˆ M ∩{ t ≥ } ∂ t ≺ ψ | γ φ ≻ dx dt = 2 ˆ b ≺ ψ | γ ψ ≻ (cid:0) T ( x ) , x (cid:1) dx − ˆ b ≺ ψ | γ ψ ≻ (0 , x ) dx , where we parametrized ∂ M + as the graph { ( T ( x ) , x ) | x ∈ [0 , b ] } . We conclude that ˆ ∂ M + (cid:12)(cid:12) ≺ γ ψ | /νγ ψ ≻ (cid:12)(cid:12) dµ ∂ M = 2 ˆ b ≺ ψ | γ ψ ≻ (cid:0) T ( x ) , x (cid:1) dx − k ψ k . Applying the pointwise estimate of Lemma 3.2, we obtain ˆ ∂ M + (cid:12)(cid:12) ≺ γ ψ | /νγ ψ ≻ (cid:12)(cid:12) dµ ∂ M ≤ b (cid:0) √ mb (cid:1) | ψ | C . We conclude that ˆ ∂ M + (cid:12)(cid:12) ≺ ψ | γ /νφ ≻ (cid:12)(cid:12) dµ ∂ M ≤ √ b (cid:0) √ mb (cid:1) k φ k | ψ | C . The integral over ∂ M − can be treated similarly. Combining these estimates with (3.12)gives the result. (cid:3) Proof of Proposition 3.4.
Using (3.9), we can write (3.11) as (cid:12)(cid:12) ( Hψ | S φ ) (cid:12)(cid:12) ≤ √ b (cid:0) √ mb (cid:1) k φ k | ψ | C . Thus for every φ ∈ H m , we have a bounded linear functional on C ([0 , b ]). The Rieszrepresentation theorem (see [38, Chapter 2] or [37, Theorem S.5] for a proof in onedimension) yields that there is a bounded regular signed Borel measure such that( Hψ | S φ ) = ˆ b ψ ( x ) dµ ( x ) (3.13)and | µ | ([0 , b ]) ≤ √ b (cid:16) √ mb (cid:17) k φ k . Choosing ψ ∈ C ∞ ((0 , b )) with compact support, we conclude that function S φ isweakly differentiable, and ( H S φ ) dx = dµ as a measure . (3.14)Moreover, choosing a function ψ with ψ (0) = 0, the vanishing of the boundary termswhen integrating by parts in (3.13). Combining these facts, we can compute S φ byintegration. Namely, writing (3.14) with the help of (3.10) in the form (cid:0) ∂ x − imγ (cid:1) S φ = − iγ γ dµ , b Figure 7.
A triangular domain.we obtain the explicit solution (cid:0) S φ (cid:1) ( x ) = e imγ x ˆ x e − imγ τ (cid:0) − iγ γ (cid:1) dµ ( τ ) . Differentiating the last equation, we obtain the estimate (cid:12)(cid:12) ( S φ ) ′ ( x ) (cid:12)(cid:12) ≤ m e mb | µ | ((0 , b )) + d | µ | ( x ) , showing that the total variation of the function S φ is bounded by a constant c = c ( m, b ).Finally, the compactness of S follows from Helly’s selection theorem (see for exam-ple [32, Section VIII.4]). (cid:3) Asymptotics of the Small Eigenvalues.
The analysis of the regularity of theimage of the fermionic signature operator (see Propositions 3.3 and 3.4) showed inparticular that S is a compact operator. Thus it has a pure point spectrum and finite-dimensional eigenspaces, and the eigenvalues can accumulate only at the origin. Inparticular, we can count the eigenvalues of S with multiplicities by λ , λ , . . . and orderthem such that | λ | ≥ | λ | ≥ · · · . (3.15)We begin with an example where S has infinite rank. Example 3.6. (A triangular domain)
We let M ⊂ R , be the triangular domainshown in Figure 7 and for simplicity the massless Dirac equation. Then the eigenvaluesof the fermionic signature operator (ordered according to (3.15) ) satisfy for n ≥ theinequalities b π n + 3 ≤ | λ n | ≤ b π n − . Proof. On H we choose the orthonormal basis of the solution space (cf. (2.2)) ψ nL ( t, x ) = 1 √ πb (cid:18) (cid:19) e πib n ( x + t ) , ψ nR ( t, x ) = 1 √ πb (cid:18) (cid:19) e πib n ( x − t ) where n ∈ Z . Then for any n, n ′ = 0, <ψ nL | ψ n ′ R > = 12 πb ˆ M e − πib n ( x + t ) e πib n ′ ( x − t ) dt dx = 14 πb ˆ b du ˆ − u dv e − πib nu e − πib n ′ v = i π n ′ ˆ b e − πib nu (cid:0) − e πib n ′ u (cid:1) du = − ib π δ n,n ′ n . (3.16)where we again chose the light-cone coordinates (2.19). The matrix elements with n =0 or n ′ = 0 are a bit more complicated, and we do not compute them here. Instead, we ORENTZIAN SPECTRAL GEOMETRY 31 only analyze S on the orthogonal complement of the two-dimensional subspace N :=span( ψ L , ψ R ). Denoting the orthogonal projection on N ⊥ by π ⊥ , a short computationusing (3.16) shows that the operator8 π b π ⊥ S π ⊥ has the eigenvalues ± , ± , ± , . . . , each of multiplicity two.We now estimate the eigenvalues of S from above and below using the min-maxprinciple. Since the spectrum is symmetric (see Proposition 2.7), we know that (cid:12)(cid:12) λ ℓ +1 (cid:12)(cid:12) = (cid:12)(cid:12) λ ℓ +2 (cid:12)(cid:12) = inf J ⊂ H , dim J = ℓ sup ψ ⊥ J, k ψ k =1 ( ψ | S ψ ) , (3.17)giving the upper bound (cid:12)(cid:12) λ ℓ +1 (cid:12)(cid:12) , (cid:12)(cid:12) λ ℓ +2 (cid:12)(cid:12) ≤ inf J ⊂ H , dim J = ℓ, J ⊃N sup ψ ⊥ J, k ψ k =1 ( ψ | S ψ )= inf K ⊂N ⊥ , dim K = ℓ − sup ψ ⊥ K, k ψ k =1 ( ψ | π ⊥ S π ⊥ ψ ) ≤ b π ℓ − . Similar, we can estimate (3.17) from below to obtain (cid:12)(cid:12) λ ℓ +1 (cid:12)(cid:12) , (cid:12)(cid:12) λ ℓ +2 (cid:12)(cid:12) ≥ inf J ⊂ H , dim J = ℓ sup ψ ⊥ J, ψ ⊥N , k ψ k =1 ( ψ | S ψ )= inf J ⊂N ⊥ , dim J = ℓ sup ψ ⊥ J, k ψ k =1 ( ψ | π ⊥ S π ⊥ ψ ) ≥ b π ℓ + 2 . This concludes the proof. (cid:3)
This example shows that in general we cannot expect a decay of | λ n | for large n faster than ∼ /n . Indeed, in the next theorem we prove this 1 /n -decay: Theorem 3.7.
