aa r X i v : . [ m a t h - ph ] O c t External Field QED on Cauchy Surfacesfor varying Electromagnetic Fields
D.-A. Deckert and F. Merkl
Mathematisches Institut der Ludwig-Maximilians-Universit¨at M¨unchenTheresienstr. 39, 80333 M¨unchen, [email protected], [email protected]
July 16, 2018
Abstract
The Shale-Stinespring Theorem (1965) together with Ruijsenaar’s criterion (1977)provide a necessary and sufficient condition for the implementability of the evolutionof external field quantum electrodynamics between constant-time hyperplanes on stan-dard Fock space. The assertion states that an implementation is possible if and onlyif the spatial components of the external electromagnetic four-vector potential A µ arezero. We generalize this result to smooth, space-like Cauchy surfaces and, for general A µ , show how the second-quantized Dirac evolution can always be implemented as amap between varying Fock spaces. Furthermore, we give equivalence classes of polar-izations, including an explicit representative, that give rise to those admissible Fockspaces. We prove that the polarization classes only depend on the tangential compo-nents of A µ w.r.t. the particular Cauchy surface, and show that they behave naturallyunder Lorentz and gauge transformations. We consider the external field model of quantum electrodynamics (QED) or no-photon QEDwhich describes a Dirac sea of electrons evolving subject to a prescribed external electromag-netic four-vector potential A µ . To infer the evolution operator of this model one attemptsto implement the one-particle Dirac evolution p i {B ´ { A q ψ “ mψ (1)in second-quantized form. Here, m ą e (having a negative sign in the case of an electron) is already absorbedin A ; units are chosen such that ~ “ c “
1. The employed relativistic notation isintroduced with all other notations in Section 1.3. For sake of simplicity we will restrict usto smooth and compactly supported A µ , i.e., A “ p A µ q µ “ , , , “ p A , A q P C c p R , R q , (2)1lthough this condition is unnecessarily strong.It is well-known [21, 18] that, on standard Fock space and for equal-time hyperplanes, asecond quantization of the one particle Dirac evolution (1) is possible if and only if A “ A is usually ignored at first which later manifestsitself in the appearance of infinities in informal perturbation series. Those infinities have tobe taken out by hand or, as for example in the case of the vacuum expectation value of thecharge current, absorbed in the coefficient of the electron charge. Nevertheless, since thesole interaction arises only from a prescribed four-vector field one may rather expect that itshould be possible to control the time evolution non-perturbatively. One way to constructa well-defined second-quantized time evolution operator, as sketched in [6], is to implementit between time-varying Fock spaces. Such constructions have been carried out, e.g., in[14, 15, 2]. While the idea of changing Fock spaces might be unfamiliar as seen from thenon-relativistic setting, in a relativistic formulation it is to be expected. A Lorentz boost forinstance may tilt an equal-time hyperplane to a space-like space-like hyperplane Σ, whichrequires a change from standard Hilbert space L p R , C q to one attached to Σ, and likewise,for the corresponding Fock spaces.In this work we extend the existing constructions in [14, 15, 2], which deal exclusivelywith equal-time hyperplanes, by implementing the second-quantized Dirac evolution fromone Cauchy surface to another. The resulting formulation of external field QED has severaladvantages: 1) Its Lorentz and gauge covariance can be made explicit; 2) as it treats theinitial value problem for general Cauchy surfaces it allows to study the evolution in theform of local deformations of Cauchy surfaces in the spirit of Tomonaga and Schwinger, e.g.,[22, 20]; 3) it gives a geometric and more general version of the implementability condition A “ F p V, H Σ q . Inthis notation H Σ is the Hilbert space of C -valued, square integrable functions on Σ (seeDefinition 1.10 below) and V P Pol p H Σ q is one of its polarizations: Definition 1.1.