Representing the boundary of ∂ M as a graph, ∂ M ± = (cid:8) ( T ± ( x ) , x ) : x ∈ [0 , b ] (cid:9) , we introduce the dimensionless constant c by c = (1 + mb ) (cid:18) X ± TV [0 ,b ] T ′± (cid:19) . (3.18) Then | λ n | ≤ cbn . Before coming to the proof, we remark that it is not clear whether the dependence ofthe constant c in (3.18) on the total variation of T ′± is only a technical assumption forour proof, or whether this assumption is needed for the theorem to hold.We again work in light-cone coordinates u and v . As in (1.4), we denote the twocomponents of the spinors by indices L and R . Then the Dirac equation can be writtenas i∂ v ψ L = m ψ R , i∂ u ψ R = m ψ L . (3.19)This allows us to rewrite the spatial derivatives (which we denote by a prime) as ψ ′ L = ∂ u ψ L + im ψ R , ψ ′ R = − ∂ v ψ L − im ψ L . (3.20) Moreover, the space-time inner product becomes <ψ | φ> = ˆ M ≺ ψ | φ ≻ dt dx = ˆ M (cid:0) ψ L φ R + ψ R φ L (cid:1) dt dx . Combining these relations, we can compute the inner product of the spatial derivativesof two Dirac solutions:
Lemma 3.8.
Let ψ, φ ∈ H m be smooth solutions of the Dirac equation. Then <ψ ′ | φ ′ > = ˆ R (cid:0) ψ L φ R + ψ R φ L (cid:1) ∂ uv χ M dt dx (3.21)+ im ˆ M (cid:16) ψ ′ R φ R − ψ R φ ′ R − ψ ′ L φ L + ψ L φ ′ L (cid:17) dt dx (3.22) (where ∂ uv χ M denotes the distributional derivative of the characteristic function).Proof. We first rewrite the spatial derivatives using (3.20) in terms of derivatives withrespect to u and v , <ψ ′ | φ ′ > = − ˆ M (cid:0) ∂ u ψ L ∂ v φ R + ∂ v ψ R ∂ u φ L (cid:1) dt dx − im ˆ M (cid:0) ∂ v ψ R φ R − ψ R ∂ v φ R + ∂ u ψ L φ L − ψ L ∂ u φ L (cid:1) dt dx − m ˆ M (cid:0) ψ L φ R + ψ R φ L (cid:1) dt dx . In the first integral, we integrate both derivatives by parts. Whenever the derivativeshit the wave functions, we apply the Dirac equation (3.19). Combining all the resultingterms, we obtain <ψ ′ | φ ′ > = ˆ R (cid:0) ψ L φ R + ψ R φ L (cid:1) ∂ uv χ M dt dx − im ˆ M (cid:0) ∂ v ψ R φ R − ψ R ∂ v φ R + ∂ u ψ L φ L − ψ L ∂ u φ L (cid:1) dt dx − m ˆ M (cid:0) ψ L φ R + ψ R φ L (cid:1) dt dx . Expressing the remaining derivatives of the wave functions with the help of (3.20) asspatial derivatives, we obtain the result. (cid:3)
Next, we need to estimate the terms (3.21) and (3.22). In (3.22) we can use Fubiniand the Schwarz inequality, | (3.24) | ≤ mb (cid:0) k ψ ′ k k φ k + k ψ k k φ ′ k (cid:1) . The analysis of the boundary terms (3.21) is more subtle. We only analyze the bound-ary terms on ∂ M + , because the past boundary can be analyzed similarly. It is againuseful to write ∂ M + as a graph of a function T ( x ) over [0 , b ]. We first consider the casethat T is smooth; the non-smooth situation will be obtained below by approximation. ORENTZIAN SPECTRAL GEOMETRY 33
The fact that ∂ M + is non-timelike implies that | T ′ ( x ) | ≤
1. Then ∂ uv χ M = 14 ( ∂ t − ∂ x ) Θ( T ( x ) − t )= 14 δ ′ (cid:0) T ( x ) − t (cid:1) (cid:0) − T ′ ( x ) (cid:1) − δ (cid:0) T ( x ) − t (cid:1) T ′′ ( x )= − ∂ t δ (cid:0) T ( x ) − t (cid:1) (cid:0) − T ′ ( x ) (cid:1) − δ (cid:0) T ( x ) − t (cid:1) T ′′ ( x ) . Using this relation in (3.21), in the term involving ∂ t δ ( T ( x ) − t ) we may integrate byparts. We thus obtain ˆ R (cid:0) ψ L φ R + ψ R φ L (cid:1) ∂ uv χ M dt dx = − ˆ b T ′′ ( x ) (cid:0) ψ L φ R + ψ R φ L (cid:1) | t = T ( x ) dx + 14 ˆ L (cid:0) − T ′ ( x ) (cid:1) ∂ t (cid:0) ψ L φ R + ψ R φ L (cid:1) | t = T ( x ) dx . Using the Dirac equation (3.19), we can rewrite the time derivatives in terms of spatialderivatives, ˆ R (cid:0) ψ L φ R + ψ R φ L (cid:1) ∂ uv χ M dt dx = − ˆ b T ′′ ( x ) (cid:0) ψ L φ R + ψ R φ L (cid:1) | t = T ( x ) dx (3.23)+ 14 ˆ b (cid:16) − T ′ ( x ) (cid:17)(cid:16) ψ ′ L φ R − ψ L φ ′ R − ψ ′ R φ L + ψ R φ ′ L (cid:17)(cid:12)(cid:12)(cid:12) t = T ( x ) dx . (3.24)In the integral (3.23) we estimate the wave functions pointwise with the help ofLemma 3.2, | (3.23) | ≤ (1 + 2 √ mb ) | ψ (0 , . ) | C | φ (0 , . ) | C TV [0 ,b ] T ′ . In order to get an idea for how to estimate (3.24), we first rewrite the scalar productas an integral over ∂ M + with integration measure dx ,( ψ | ψ ) = − ˆ R ≺ ψ | γ j ψ ≻ ∂ j Θ( T ( x ) − t ) dt dx = ˆ R ≺ ψ | ( γ − T ′ ( x ) γ ) φ ≻ δ ( T ( x ) − t )) dt dx = ˆ b ≺ ψ | ( γ − T ′ ( x ) γ ) ψ ≻| t = T ( x ) dx = ˆ b (cid:16)(cid:0) T ′ ( x ) (cid:1) | ψ L | + (cid:0) − T ′ ( x ) (cid:1) | ψ R | (cid:17)(cid:12)(cid:12)(cid:12) t = T ( x ) dx . Writing the factor (1 − T ′ ( x ) ) in (3.24) as (1 − T ′ )(1 + T ′ ), we can always group thefactors 1 + T ′ and 1 − T ′ together with the components L respectively R . Applyingthe Schwarz inequality, we obtain | (3.24) | ≤ (cid:0) k ψ ′ k k φ k + k ψ k k φ ′ k (cid:1) . In the above estimates we made use of the fact that T is twice differentiable. However,the estimates can be extended by approximation to the situation when T is differen-tiable almost everywhere and T ′ has bounded total variation.Combining all the terms gives the following estimate: Lemma 3.9.