Let
Pol p H Σ q denote the set of all closed, linear subspaces V Ă H Σ suchthat V and V K are both infinite dimensional. Any V P Pol p H Σ q is called a polarization of H Σ . For V P Pol p H Σ q , let P V Σ : H Σ Ñ V denote the orthogonal projection of H Σ onto V . The Fock space corresponding to polarization V on Cauchy surface Σ is then defined by F p V, H Σ q : “ à c P Z F c p V, H Σ q , F c p V, H Σ q : “ à n,m P N c “ m ´ n p V K q ^ n b V ^ m , (3)where À denotes the Hilbert space direct sum, ^ the antisymmetric tensor product ofHilbert spaces, and V denotes the conjugate complex vector space of V , which coincideswith V as a set and has the same vector space operations as V with the exception of thescalar multiplication, which is redefined by p z, ψ q ÞÑ z ˚ ψ for z P C , ψ P V .2ach polarization V splits the Hilbert space H Σ into a direct sum, i.e., H Σ “ V K ‘ V . Theso-called standard polarizations H ` Σ and H ´ Σ are determined by the orthogonal projectors P ` Σ and P ´ Σ onto the free positive and negative energy Dirac solutions, respectively, restrictedto Σ: H ` Σ : “ P ` Σ H Σ “ p ´ P ´ Σ q H Σ , H ´ Σ : “ P ´ Σ H Σ . (4)Loosely speaking, in terms of Dirac’s hole theory, the polarization V P Pol p H Σ q indicatesthe “sea level” of the Dirac sea, and electron wave functions in V K and V are consideredto be “above” and “below” sea level, respectively. However, it has to be stressed that themathematical structure of the external field problem in QED does not seem to discriminatebetween particular choices of polarizations V . Hence, unless an additional physical conditionis delivered, the V -dependent labels “electron” and “positron” are somewhat arbitrary, and V should rather be regarded as a choice of coordinate system w.r.t. which the states of theDirac sea are represented. To describe pair-creation on the other hand it is necessary tohave a distinguished V , and the common (and seemingly most natural) ad hoc choice insituations when the external field vanishes is V “ H ´ Σ . Nevertheless, it is conceivable thatonly a yet to be found full version of QED, including the interaction with the photon field,may distinguish particular polarizations V in general situations.Given two Cauchy surfaces Σ , Σ and two polarizations V P Pol p H Σ q and W P Pol p H Σ q a sensible lift of the one-particle Dirac evolution U A Σ Σ : H Σ Ñ H Σ (see Definition 1.13)should be given by a unitary operator r U A Σ Σ : F p V, H Σ q Ñ F p W, H Σ q that fulfills r U A Σ Σ ψ V, Σ p f q p r U A Σ Σ q ´ “ ψ W, Σ p U A Σ Σ f q , @ f P H Σ . (5)Here, ψ V, Σ denotes the Dirac field operator corresponding to Fock space F p V, Σ q , i.e., ψ V, Σ p f q : “ b Σ p P V K Σ f q ` d ˚ Σ p P V Σ f q , @ f P H Σ . (6)Here, b Σ , d ˚ Σ denote the annihilation and creation operators on the V K and V sectors of F c p V, H Σ q , respectively. Note that P V Σ : H Ñ V is anti-linear ; thus, ψ V, Σ p f q is anti-linearin its argument f . The condition under which such a lift r U A Σ Σ exists can be inferred from astraight-forward application of Shale and Stinespring’s well-known theorem [21]: Theorem 1.2 (Shale-Stinespring) . The following statements are equivalent:(a) There is a unitary operator r U A ΣΣ : F p V, H Σ q Ñ F p W, H Σ q which fulfills (5) .(b) The off-diagonals P W K Σ U A Σ Σ P V Σ and P W Σ U A Σ Σ P V K Σ are Hilbert-Schmidt operators. Note that the phase of the lift is not fixed by condition (5). Even worse, as indicatedearlier, depending on the external field A this condition is not always satisfied; see [18]. Onthe other hand, the choices made for the polarizations V and W were completely arbitrary.We shall see next that adapting these choices carefully will however yield an evolution of theDirac sea in the corresponding Fock space representations.There is a trivial but not so useful choice. Pick a Σ in in the remote past of the supportof A fulfilling Σ in is a Cauchy surface such that supp A X Σ in “ H . (7)3hen the choices V “ U A ΣΣ in H ´ Σ in and W “ U A Σ Σ in H ´ Σ in fulfill (b) of Theorem 1.2 as theoff-diagonals are zero. The drawback of these choices is that the resulting lift depends onthe whole history of A between Σ in and Σ , Σ . Moreover, such V and W are rather implicit.But statement (b) in Theorem 1.2 also allows to differ from the projectors P V Σ and P W Σ by aHilbert-Schmidt operator. Hence, it lies near to characterize polarizations according to thefollowing classes: Definition 1.3 (Physical Polarization Classes) . For a Cauchy surface Σ we define C Σ p A q : “ “ U A ΣΣ in H ´ Σ in ‰ « , (8) where for V, V P Pol p H Σ q , V « V means that P V Σ ´ P V Σ P I p H Σ q , i.e., is a Hilbert-Schmidtoperator H Σ ý . The equivalence relation « can be refined to give another equivalence relation « de-scribing polarization classes of equal charge; c.f. [2] and Remark 1.8. As a simple corollaryof Theorem 1.2 one gets: Corollary 1.4 (Dirac Sea Evolution) . Let Σ , Σ be Cauchy surfaces. Then any choice V P C Σ p A q and W P C Σ p A q implies condition (b) of Theorem 1.2 and therefore the existence ofa lift r U A Σ Σ : F p V, H Σ q Ñ F p W, H Σ q obeying (5) . Consequently, any choice V P C Σ p A q and W P C Σ p A q gives rise to a lift of the one-particleDirac evolution between the corresponding F p V, H Σ q and F p W, H Σ q that is unique up to aphase. The crucial questions are: 1) On which properties of A and Σ do these polarizationclasses depend? 2) How do they behave under Lorentz and gauge transforms? 3) Is there anexplicit representative for each class? These question will be answered by our main resultsgiven in the next section. The next important question is about the unidentified phase.Although transition probabilities are independent of this phase, dynamic quantities like thecharge current will depend directly on it. We briefly discuss this in Section 1.2 and givean outlook of what needs to be done to derive the vacuum expectation of the polarizationcurrent. The definition (8) of the physical polarization classes involves the one-particle Dirac evolutionoperator and is therefore not very useful in finding an explicit description of admissible Fockspaces for the implementation of the second-quantized Dirac evolution. In our main resultsTheorems 1.5-1.7 we give a more direct identification of the polarization classes classes C Σ p A q and state some of their fundamental geometric properties.The first one ensures that the classes C Σ p A q are independent of the history of A , insteadthey depend on the tangential components of A on Σ only. Theorem 1.5 (Identification of Polarization Classes) . Let Σ be a Cauchy surface and let A and r A be two smooth and compactly supported external fields. Then C Σ p A q “ C Σ p r A q ô A | T Σ “ r A | T Σ (9) where A | T Σ “ r A | T Σ means that for all x P Σ and y P T x Σ we have A µ p x q y µ “ r A µ p x q y µ . r A “ t fixed, Σ “ Σ t “ t x P R | x “ t u being an equal-time hyperplane.Furthermore, the polarization classes transform naturally under Lorentz and gauge trans-formations: Theorem 1.6 (Lorentz and Gauge Transforms) . Let V P Pol p H Σ q be a polarization.(i) Consider a Lorentz transformation given by L p S, Λ q Σ : H Σ Ñ H ΛΣ for a spinor transfor-mation matrix S P C ˆ and an associated proper orthochronous Lorentz transformationmatrix Λ P SO Ò p , q , cf. [3, Section 2.3]. Then: V P C Σ p A q ô L p S, Λ q Σ V P C ΛΣ p Λ A p Λ ´ ¨qq . (10) (ii) Consider a gauge transformation A ÞÑ A ` B Ω for some Ω P C c p R , R q given by themultiplication operator e ´ i Ω : H Σ Ñ H Σ , ψ ÞÑ ψ “ e ´ i Ω ψ . Then: V P C Σ p A q ô e ´ i Ω V P C Σ p A ` B Ω q . (11)As we are mainly interested in a local study of the second-quantized Dirac evolution,we only allow compactly supported vector potentials A , and therefore, have to restrict thegauge transformations e ´ i Ω to compactly supported Ω as well. Treating more general vectorpotentials A and gauge transforms e ´ i Ω would require an analysis of decay properties atinfinity which is not our focus here.Finally, given Cauchy surface Σ, there is an explicit representative of the equivalenceclass of polarizations C Σ p A q which can be given in terms of a compact, skew-adjoint linearoperator Q A Σ : H Σ ý , as defined in (56) below. With it the polarization class can be identifiedas follows: Theorem 1.7.
Given Cauchy surface Σ , we have C Σ p A q “ ” e Q A Σ H ´ Σ ı « . Other representatives for polarization classes C Σ p A q beyond the “interpolating represen-tation” U A ΣΣ in H ´ Σ in , as used in Definition 1.3, can be inferred from the so-called Furry picture,as worked out for equal-time hyperplanes in [6], and from the global constructions of thefermionic projector given in [11, 10]. In contrast to global constructions, the representationgiven in Theorem 1.7 uses only local geometric information of the vector potential A at Σ;cf. (56), (39), and Lemma 2.3 below.The implications on the physical picture can be seen as follows. The Dirac sea on Cauchysurface Σ can be described in any Fock space F p V, H Σ q for any choice of polarization V P C Σ p A q . The polarization class C Σ p A q is uniquely determined by the tangential components ofthe external potential A on Σ. This is an object that transforms covariantly under Lorentzand gauge transformations. The choice of the particular polarization can then be seen asa “choice of coordinates” in which the Dirac sea is described. When regarding the Diracevolution from one Cauchy surface Σ to Σ another “choice of coordinates” W P C Σ p A q has to be made. Then one yields an evolution operator r U A Σ Σ : F p V, H Σ q Ñ F p W, H Σ q which is unique up to an arbitrary phase Corollary 1.4. Transition probabilities of the kind |x Ψ , r U A Σ Σ Φ y| for Ψ P F p W, H Σ q and Φ P F p V, H Σ q are well-defined and unique without5he need of a renormalization method. Finally, for a family of Cauchy surfaces p Σ t q t P R that interpolates smoothly between Σ and Σ we also give an infinitesimal version of howthe external potential A changes the polarization in terms of the flow parameter t ; seeTheorem 2.8 below. Remark 1.8 (Charge Sectors) . Given two polarizations