Suppose that the future and past boundaries ∂ M ± are parametrized byfunctions T ± ∈ C ((0 , b )) . Then (cid:12)(cid:12) <ψ ′ | φ ′ > (cid:12)(cid:12) ≤
12 (1 + mb ) (cid:0) k ψ ′ k k φ k + k ψ k k φ ′ k (cid:1) + (1 + 2 √ mb ) | ψ (0 , . ) | C | φ (0 , . ) | C X ± TV [0 ,b ] T ′± . Proof of Theorem 3.7.
We apply the min-max principle in the form | λ n +1 | = inf I,J ⊂ H m, dim I =dim J = n k π J ⊥ S π I ⊥ k , where k . k denotes the sup-norm. In fact, the infimum is attained if I and J areinvariant subspaces of S which together span the spectral subspace corresponding tothe eigenvalues λ , . . . , λ n . We choose an orthonormal basis ( e k,s ) k ∈ Z ,s ∈{±} of H m formed of plane-wave solutions, e k, ± ( t, x ) = 1 √ πb ω ( ± ωγ − kγ + m ) χ e ∓ iωt + i kxb , where ω ( k ) := p k /b + m , and χ is the fixed spinor χ = (1 , i ) / √
2. We choose I and J as the (4 k + 2)-dimensional subspace H ( k ) := span (cid:0) e − k, ± , . . . , e k, ± (cid:1) . Then | λ k +5 | ≤ k π H ⊥ ( k ) S π H ⊥ ( k ) k = sup ψ ∈ H ⊥ ( k ) , k ψ k =1 (cid:12)(cid:12) h ψ | S ψ i (cid:12)(cid:12) , and applying Lemma 3.9 gives | λ k +5 | ≤ sup ψ ∈ H ⊥ ( k ) , k ψ k =1 (cid:18) (1 + mb ) k θ k + (1 + 2 √ mb ) | θ | C X ± TV [0 ,b ] T ′± (cid:19) , (3.25)where θ is a primitive of θ , θ ( x ) = ˆ x ψ (0 , y ) dy . In order to estimate θ , we expand ψ in the basis ( e k, ± ) and integrate, ψ ( t, x ) = X | ℓ | >k, s c ℓ,s e ℓ,s ( t, x ) θ ( x ) = X | ℓ | >k, s c ℓ,s √ πb ω ( ± ωγ − kγ + m ) χ bik (cid:16) e ikx − (cid:17) . ORENTZIAN SPECTRAL GEOMETRY 35
As a consequence, k θ k = X | ℓ | >k, s | c ℓ,s | b ℓ ≤ b k k ψ k | θ | C ≤ X | ℓ | >k, s | c ℓ,s |√ πb b | ℓ | ≤ r b π (cid:18) X | ℓ | >k, s | c ℓ,s | (cid:19) (cid:18) X | ℓ | >k, s ℓ (cid:19) = r b π k ψ k (cid:18) X | ℓ | >k, s ℓ (cid:19) ≤ r b π k ψ k √ k . Using this inequality in (3.25), we conclude that | λ k +5 | ≤ bk (cid:18) (1 + mb ) + (1 + 2 √ mb ) π X ± TV [0 ,b ] T ′± (cid:19) . Simplifying the constant with the Schwarz inequality gives the result. (cid:3)
Representation of S as an Integral Operator. We now generalize methodsand results of Sections 2.2 and 2.4 to the massive case.
Proposition 3.10.
The statements of Proposition 2.5 and Corollary 2.6 also hold inthe massive case.Proof.
According to Lemma 3.1, the solution of the Cauchy problem can be writtenas ψ ( t, x ) = (cid:18) ψ L (0 , x + t ) ψ R (0 , x − t ) (cid:19) + ˆ b K ( t, x − x ′ ) ψ (0 , x ′ ) dx ′ with a bounded kernel K , | K ( t, x ) | < c for all t, x ∈ R . Using this representation in (1.9), multiplying out and estimating each term gives (cid:12)(cid:12) <ψ | φ> (cid:12)(cid:12) ≤ ˆ M ( k ψ R (0 , x − t ) k k φ L (0 , x + t ) k + k ψ L (0 , x + t ) k k φ R (0 , x − t ) k ) dt dx + c (cid:18) ˆ L k ψ (0 , x ′ ) k dx ′ (cid:19) ˆ M ( k φ R (0 , x − t ) k + k φ L (0 , x + t ) k ) dt dx + c (cid:18) ˆ L k φ (0 , x ′ ) k dx ′ (cid:19) ˆ M ( k ψ R (0 , x − t ) k + k ψ L (0 , x + t ) k ) dt dx + c µ ( M ) (cid:18) ˆ b k φ (0 , x ) k dx (cid:19) (cid:18) ˆ b k ψ (0 , x ′ ) k dx ′ (cid:19) . Now the first integral can be estimated as in the proof of Proposition 2.5 by the integralover the whole causal diamond. We conclude that there is a constant C (which dependsonly on m and the geometry of M ) such that for all ψ, φ ∈ H m , (cid:12)(cid:12) <ψ | φ> (cid:12)(cid:12) ≤ C (cid:18) ˆ b k ψ (0 , x ) k dx (cid:19) (cid:18) ˆ b k φ L ( x ′ ) k dx ′ (cid:19) . Now we can proceed exactly as in the proof of Proposition 2.5 and Corollary 2.6. (cid:3)
Using the solution of the Cauchy problem in Lemma 3.1, we can immediately gen-eralize Lemma 2.9:
Lemma 3.11.
The fermionic signature operator can be written as an integral operator ( S ψ )( x ) = ˆ b S ( x, y ) ψ ( y ) dy with the distributional kernel S ( x, y ) = 2 π ˆ M k m ( − t, x − z ) k m ( t, z − y ) γ dt dz . By iterating this integral representation, one can form composite expressions inthe fermionic signature operator, just as explained in Section 2.9. In particular, theformula (2.24) generalizes totr (cid:0) S q (cid:1) = ˆ M d ζ · · · ˆ M d ζ q × Tr (cid:16) k m (cid:0) ζ − ζ (cid:1) · · · k m (cid:0) ζ q − − ζ q (cid:1) k m (cid:0) ζ q − ζ (cid:1)(cid:17) . (3.26)3.5. Symmetry of the Spectrum.