V, W P Pol p H Σ q such that P V Σ ´ P W Σ is a compact operator, e.g., as in the case V « W as defined in (8) , one can define theirrelative charge, denoted by charge p V, W q , to be the Fredholm index of P W Σ | V Ñ W ; cf. [2].The equivalence relation « in the claim of Theorem 1.7 can then be replaced by the finerequivalence relation « , which is defined as follows: V « W if and only if V « W and charge p V, W q “ . This is shown as an addendum to the proof of Theorem 1.7. As indicated at the end of the introduction the current operator depends directly on theunspecified phase of r U A Σ Σ . This can be seen from Bogolyubov’s formula j µ p x q “ i r U A Σ in Σ out δ r U A Σ out Σ in δA µ p x q (12)where Σ out is a Cauchy surfaces in the remote future of the support of A such that Σ out X supp A “ H . Hence, without identification of the derivative of the phase of r U A Σ Σ the physicalcurrent is not fully specified. Nevertheless, now the situation is slightly better than in thestandard perturbative approach. As for each choice of admissible polarizations in C Σ p A q and C Σ p A q , identified above, there is a well-defined lift r U A Σ Σ of the Dirac evolution operator U A Σ Σ and therefore also a well-defined current (12). Now it is only the task to select thephysical relevant one. One way of doing so is to impose extra conditions on the (12), andhence, the phase, so that the set of admissible phases shrinks to one that produces thesame currents up to the known freedom of charge renormalization; see [5, 19, 15, 12]. Inthe case of equal-time hyperplanes a choice of the unidentified phase was given by paralleltransport in [16]. On top of the geometric construction and despite the fact that there arestill degrees of freedom left, Mickelsson’s current is particularly interesting because it agreeswith conventional perturbation theory up to second order. Yet the open question remainswhich additional physical requirements may constraint these degree of freedoms up to theone of the numerical value of the elementary charge e fixed by the experiment.The issue of the unidentified phase particularly concerns the so-called phenomenon of“vacuum polarization” as well as the dynamical description of pair creation processes forwhich only a few rigorous treatments are available; e.g., see [13] for vacuum polarization inthe Hartree-Fock approximation for static external sources, [17] for adiabatic pair creation,and for a more fundamental approach the so-called “Theory of Causal Fermion Systems”[7, 8, 9], which is based on a reformulation of quantum electrodynamics by means of anaction principle. In this section we briefly review the notation and results about the one-particle Dirac evo-lution on Cauchy surfaces provided in a previous work [3]. The present article, dealing with6he second-quantization Dirac evolution, is based on this work.Space-time R is endowed with metric tensor g “ p g µν q µ,ν “ , , , “ diag p , ´ , ´ , ´ q ,and its elements are denoted by four-vectors x “ p x , x , x , x q “ p x , x q “ x µ e µ , for e µ being the canonical basis vectors. Raising and lowering of indices is done w.r.t. g . Moreover,we use Einstein’s summation convention, the standard representation of the Dirac matrices γ µ P C ˆ that fulfill t γ µ , γ ν u “ g µν , and Feynman’s slash-notation {B “ γ µ B µ , { A “ γ µ A µ .When considering subsets of space-time R we shall use the following notations: Causal : “t x P R | x µ x µ ě u and Past : “ t x P R | x µ x µ ą , x ă u .The central geometric objects for posing the initial value problem for (1) are Cauchysurfaces defined as follows: Definition 1.9 (Cauchy Surfaces) . We define a Cauchy surface Σ in R to be a smooth,3-dimensional submanifold of R that fulfills the following three conditions:(a) Every inextensible, two-sided, time- or light-like, continuous path in R intersects Σ ina unique point.(b) For every x P Σ , the tangential space T x Σ is space-like.(c) The tangential spaces to Σ are bounded away from light-like directions in the followingsense: The only light-like accumulation point of Ť x P Σ T x Σ is zero. In coordinates, every Cauchy surface Σ can be parametrized asΣ “ t π Σ p x q : “ p t Σ p x q , x q | x P R u (13)with a smooth function t Σ : R Ñ R . For convenience and without restricting generality ofour results we keep a global constant 0 ă V max ă x P R | ∇ t Σ p x q| ď V max . (15)The assumption (c) in Definition 1.9 and (15) can be relaxed to | ∇ t Σ p x q| ă x P R due to the causal structure of the solutions to the Dirac equation, although this is not workedout in this paper.The standard volume form over R is denoted by d x “ dx dx dx dx ; the product offorms is understood as wedge product. The symbols d x and d x mean the 3-form d x “ dx dx dx on R and on R , respectively. Contraction of a form ω with a vector v isdenoted by i v p ω q . The notation i v p ω q is also used for the spinor matrix valued vector γ “p γ , γ , γ , γ q “ γ µ e µ : i γ p d x q “ γ µ i e µ p d x q . (16)Furthermore, for a 4-spinor ψ P C (viewed as column vector), ψ stands for the row vector ψ ˚ γ , where ˚ denotes hermitian conjugation.7mooth families p Σ t q t P T of Cauchy surfaces, indexed by an interval T Ď R and fulfilling(15), are denoted by Σ : “ tp x, t q| t P T, x P Σ t u . (17)Given the external electromagnetic vector potential A P C c p R , R q of interest, we assumethat the set tp x, t q P Σ | x P supp p A qu is compact. This condition is trivially fulfilled in theimportant case of a compact interval T “ r t , t s with Σ interpolating between two Cauchysurfaces Σ t and Σ t . The compactness condition is also automatically fulfilled in the casethat T “ R with Σ being a smooth foliation of the Minkowski space-time R .We assume furthermore that the family p Σ t q t P T is driven driven by a (Minkowski) normalvector field vn : Σ Ñ R , where n : Σ Ñ R , p x, t q ÞÑ n t p x q , denotes the future-directed(Minkowski) normal unit vector field to the Cauchy surfaces and v : Σ Ñ R , p x, t q ÞÑ v t p x q ,denotes the speed at which the Cauchy surfaces move forward in normal direction. Fortechnical reasons, in particular when using the chain rule, it is convenient to extend the“speed” v and the unit vector field n in a smooth way to the domain R ˆ T . In the casethat Σ is a foliation of space-time, we may even drop the t –dependence of v and n . In thisimportant case, some of the arguments below become slightly simpler. Definition 1.10 (Spaces of Initial Data) . For any Cauchy surface Σ we define the vectorspace C Σ : “ C c p Σ , C q . For a given Cauchy surface Σ , let H Σ “ L p Σ , C q denote the vectorspace of all 4-spinor valued measurable functions φ : Σ Ñ C (modulo changes on null sets)having a finite norm } φ } “ a x φ, φ y ă 8 w.r.t. the scalar product x φ, ψ y “ ż Σ φ p x q i γ p d x q ψ p x q . (18)For x P Σ, the restriction of the spinor matrix valued 3-form i γ p d x q to the tangentialspace T x Σ is given by i γ p d x q “ { n p x q i n p d x q “ ˜ γ ´ ÿ µ “ γ µ B t Σ p x qB x µ ¸ d x “ : Γ p x q d x on p T x Σ q . (19)As a consequence of the (15), there is a positive constant Γ max “ Γ max p V max q such that } Γ p x q} ď Γ max , @ x P R . (20)The class of solutions to the Dirac equation (1) considered in this work is defined by: Definition 1.11 (Solution Spaces) . (i) Let C A denote the space of all smooth solutions ψ P C p R , C q of the Dirac equation (1) which have a spatially compact causal support in the following sense: There is a compactset K Ă R such that supp ψ Ď K ` Causal .(ii) We endow C A with the scalar product given in (18) ; note that due to conservation ofthe 4-vector current φγ µ ψ , the scalar product x¨ , ¨y : C A ˆ C A Ñ C is independent of theparticular choice of Σ . iii) Let H A be the Hilbert space given by the (abstract) completion of C A . Theorem 2.21 in [3] ensures:
Theorem 1.12 (Initial Value Problem and Support) . Let Σ be a Cauchy surface and χ Σ P C c p Σ , C q be given initial data. Then, there is a ψ P C A such that ψ | Σ “ χ Σ and supp ψ Ď supp χ Σ ` Causal . Moreover, suppose r ψ P C p R , C q solves the Dirac equation (1) for initialdata r ψ | Σ “ χ Σ , then r ψ “ ψ . This theorem gives rise to the following definition in which we use the notation ψ | Σ P C Σ to denote the restriction of a ψ P C A to a Cauchy surface Σ. Definition 1.13 (Evolution Operators) . Let Σ , Σ be Cauchy surfaces. In view of Theo-rem 1.12 we define the isomorphic isometries U Σ A : C A Ñ C Σ ,U A Σ : C Σ Ñ C A ,U A Σ Σ : C Σ Ñ C Σ , U Σ A φ : “ φ | Σ ,U Σ A χ Σ : “ ψ,U A Σ Σ : “ U Σ A U A Σ , (21) where χ Σ P C Σ , φ P C A , and ψ is the solution corresponding to initial value χ Σ as in Theo-rem 1.12. These maps extend uniquely to unitary maps U A Σ : H Σ Ñ H A , U Σ A : H A Ñ H Σ and U A Σ Σ : H Σ Ñ H Σ . Here we differ from the notation used in Theorem 2.23 in [3] where U A Σ Σ was denoted by F A Σ Σ . Furthermore, it will be useful to express the orthogonal projector P ´ Σ in an momentumintegral representation over the mass shell M “ t p P R | p µ p µ “ m u “ M ` Y M ´ , M ˘ “ t p P M | ˘ p ą u ; (22)cf. Lemma 2.1 and the definition of F M Σ in [3]. We endow M with the orientation thatmakes the projection M Ñ R , p p , p q ÞÑ p positively oriented. One finds that i p p d p q “ m p dp dp dp “ m p d p on p T p M q . (23) General Notation.
Positive constants and remainder terms are denoted by C , C , C , . . . and r , r , r , . . . , respectively. They keep their meaning throughout the whole article. Anyfixed quantity a constant depends on (except numerical constants like electron mass m andcharge e ) is displayed at least once when the constant is introduced. Furthermore, we classifythe behavior of functions using the following variant of the Landau symbol notation. Definition 1.14.