The symmetry argument of Proposition 2.7 nolonger applies in the massive case, because if ψ solves the Dirac equation for mass m ,then Γ ψ is a solution corresponding to the mass − m . But we now given anothertransformation of the spinors involving complex conjugation which again shows thatthe spectrum of S is symmetric. Proposition 3.12.
The spectrum of S is symmetric with respect to the origin.Proof. We introduce the anti-linear mapping A : ψ Γ ψ , (3.27)where the bar denotes complex conjugation and Γ is again the matrix in (2.12). Sup-pose that ψ ∈ H m is a solution of the Dirac equation (1.5) Using that the Diracmatrices (1.6) have real entries, we obtain( D − m ) Γ ψ = Γ ( − D − m ) ψ = Γ ( D − m ) ψ = 0 , showing that A : H m → H m maps solutions to solutions. Moreover, using (1.7),(1.9) and (2.1), one readily verifies that A preserves the norm but flips the sign of thespace-time inner product,( A ψ | A φ ) = ( φ | ψ ) , < A ψ | A φ> = − <φ | ψ> . (3.28)Using the orthogonality of the eigenspaces, the eigenvalue equation S ψ = λψ can bewritten in the equivalent form( ψ | S ψ ) = λ ( ψ | ψ ) and ( φ | S ψ ) = 0 ∀ φ ⊥ ψ . By definition of the fermionic signature operator (1.11), this can be written equivalentlyas <ψ | ψ> = λ ( ψ | ψ ) and <φ | ψ> = 0 ∀ φ ⊥ ψ . (3.29)Suppose that ψ ∈ H m is an eigenvector corresponding to the eigenvalue λ . Then (3.29)holds. The relations (3.28) imply that < A ψ | A ψ> = − λ ( A ψ | A ψ ) and <φ | A ψ> = 0 ∀ φ ⊥ A ψ . Hence SA ψ = − λ A ψ , completing the proof. (cid:3) ORENTZIAN SPECTRAL GEOMETRY 37
In the physics literature, the analog of the transformation (3.27) in four space-timedimensions is referred to as charge conjugation (see for example [6, Section 5.2] or [35,Section 3.6]). The interesting point is that the symmetry under charge conjugationsis broken if external potentials (like an electromagnetic potential) are present. In thiscase, the spectrum of the fermionic signature operator will in general no longer besymmetric. The deviation from charge conjugation symmetry could be detected forexample by computing traces of odd powers of S .3.6. Computation of tr( S ) . In order to see the effect of the mass on the traces, wenow compute the trace of S . Proposition 3.13.
The Hilbert-Schmidt norm of the fermionic signature operator isgiven by tr( S ) = µ ( M )4 π + m π ¨ M × M (cid:0) J + J (cid:1)(cid:0) m p ( ζ − ζ ′ ) (cid:1) Θ (cid:0) ( ζ − ζ ′ ) (cid:1) d ζ d ζ ′ . For small m , we have the expansion tr( S ) = µ ( M )4 π + m π ¨ M × M Θ (cid:0) ( ζ − ζ ′ ) (cid:1) d ζ d ζ ′ (3.30)+ m π ¨ M × M ( ζ − ζ ′ ) Θ (cid:0) ( ζ − ζ ′ ) (cid:1) d ζ d ζ ′ + O ( m ) . (3.31) For large m , we have the asymptotics tr( S ) = m π ¨ M × M Θ (cid:0) ( ζ − ζ ′ ) (cid:1)p ( ζ − ζ ′ ) d ζ d ζ ′ + O (cid:16) m (cid:17) . (3.32) Proof.
We again work in light-cone coordinates ( u, v ). Then the causal fundamentalsolution of Lemma 3.1 can be written as k m ( u, v ) = 14 π (cid:16) γ u δ ( u ) + γ v δ ( v ) (cid:17) − im π J (cid:0) m √ uv (cid:1) ǫ ( u + v ) Θ( uv ) − m π (cid:0) vγ u + uγ v (cid:1) J (cid:0) m √ uv (cid:1) √ uv ǫ ( u + v ) Θ( uv ) . Now the result follows from (3.26) by a straightforward computation using asymptoticexpansion of the Bessel functions. (cid:3)
Compared to the formula of Proposition 2.10, the dependence on the mass parame-ter m gives additional geometric information: The term ∼ m in (3.30) has the samestructure as the formula (2.27) for tr( S ) in the massless case. The term ∼ m , on theother hand, involves an additional weight factor ( ζ − ζ ′ ) . The formula for large m in (3.32) again has a similar structure, but with yet another weight factor 1 / p ( ζ − ζ ′ ) .For brevity, we do not work out the geometric meaning of these different integrals.4. Lorentzian Surfaces in the Massless Case
Conformal Embedding into Minkowski Space.
Let ( M , g ) be a two-dimen-sional time-oriented Lorentzian manifold. The manifold is globally hyperbolic if it doesnot contain closed causal curves and if the causal diamonds D ( ζ, ζ ′ ) (see (1.13)) arecompact for all space-time points ζ, ζ ′ ∈ M (for details see [4]). It is proven in [3] thatany globally hyperbolic space-time admits a smooth foliation ( N t ) t ∈ R by spacelikeCauchy hypersurfaces, defined as follows. Definition 4.1.
A subset of a time-oriented Lorentzian manifold ( M , g ) is called Cauchy surface if it is intersected exactly once by every C -inextendible future causalcurve in M . It is a well-known fact that any two-dimensional Lorentzian manifold is locally con-formally flat, in the sense that any point of M has a neighborhood which is conformalto an open subset of Minkowski space. It is less well-known that a globally hyperbolicLorentzian manifolds admits even a global conformal embedding to Minkowski space: Proposition 4.2.
Let ( M , g ) be a globally hyperbolic two-dimensional manifold witha non-compact Cauchy surface N . Then there is a conformal map Φ : ( M , g ) → R , whose image is open, relatively compact and causally convex (meaning that no future-directed causal curve can leave and reenter Φ( M ) ), and such that N is mapped to { } × (0 , .Proof. Since N is non-compact, it is diffeomorphic to R . We introduce a new metric h on M obtained by the conformal change h = e u g with u ∈ C ∞ ( M ). A direct computation shows that the scalar curvatures of g and h are related by 2∆ g w + s g = e w s h . This shows that the equation s h = 0 is equivalent to the linear normally hyperbolicequation 2∆ g w + s g = 0 . (4.1)We impose the initial conditions w (0 , r ) = w ( r ) , ∇ ν w (0 , r ) = w ( r ) (4.2)(where r ∈ R parametrizes N , and ν is again the future-directed normal vector fieldon N ). The resulting Cauchy problem (4.1), (4.2) is globally well-posed (see forexample [28, 40, 1]).We next choose the initial conditions w and w such that N becomes a h -pregeodesicof length one (a pregeodesic is a geodesic up to reparametrizations): Specializing thegeneral formulas in [5, Theorem 1.159], the condition for being a pregeodesic is0 = ∇ h∂ r ( e − u ν ) = ∇ g∂ r ( e − u ν ) + e − u ∂u∂ r ν + e − u ν ( u ) ∂ r , which is equivalent to the equation ∇ ν u = −∇ g∂ r ν = − H g , where H g is mean curvature. This equation can be satisfied by suitably choosing w .We still have the freedom to choose w arbitrarily. We use this freedom to give N length one.Solving the above Cauchy problem, we obtain a flat metric in which N is totallygeodesic. The proof is completed by applying Lemma 4.3 below. (cid:3) Lemma 4.3.