For lists of variables x, y, z we use the notation f p x, y, z q “ O y p g p x qq , for all p x, y, z q P domain (24) to mean the following: There exists a constant C p y q depending only on the parameters y , butneither on x nor on z , such that | f p x, y, z q| ď C p y q| g p x q| , for all p x, y, z q P domain , (25) where |¨| stands for the appropriate norm applicable to f . Note that the notation (24) doesnot mean that f p x, y, z q “ f p x, y q , i.e., that the value of f is independent of z . Rather, itjust means that the bound is uniform in z . Proofs
The key idea in the proofs of our main results Theorem 1.5, 1.6, and 1.7 is to guess a simpleenough operator P A Σ : H Σ ý so that U A ΣΣ in P ´ Σ in U A Σ in Σ ´ P A Σ P I p H Σ q . (26)It turns out that all claims about the properties of the polarization classes C Σ p A q above canthen be inferred from the properties of P A Σ . This is due to the fact that (26) is compatiblewith the Hilbert-Schmidt operator freedom encoded in the « equivalence relation that wasused to define the polarization classes C Σ p A q ; see Definition 1.3.The intuition behind our guess of P A Σ comes from gauge transforms. Imagine the specialsituation in which an external potential A could be gauged to zero, i.e., A “ B Ω for a givenscalar field Ω. In this case e ´ i Ω P ´ Σ e i Ω is a good candidate for P A Σ . Now in the case of generalexternal potentials A that cannot be attained by a gauge transformation of the zero potential,the idea is to implement different gauge transforms locally near to each space-time point. Forexample, if p ´ p y ´ x q denotes the informal integral kernel of the operator P ´ Σ , one could try todefine P A Σ as the operator corresponding to the informal kernel p A p x, y q “ e ´ iλ A p x,y q p ´ p y ´ x q for the choice λ A p x q “ p A p x q ` A p y qq µ p y ´ x q µ . Due to this choice, the action of λ A p x, y q can be interpreted as a local gauge transform of p ´ p y ´ x q from the zero potential to thepotential A µ p x q at space-time point x . It turns out that these local gauge transforms giverise to an operator P A Σ that fulfills (26). Section Overview
In Section 2.1 we define the operators P ´ Σ and P A Σ and state their mainproperties. Assuming these properties we prove our main results in Section 2.2. The proofsof those employed properties are delivered afterwards in Sections 2.3 and 2.4. P ´ Σ and P A Σ As described in the previous section, the central objects of our study are the operators P ´ Σ and operators which are derived from them like the discussed P A Σ . Lemma 2.1 describes theintegral representation of the orthogonal projector P ´ Σ . For this we introduce the notation r p w q : “ a ´ w µ w µ for w P domain p r q : “ t w P C | ´ w µ w µ P C z R ´ u . (27)The square root is interpreted as its principal value ? r e iϕ “ re iϕ for r ą ´ π ă ϕ ă π .We note that for a Cauchy surface Σ fulfilling (15) and 0 ‰ z “ y ´ x with x, y P Σ one has a ´ V max2 | z | ď r p z q ď | z | ď | z | ď a ` V max2 | z | . (28)To deal with the singularity of the informal integral kernel p ´ p y ´ x q of the projection operator P ´ Σ at the diagonal x “ y , we use a regularization shifting the argument y ´ x a little indirection of the imaginary past. Lemma 2.1.
For φ, ψ P C Σ and any past-directed time-like vector u P Past one has @ φ, P ´ Σ ψ D “ lim ǫ Ó ż x P Σ φ p x q i γ p d x q ż y P Σ p ´ p y ´ x ` iǫu q i γ p d y q ψ p y q , (29)10 here p ´ : R ` i Past Ñ C x , p ´ p w q “ p π q m ż M ´ { p ` m m e ipw i p p d p q “ ´ i {B ` m m D p w q , (30) D : R ` i Past Ñ C , D p w q “ p π q m ż M ´ e ipw i p p d p q “ ´ m π K p mr p w qq mr p w q , (31) K : R ` ` i R Ñ C , K p ξ q “ ξ ż e ´ ξs ? s ´ ds. (32) K is the modified Bessel function of second kind of order one. The functions D and p ´ haveanalytic continuations defined on domain p r q . The corresponding continuations are denotedby the same symbols. The proof is given in Section 2.3. It is based on the momentum integral representationgiven in Theorem 2.15 in [3]. In the following we define several candidates for P A Σ fulfillingthe key property (26) as discussed in the beginning of Section 2. We will denote theseoperators by P λ Σ : H Σ ý where the superscript λ denotes an element out of the followingclass of “local” gauge functions: Definition 2.2.
For A P C c p R , R q let G p A q denote the set of all functions λ : R ˆ R Ñ R with the following properties:(i) λ P C p R ˆ R , R q .(ii) There is a compact set K Ă R such that supp λ Ď K ˆ R Y R ˆ K .(iii) λ vanishes on the diagonal, i.e., λ p x, x q “ for x P R .(iv) On the diagonal the first derivatives fulfill B x λ p x, y q “ ´B y λ p x, y q “ A p x q for x “ y P R . (33)Given a “local” gauge transform λ P G p A q we define the corresponding operator P λ Σ usingthe heuristic idea behind P A Σ we discussed in the beginning of Section 2. Lemma 2.3.