Let ( M , h ) be a two-dimensional, flat Lorentzian manifold which containsa totally geodesic Cauchy surface of length one. Then ( M , h ) is isometric to an openneighborhood of { } × (0 , in R , . ORENTZIAN SPECTRAL GEOMETRY 39
We point out that in the Riemannian case, this proposition does not hold, as thereare examples of flat contractible two-dimensional manifolds which do not admit anisometric embedding into R : Take any periodic immersed curve c : S → R n self-intersecting exactly once at p ∈ S , like for example the Lemniscate of Bernoulli.As it is immersed, it has a normal neighborhood N such that c extends to a localdiffeomorphism C : S × R n − → N . We pull back the Euclidean metric to a flatmetric G on S × R n − and restrict it to the open subset N := ( S \ { q } ) × R n +1 where q ∈ S \ { p } . Then the usual rigidity arguments ensure that any other isometricimmersion of ( N , G ) into Euclidean R n coincides with C up to rigid motions and thusis not an embedding. Proof of Proposition 4.3.
Parametrizing the Cauchy surface by arc length, we obtaina h -geodesic c : (0 , → M with N = c ((0 , T ⊥ N := { v ∈ T n M | n ∈ N , v ⊥ T N } , the normal exponentialmap E := exp M | T ⊥ N on N is injective : Consider any two timelike geodesics c , c starting at different points x , x ∈ N , in the direction of the normal ν . These geodesicscannot intersect at a point p as that would be in contradiction to the Ambrose-Singertheorem (see [5, Theorem 10.58]). Namely, assume conversely that these geodesicsintersect at a point p . Let ∆ be the geodesic triangle with vertices x , x and p . Since N is totally geodesic, its normal vector field ν is parallel along N . Moreover, since the twogeodesics c and c must intersect transversely, the parallel transport of ν along thesegeodesics gives two different vectors at p . Hence the corresponding holonomy of alongthe triangle ∆ is non-zero. On the other hand, the triangle ∆ clearly is contractible.But since ( M , g ) is flat, the Ambrose-Singer theorem implies that the the Lie algebraof the connected Lie group Hol ( M , g ) is trivial and thus Hol ( M , g ) = { } . Thisimplies that the holonomy along any contractible curve in M is the identity. This is acontradiction.We next show that E is also surjective : For a point p ∈ J ∨ ( N ) in the future of N ,we let d p : M → [0 , ∞ ) be the distance function from p (set to zero for spatially sepa-rated points). In globally hyperbolic space-times, this distance function is continuous(see [34, Lemma 14.21]). Moreover, the global hyperbolicity of M implies that theset R p := J ∧ ( p ) ∩ N is compact. Hence the restriction of d p to N attains a maximumin R p at a point q . Again due to global hyperbolicity, there is a geodesic curve γ joining q and p . The first variational formula implies that γ is perpendicular to N at q . Consequently, p has to be in the image of the normal exponential map.We conclude that we have global Fermi coordinates in ( M , h ) in which the metrictakes the form h = dt − b ( t, x ) dx . A short computation of the curvature tensor shows that b must not depend on t .Since b (0 , x ) ≡ c by arc length), we find that b ≡ M . Weconclude that the metric in Fermi coordinates simply is the Minkowski metric.The above argument can be applied just as well to the past of N . Combining theresults for the past and future of N , we find that E gives an isometric diffeomorphismfrom an open subset Ω of R endowed with the Minkowski metric to ( M , h ). It remainsto show that Ω is a globally hyperbolic subset of R , with Cauchy surface { } × (0 , E − : ( M , h ) → Ω ⊂ R , mapsCauchy surfaces isometrically to Cauchy surfaces. (cid:3) Conformal Transformation of the Fermionic Signature Operator.
Weagain let ( M , g ) be a time-oriented, globally hyperbolic Lorentzian surface with agiven Cauchy surface N . According to Proposition 4.2, we can identify M with anopen subset of Minkowski space R , , endowed with the conformal metric g = f ( t, x ) (cid:0) dt − dx (cid:1) , f ∈ C ∞ ( M ) . (4.3)Moreover, this proposition allows us to arrange that the Cauchy surface N is theset { } × (0 , g for clarity with atilde, whereas the quantities without a tilde refer to the flat Minkowski metric. Weconsider the massless Dirac equation ˜ D ˜ ψ = 0 . This equation as well as its solutions can be described most conveniently using the conformal invariance of the massless Dirac equation (see for example [27, 26]), whichimplies that ˜ D = f − D f , ˜ ψ = f − ψ , (4.4)where D is again the Dirac operator in Minkowski space (1.5), and ψ is a solution ofthe form (2.2). The scalar product on the solutions becomes( ˜ ψ | ˜ φ ) = ˆ N ≺ ˜ ψ | /ν ˜ φ ≻| x dµ N ( x )= ˆ ≺ ˜ ψ | γ ˜ φ ≻| (0 ,x ) f (0 , x ) dx = ˆ ≺ ψ | γ φ ≻| (0 ,x ) dx , showing that the scalar product on the solutions is conformally invariant. We againdenote the corresponding Hilbert space of solutions by ( H , ( . | . )). The space-timeinner product (1.9), however, does involve the conformal factor, because < ˜ ψ | ˜ φ> = ˆ M ≺ ˜ ψ | ˜ φ ≻ dµ = ˆ M ≺ ˜ ψ | ˜ φ ≻ f ( t, x ) dt dx = ˆ M ≺ ψ | φ ≻ f ( t, x ) dt dx . As a consequence, the fermionic signature operator has a non-trivial dependence onthe conformal factor.Before we can define the fermionic signature operator again by (1.11), we need tomake sure that the space-time inner product is bounded (1.10). To this end, we as-sume that ( M , g ) has finite lifetime in the sense that it admits a foliation ( N t ) t ∈ ( t ,t ) by Cauchy surfaces with a bounded time function t such that the function h ν, ∂ t i is bounded on M (where ν denotes the future-directed normal on N t and h ν, ∂ t i ≡ g ( ν, ∂ t )). Then (1.10) holds for a suitable constant c (see [20, Proposition 3.5]).Thus (1.11) defines S as a bounded symmetric operator on the Hilbert space H .In the next lemma we again represent the fermionic signature operator as an integraloperator. Lemma 4.4.
The fermionic signature operator can be written as an integral operator ( S ˜ ψ )( x ) = ˆ S ( x, y ) ˜ ψ ( y ) f (cid:0) , y ) dy (4.5) ORENTZIAN SPECTRAL GEOMETRY 41 with the distributional kernel S ( x, y ) = 2 π f (0 , x ) − f (0 , y ) − ˆ M f ( t, z ) k ( − t, x − z ) k ( t, z − y ) γ dt dz , (4.6) where k is the causal fundamental solution of Minkowski space (2.16) .Proof. In view of the transformation of the Dirac operator in (4.4), the advancedGreen’s function ˜ s ∨ (defined by the equation ˜ D ˜ s ∨ = 11) transforms conformally as ˜ s ∨ = f − s ∨ f . Writing this operator with an integral kernel and keeping in mind thetransformation of the volume forms, one finds˜ s ∨ ( ζ, ζ ′ ) = f ( ζ ) − s ∨ ( ζ, ζ ′ ) f ( ζ ′ ) − . The retarded Green’s function transforms in the same way. Thus, introducing thecausal fundamental solution k similar to (3.3), we obtain˜ k ( ζ, ζ ′ ) = 12 πi (cid:0) ˜ s ∨ − ˜ s ∧ (cid:1) ( ζ, ζ ′ ) = f ( ζ ) − k ( ζ, ζ ′ ) f ( ζ ′ ) − . The solution formula for the Cauchy problem (2.15) and (3.1) generalizes to (see [20,Lemma 2.1]) ˜ ψ ( ζ ) = ˆ N ˜ k (cid:0) ζ, ζ ′ (cid:1) /ν ( ζ ′ ) ˜ ψ ( ζ ′ ) dµ N ( ζ ′ )= f ( ζ ) − ˆ k (cid:0) ζ − (0 , x ) (cid:1) γ ˜ ψ (0 , x ) f (0 , x ) dx . Modifying the proof of Lemma 2.9 in an obvious manner, one obtains the integralrepresentation (4.5) with S ( x, y ) = 2 π ˆ M ˜ k (cid:0) (0 , x ) , ζ (cid:1) ˜ k (cid:0) ζ, (0 , y ) (cid:1) /ν (cid:0) (0 , y ) (cid:1) dµ ( ζ )= 2 π f (0 , x ) − ˆ M k (cid:0) (0 , x ) − ζ (cid:1) f ( ζ ) − k (cid:0) ζ − (0 , y ) (cid:1) γ f (0 , y ) − f ( ζ ) d ζ . This concludes the proof. (cid:3)
We now compute the kernel more explicitly by transforming to light-cone coordi-nates (2.19) and using the form the distribution k in (2.23). We extend the function f by zero to all of R , and denote this function for clarity by χ M f . Lemma 4.5.
The integral kernel (4.6) can be written as S ( x, y ) = 116 π f (0 , x ) − f (0 , y ) − × (cid:26) γ u γ v ( χ M f ) (cid:0) i + ( x, y ) (cid:1) + γ v γ u ( χ M f ) (cid:0) i − ( x, y ) (cid:1)(cid:27) γ , (4.7) where i ± are the upper and lower points of the corresponding causal diamond definedby i + ( x, y ) = (cid:16) x − y , x + y (cid:17) , i − ( x, y ) = (cid:16) − x − y , x + y (cid:17) . (4.8) Proof.
Transforming to light-cone coordinates and using (2.23), the kernel (4.6) canbe written as S ( x, y ) = π f (0 , x ) − f (0 , y ) − × ˆ M f (cid:16) u + v , u − v (cid:17) k ( x − u, − x − v ) k ( u − y, v + y ) γ du dv , where k ( u, v ) is given by (2.16). Using the explicit form of this distribution, we cancarry out the u and v -integrations to obtain S ( x, y ) = 116 π f (0 , x ) − f (0 , y ) − ˆ M f (cid:16) u + v , u − v (cid:17) × (cid:16) γ u γ v δ ( x − u ) δ ( v + y ) + γ v γ u δ ( − x − v ) δ ( u − y ) (cid:17) γ du dv = 116 π f (0 , x ) − f (0 , y ) − × (cid:26) γ u γ v ( χ M f ) (cid:16) x − y , x + y (cid:17) + γ v γ u ( χ M f ) (cid:16) y − x , y + x (cid:17)(cid:27) γ . This gives the result. (cid:3)
Computation of tr( S ) and tr( S ) : Volume and Curvature. Having derivedexplicit formulas for the integral kernel S , the spectral properties of the fermionicsignature operator can be analyzed similarly as described in Sections 2.3–2.6 for sub-sets of Minkowski space. Some results (like Proposition 2.13) generalize immediately,whereas for other results (like Proposition 2.12) the generalization is less obvious. Forbrevity, we shall not reconsider all the results for the Minkowski drum. Instead, werestrict attention to generalizing Propositions 2.10 and 2.11 to curved surfaces. Themain point of interest is that the resulting formula for tr( S ) involves scalar curvature(see Proposition 4.8 below).In preparation, we show that the statement of Corollary 2.6 still holds, providedthat the space-time volume is finite. Lemma 4.6.
Let ( M , g ) be a globally hyperbolic Lorentzian surface of finite lifetime.If the total g -volume µ ( M ) is finite, then the fermionic signature operator is Hilbert-Schmidt. Moreover, the traces of even powers of S q , q ∈ N , are given by the integrals tr( S q ) = ˆ f (0 , x ) dx . . . ˆ f (0 , x q ) dx q Tr (cid:0) S ( x , x ) · · · S ( x q , x ) (cid:1) . (4.9) Proof.
Following the method in the proof of Corollary 2.6, the Hilbert-Schmidt prop-erty as well as the formula (4.9) in case q = 1 follows immediately once we know thatthe kernel of the fermionic signature operator is square integrable in the sense that ˆ ˆ k S ( x, y ) k f (0 , x ) dx f (0 , y ) dy < ∞ . (4.10)Estimating (4.7), we obtain k S ( x, y ) k ≤ π f (0 , x ) − f (0 , y ) − (cid:16) ( χ M f ) (cid:0) i + ( x, y ) + ( χ M f ) (cid:0) i − ( x, y ) (cid:1)(cid:17) k S ( x, y ) k ≤ π ) f (0 , x ) − f (0 , y ) − (cid:16) ( χ M f ) (cid:0) i + ( x, y ) + ( χ M f ) (cid:0) i − ( x, y ) (cid:1)(cid:17) ORENTZIAN SPECTRAL GEOMETRY 43 and thus ˆ ˆ k S ( x, y ) k f (0 , x ) dx f (0 , y ) dy ≤ π ) ˆ ˆ (cid:16) ( χ M f ) (cid:0) i + ( x, y ) + ( χ M f ) (cid:0) i − ( x, y ) (cid:1)(cid:17) dx dy . Using (4.8), one can rewrite the integrals as a space-time integral to obtain ˆ ˆ k S ( x, y ) k f (0 , x ) dx f (0 , y ) dy ≤ π ) ˆ M f ( t, z ) dt dz = 8(16 π ) µ ( M ) , where in the last step we used that dµ = f dx dy . This shows (4.10) and concludesthe proof in the case q = 1.In order to treat the case q >
1, by iterating (4.5) and using Fubini’s theorem, oneobtains an integral representation of S q with a kernel which is again square integrable.Again arguing as in the proof of Corollary 2.6, we obtain the result. (cid:3) Proposition 4.7.