Given A P C c p R , R q and λ P G p A q there is a unique bounded operator P λ Σ : H Σ ý with matrix elements @ φ, P λ Σ ψ D “ lim ǫ Ó A φ, P λ,ǫu Σ ψ E with (34) A φ, P λ,ǫu Σ ψ E : “ ż x P Σ φ p x q i γ p d x q ż y P Σ e ´ iλ p x,y q p ´ p y ´ x ` iǫu q i γ p d y q ψ p y q . (35) for any given φ, ψ P C Σ and any past-directed time-like vector u P Past . In particular, thelimit in (34) does not depend on the choice of u P Past . For ∆ P λ Σ : “ P λ Σ ´ P ´ Σ , ψ P H Σ , andalmost all x P Σ it holds ` ∆ P λ Σ ψ ˘ p x q “ ż y P Σ p e ´ iλ p x,y q ´ q p ´ p y ´ x q i γ p d y q ψ p y q , (36) and furthermore: i) The operator norm of P λ Σ is bounded by a constant C p V max , λ q ; cf. (15) ;(ii) ∆ P λ Σ is a compact operator;(iii) | ∆ P λ Σ | is a Hilbert-Schmidt operator.(iv) If λ p x, y q “ ´ λ p y, x q for all x, y P Σ , then P λ Σ is self-adjoint. This lemma is proven in Section 2.3. Two important examples of elements in G p A q are: • The choice λ p x, y q “ Ω p x q ´ Ω p y q for Ω P C c p R , R q fulfills λ P G pB Ω q . Such a λ delivers a good candidate for the operator P A Σ fulfilling (26) if the external field A canbe attained from the zero field via a gauge transform A “ ÞÑ A “ B Ω. We observefor any path C y,x from y to xλ p x, y q “ ż C y,x A µ p u q du µ “ p A µ p x q ` A µ p y qqp x µ ´ y µ q ` O A p| x ´ y | q . (37) • For an arbitrary vector potential A P C c p R , R q also λ A p x, y q : “ p A µ p x q ` A µ p y qqp x µ ´ y µ q (38)fulfills λ A P G p A q . This choice is motivated by the special case (37). It will be partic-ularly convenient for our work. Note that it has the symmetry λ A p x, y q “ ´ λ A p y, x q ;cf. part (iv) in Lemma 2.3. In particular, the operator P A Σ from the discussion will begiven by P A Σ : “ P λ A Σ . (39)We shall show that for λ P G p A q the operators P λ Σ obey the key property (26). Our firstresult about P λ Σ for a λ P G p A q is that, up to a Hilbert-Schmidt operator, it depends onlyon the restriction of the 1-form A to the tangent bundle T Σ of the Cauchy surface Σ.
Theorem 2.4.
Given A, r A P C c p R , R q and λ P G p A q , r λ P G p r A q , the following is true: P λ Σ ´ P r λ Σ P I p H Σ q ô A | T Σ “ r A | T Σ . (40)This theorem is also proven in Section 2.3. From our next result we can infer that theoperators P λ Σ obey the key property (26). Theorem 2.5.
Given A P C c p R , R q , λ P G p A q , and two Cauchy surfaces Σ , Σ , one has U A Σ P λ Σ U Σ A ´ U A Σ P λ Σ U Σ A P I p H A q , (41) where U A Σ and U Σ A are the Dirac evolution operators given in Definition 1.13. p Σ t q t P T of Cauchy surfaces en-coded by Σ , see (17), such that Σ “ Σ t and Σ “ Σ t . In addition we need the followinghelper object s A Σ defined in Definition 2.6 below as well as the following notation. Given anelectromagnetic potential A P C c p R , R q and a Cauchy surface Σ with future-directed unitnormal vector field n , we define the electromagnetic field tensor F µν “ B µ A ν ´ B ν A µ and E µ : “ F µν n ν (42)referred to as the “electric field” with respect to the local Cauchy surface Σ. In the specialcase n “ e “ p , , , q , this encodes just the electric part of the electromagnetic fieldtensor.Recall from the paragraph preceding Definition 1.10 that we extended the unit normalfield n on the Cauchy surface to a smooth unit normal field n : R ˆ T Ñ R and velocityfield v : R ˆ T Ñ R , which induces the “electric field” E to be defined on R ˆ T as well. Inparticular, after this extension, the partial derivative B E µ p x, t q{B t “ F µν p x q B n tν p x q{B t thenmakes sense. Definition 2.6.
Recall the definitions of r p w q and D p w q given in (27) and (31) , respectively.For ǫ ą , u P Past , and x, y P R , we define the integral kernel s A,ǫu Σ p x, y q : “ m { n p x q { E p x q r p w q {B D p w q , where w “ y ´ x ` iǫu. (43) Furthermore, for x ´ y being space-like (in particular x ‰ y ), we also define the integralkernel s A Σ p x, y q “ s A, p x, y q : “ lim ǫ Ó s A,ǫu Σ p x, y q “ m { n p x q { E p x q r p y ´ x q {B D p y ´ x q . (44)We remark that restricted to x and y within a single Cauchy surface Σ, the value of thekernel s A,ǫu Σ p x, y q depends only on Σ through its normal field n : Σ Ñ R . In this case thedefinition makes sense without specifying neither the velocity field v nor the extension of n and v to R ˆ T . In particular, s A,ǫu Σ p x, y q depends only on the Cauchy surface Σ but noton the choice of a family p Σ t q t P T . This stands in contrast to the derivative B s A,ǫu Σ t {B t , whichmakes sense everywhere only given a family p Σ t q t P T and the extended version of n .Exploiting the properties of D p w q given in Lemma 2.1 and in Corollary A.1 in the ap-pendix we shall find: Lemma 2.7.