Let ( M , g ) be a globally hyperbolic Lorentzian surface of finite life-time and finite volume. Then tr (cid:0) S (cid:1) = µ ( M )4 π . Proof.
Evaluating (4.9) for the kernel (4.6) and (4.7), in generalization of (2.24)and (2.26) we obtaintr (cid:0) S q (cid:1) = ˆ M f ( ξ ) d ξ · · · ˆ M f ( ξ q ) d ξ q × Tr (cid:16) k m (cid:0) ξ − ξ (cid:1) · · · k m (cid:0) ξ q − − ξ q (cid:1) k m (cid:0) ξ q − ξ (cid:1)(cid:17) = 2(2 π ) q ˆ M f ( ζ ) d ζ ˆ M f ( η ) d η · · · ˆ M f ( ζ q ) d ζ q ˆ M d f ( η q ) η q × δ ( u − ˜ u ) δ (˜ v − v ) · · · δ ( u q − ˜ u q ) δ (˜ v q − v ) , (4.11)where in the last line we set ζ j = ξ j − (having the light-cone coordinates ( u j , v j ))and η j = ξ j (having the light-cone coordinates (˜ u j , ˜ v j )). In particular,tr (cid:0) S (cid:1) = 12 π ˆ M f ( ζ ) d ζ ˆ M f ( η ) d η δ ( u − ˜ u ) δ (˜ v − v )= 14 π ˆ M f ( ζ ) d ζ = µ ( M )4 π , giving the result. (cid:3) Proposition 4.8.
Let ( M , g ) be a globally hyperbolic Lorentzian surface of finite life-time and finite volume. Then tr (cid:0) S (cid:1) = 18 π ˆ M dµ ( ζ ) ˆ J ( ζ ) exp (cid:18) ˆ D ( ζ,ζ ′ ) R dµ (cid:19) dµ ( ζ ′ ) , where D ( ζ, ζ ′ ) is the causal diamond (1.13) , and R denotes scalar curvature. Proof.
We evaluate (4.11) in the case q = 2. For two causally separated points ζ and ζ ′ , we again let D ( ζ, ζ ′ ) be the causal diamond whose upper and lower cornersare ζ and ζ ′ . The left and right corners of this causal diamond are denoted by η and η ′ ,respectively (similar as in the left of Figure 5). Thentr (cid:0) S (cid:1) = 18 π ˆ M f ( ζ ) d ζ ˆ M f ( η ) d η ˆ M f ( ζ ) d ζ ˆ M d f ( η ) η × δ ( u − ˜ u ) δ (˜ v − v ) δ ( u − ˜ u ) δ (˜ v − v )= 18 π ˆ M d ζ ˆ J ( ζ ) d ζ ′ f ( ζ ) f ( η ) f ( ζ ′ ) f ( η ′ ) . (4.12)The interesting point is that this not the same as the coordinate invariant quantity18 π ˆ M dµ ( ζ ) ˆ J ( ζ ) dµ ( ζ ′ ) = 18 π ˆ M d ζ ˆ J ( ζ ) d ζ ′ f ( ζ ) f ( ζ ′ ) , because the factors f appear in a different combination. In order to express thisdifference geometrically, we first note that scalar curvature is given by R = 2 K = − f (cid:3) log( f ) = − f (cid:3) log f (see for example [8, page 237], where (cid:3) denotes the wave operator in Minkowski space).Integrating this formula for scalar curvature over the causal diamond, introducinglight-cone coordinates and integrating by parts, we obtain ˆ D ( ζ,ζ ′ ) R dµ = − ˆ D ( ζ,ζ ′ ) (cid:3) log f dt dx = − ˆ D ( ζ,ζ ′ ) ∂ uv log f du dv = − (cid:0) log f ( ζ ) + log f ( ζ ′ ) − log f ( η ) − log f ( η ′ ) (cid:1) . Hence exp (cid:18) ˆ D ( zη,ζ ′ ) R dµ (cid:19) = f ( η ) f ( η ′ ) f ( ζ ) f ( ζ ′ )and thus f ( ζ ) f ( η ) f ( ζ ′ ) f ( η ′ ) = f ( ζ ) f ( ζ ′ ) exp (cid:18) ˆ D ( ζ,ζ ′ ) R dµ (cid:19) . Using this relation in (4.12) concludes the proof. (cid:3)
A Reconstruction Theorem.
The goal of this section is to prove Theorem 2.14as well as its generalization to curved surfaces. Thus we again let ( M , g ) be atime-oriented, globally hyperbolic Lorentzian surface of finite lifetime together witha Cauchy surface N . Just as described at the beginning of Section 4.2, we can con-sider M as a subset of R , with the conformally flat metric (4.3). Moreover, wecan arrange that N = { } × (0 , I ⊂ (0 ,
1) and a chiral in-dex c ∈ { L, R } we introduce π c,I := χ c χ I , where χ I is the characteristic function, and χ c are the projections on the left- orright-handed components, χ L = (cid:18) (cid:19) , χ R = (cid:18) (cid:19) . (4.13) ORENTZIAN SPECTRAL GEOMETRY 45
We consider π c,I : H → H as a multiplication operator on the wave functions ˜ ψ onthe Cauchy surface N . Obviously, π c,I is a projection operator on H .Next, for an open subset I ⊂ (0 ,
1) we introduce the sets K L ( I ) , K R ( I ) ⊂ M ⊂ R , obtained by propagating I with velocity one to the left respectively right, K L ( I ) = (cid:8) ( t, x ) ∈ M | x + t ∈ I (cid:9) , K R ( I ) = (cid:8) ( t, x ) ∈ M | x − t ∈ I (cid:9) . Lemma 4.9.
Assume that for two open subsets
I, J ⊂ (0 , , the following integralexists, ˆ K L ( I ) ∩ K R ( J ) f d ζ < ∞ . Then the operator product π L,I S π R,J is Hilbert-Schmidt, and its Hilbert-Schmidt normis given by (cid:13)(cid:13) π L,I S π R,J (cid:13)(cid:13) HS = 18 π ˆ K L ( I ) ∩ K R ( J ) f d ζ . Proof.
We first compute the kernel of the operator π L,I S π R,J . Combining (4.13)and (2.20) with (1.6), one sees that γ u γ v = 4 χ L and γ v γ u = 4 χ R . Using Lemma 4.5, we obtain π L,I S ( x, y ) π R,J = 14 π f (0 , x ) − f (0 , y ) − × χ I ( x ) χ J ( y ) ( χ M f ) (cid:0) i + ( x, y ) (cid:1) χ L γ (4.14) k π L,I S ( x, y ) π R,J k = 116 π f (0 , x ) − f (0 , y ) − × χ I ( x ) χ J ( y ) (cid:16) ( χ M f ) (cid:0) i + ( x, y ) (cid:1)(cid:17) . (4.15)Using the Fourier series method in the proof of Corollary 2.6, one concludes that theoperator π L,I S π R,J is Hilbert-Schmidt if and only if the function (4.15) is integrablewith respect to the measure f (0 , x ) dx f (0 , y ) dy . In this case, the Hilbert-Schmidtnorm satisfies the equality (cid:13)(cid:13) π L,I S π R,J (cid:13)(cid:13) = 116 π ˆ ˆ χ I ( x ) χ J ( y ) (cid:16) ( χ M f ) (cid:0) i + ( x, y ) (cid:1)(cid:17) dx dy . It remains to interpret the integrand geometrically with the help of the definitionof i + in (4.8). First of all, due to the factor χ M , it suffices to consider the casethat i + ∈ M . Then the condition x ∈ I means that the space-time point i + ( x, y ) mustlie in K L ( I ). Similarly, the condition y ∈ J means that i − ( x, y ) ∈ K R ( J ). Finally,denoting the components of i + ( x, y ) by ( t, x ) and transforming the integration measureaccording to dx dy = 2 dt dx , the result follows. (cid:3) The result of this lemma has a simple geometric interpretation which does not relyon our embeddings. In order to make this point clear, we now consider ( M , g ) as anabstract oriented, time-oriented, globally hyperbolic manifold of finite lifetime with agiven non-compact Cauchy surface N . We identify the Hilbert space of solutions withthe initial values on the Cauchy surface, i.e. H = L ( N , S M ) ≃ L ( N , S N ) ⊕ L ( N , S N ) , where the two direct summands describe the left- and right-handed components ofthe spinors, respectively. Let I and J be open subsets of N . Then the multiplicationoperators χ L,I and χ R,J are defined on H in an obvious way. Moreover, the sets K L ( I )can be defined as all points of M which can be reached from I by a lightlike geodesicpropagating to the left. Similarly, K R ( J ) ⊂ M is the set of all points which canbe reached from J by a lightlike geodesic propagating to the right. Moreover, theintegrand f d ζ is the same as the volume measure dµ corresponding to the metric g .We thus obtain the following result. Proposition 4.10.
Assume that for two open subsets
I, J ⊂ N , the set K L ( I ) ∩ K R ( J ) has finite volume. Then the operator product π L,I S π R,J is Hilbert-Schmidt, and itsHilbert-Schmidt norm is given by (cid:13)(cid:13) π L,I S π R,J (cid:13)(cid:13) HS = 18 π µ (cid:16) K L ( I ) ∩ K R ( J ) (cid:17) . This lemma is very useful because if S is given as an operator on the Hilbert space ofsections of the spinor bundle on a Cauchy surface, then the volume of the sets K L ( I ) ∩ K R ( J ) can be recovered for any open subsets I, J ⊂ N . In particular, by choosing thesets I and J as small neighborhoods of points x, y ∈ N , one may find out whether thenull geodesics through x and y meet at a space-time point i + ( x, y ). If they do, one caneven determine the volume form at this space-time point. For subsets of Minkowskispace, we thus obtain the statement of Theorem 2.14.We finally formulate the reconstruction theorem for general surfaces. For the sakeof conceptual clarity, we formulate this result in the language of categories. Let X be a locally compact Hausdorff space. By C ( X ) we denote the continuous functionson X which vanish at infinity in the sense that for every ε > K ⊂ X such that | f ( x ) | < ε for all x ∈ X \ K . Then the celebrated Gelfand-Naimark theorem states that X can be reconstructed (modulo homeomorphisms) fromthe single datum of the C ∗ algebra C ( X ). More specifically, for a commutative C ∗ -algebra A with the property k a k = k a k , the spectrum of A is defined as the set s ( A ) of all non-zero ∗ -homomorphisms from A to C with the topology of pointwiseconvergence. Then the Gelfand-Naimark theorem states that A and C ( s ( A )) (thelatter equipped with the sup-metric) are ∗ -isometric by evaluation. Thus C ◦ s is theidentity on the family of C ∗ algebras, modulo ∗ -isomorphisms. Moreover, applying s once again, one finds that s ◦ C is the identity on the family of locally compactHausdorff topological spaces, modulo homeomorphisms.Various attempts to extend this approach such as to include geometrical data in thereconstruction have received much attention in the past decades (see for example [25]or [7]). In the same spirit, we define G as the category of all tuples ( M , N ), where M isan oriented globally hyperbolic surface of finite life-time with a spatially non-compactCauchy surface N , and the morphisms given by pair isometries. Next, let H be thecategory of isomorphism classes of triples ( A , H ⊕ H , S ) where H is a Hilbert space, A is a C ∗ -algebra of bounded linear operators on H , and S is a bounded linear operatoron H ⊕ H . We now construct a functor from G to H . Given ( M , N ), the fibers S x N of the spinor bundle S N are isomorphic to C . Moreover, the bundle S N is canonicallyisomorphic to the restriction of S M to N and projecting to the left- or right-handedcomponent. We choose H = L ( N , S N ). Moreover, we choose A as C ( N ) actingby multiplication on H . Noting that H = L ( N , S M ) ≃ H ⊕ H , we let S be the ORENTZIAN SPECTRAL GEOMETRY 47 fermionic signature operator on H ⊕ H . This gives rise to the functor S | N : G → H . Theorem 4.11.
The functor S | N is injective.Proof. First of all, assume that two elements ( M , N ) and ( M , N ) of G are mappedby S | N to one and the same element ( A , H ⊕ H , S ) of H . We need to show thatthere is a pair isometry between ( M , N ) and ( M , N ). First, the Gelfand-Naimarktheorem tells us that A is homeomorphic to N and N , giving rise to a homeomor-phism h : N → N . Taking the pull-back of h , we obtain an isomorphism ι betweenthe corresponding spaces of L -sections H and H . Applying Proposition 4.10 andchoosing I = J as small neighborhoods of a point x ∈ N ≃ N , we see that h isactually an isometry (in particular, a diffeomorphism). Next, we identify M and M via identification of the corresponding sets K L ( I ) ∩ K R ( J ). Since this identificationobviously preserves the conformal structure, the resulting metrics coincide up to aconformal factor. Applying Proposition 4.10 once again, we conclude that the volumeof the sets K L ( I ) ∩ K R ( J ) coincides, proving that the conformal factor is equal to one.We thus obtain an extension of the above isometry N → N to an isometry M → M .This concludes the proof. (cid:3) Acknowledgments:
We would like to thank Moritz Reintjes for helpful discussions.
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