Let u P Past .(i) The integral kernels s A,ǫu Σ , ǫ ě , give rise to Hilbert-Schmidt operators S A,ǫu Σ : H Σ ý , S A,ǫu Σ ψ p x q : “ ż Σ s A,ǫu Σ p x, y q i γ p d y q ψ p y q for almost all x P Σ , (45) S A Σ : “ S A, , with the property that } S A Σ ´ S A,ǫu Σ } I p H Σ q ǫ Ó ÝÑ . ii) Similarly, for t P T , the integral kernels B s A,ǫu Σ t {B t , ǫ ě , give rise to Hilbert-Schmidtoperators S A,ǫu Σ t : H Σ ý , S A,ǫu Σ t ψ p x q : “ ż Σ t B s A,ǫu Σ t B t p x, y q i γ p d y q ψ p y q for almost all x P Σ t , (46) S A Σ t : “ S A, t , with the property that sup t P T } S A Σ t } I p H Σ t q ă 8 and } S A Σ t ´ S A,ǫu Σ t } I p H Σ t q ǫ Ó ÝÑ for all t . With this ingredient our infinitesimal version of Theorem 2.5 can be formulated as follows;for technical convenience, we phrase it only for the special choice λ A P G p A q defined in (38). Theorem 2.8.
Given A P C c p R , R q , any smooth family of Cauchy surfaces Σ , cf. (17) ,and t , t P T , and one has U A Σ t ´ P A Σ t ` S A Σ t ¯ U Σ t A ´ U A Σ t ´ P A Σ t ` S A Σ t ¯ U Σ t A “ ż t t U A Σ t R p t q U Σ t A dt (47) for a family of Hilbert-Schmidt operators R p t q , t P T , with sup t P T } R p t q} I p H Σ t q ă 8 . Theintegral in (47) is understood in the weak sense. Note that for the choice λ P G p A q , Σ t “ Σ, Σ t “ Σ in one has P λ Σ in “ P ´ Σ in , and therestriction of (41) to Cauchy surface Σ yields property U A ΣΣ in P ´ Σ in U A Σ in Σ ´ P λ Σ P I p H Σ q , i.e.,the key property (26). The proof of Theorem 2.8 given in Section 2.4 is the heart of thiswork. In this section, we prove the main results under the assumption that the claims in Section 2.1are true. The proofs of these assumed claims are then provided in Sections 2.3-2.4. Theconnection of how to infer the properties of C Σ p A q from the properties of the operators P λ Σ is given by the following lemma. Lemma 2.9.
Let A P C c p R , R q , Σ be a Cauchy surface, and λ P G p A q . Then for everypolarization V in H Σ , we have V P C Σ p A q ô P V Σ ´ P λ Σ P I p H Σ q . (48) Proof.
By Definition 1.3, V P C Σ p A q is equivalent to P V Σ ´ U A ΣΣ in P ´ Σ in U A Σ in Σ P I p H Σ q . (49)On the other hand, Theorem 2.5 implies P λ Σ ´ U A ΣΣ in P ´ Σ U A Σ in Σ P I p H Σ q . (50)Thus, statement (49) is equivalent to P V Σ ´ P λ Σ P I p H Σ q .14 roof of Theorem 1.5. C Σ p A q “ C Σ p r A q holds true if and only if there are V P C Σ p A q and W P C Σ p r A q such that P V Σ ´ P W Σ P I p H Σ q . (51)Let λ P G p A q and r λ P G p r A q . In view of Lemma 2.9, statement (51) is equivalent to P λ Σ ´ P r λ Σ P I p H Σ q . Due to Theorem 2.4 the latter is equivalent to A | T Σ “ r A | T Σ , which proves theclaim. Proof of Thorem 1.6.
Claim (i): Is is sufficient to prove that there exist V P C Σ p A q and W P C ΛΣ p Λ A p Λ ´ ¨qq such that L p S, Λ q P V Σ p L p S, Λ q q ´ ´ P W ΛΣ P I p H ΛΣ q . We remark that for the linear form A , Λ A stands for the linear form with coordinates Λ µν A ν , while for a vector x ,the term Λ x stands for the vector with coordinates Λ µν x ν . We take λ P G p A q , e.g., λ “ λ A from (38). Thanks to Lemma 2.9, for all V P C Σ p A q we have P V Σ ´ P λ Σ P I p H Σ q . First,let us discuss how such a P λ Σ behaves under the Lorentz transforms L p S, Λ q . For ǫ ą u P Past, the integral kernel p λ,ǫu Σ p x, y q “ e ´ iλ p x,y q p ´ p y ´ x ` iǫu q of P λ,ǫu Σ , cf. (35), transformsas follows: The integral kernel of L p S, Λ q Σ P λ,ǫu Σ p L p S, Λ q Σ q ´ is given by Sp λ,ǫu Σ p Λ ´ x, Λ ´ y q S ˚ “ e ´ iλ p Λ ´ x, Λ ´ y q S p ´ p Λ ´ p y ´ x q ` iǫu qq S ˚ “ e ´ iλ p Λ ´ x, Λ ´ y q p ´ p y ´ x ` iǫ Λ u q “ p λ,ǫ Λ u ΛΣ p x, y q , (52)where λ p x, y q “ λ p Λ ´ x, Λ ´ y q . We claim λ P G p Λ A p Λ ´ ¨qq . Indeed, λ clearly fulfills condi-tions (i)-(iii) of the Definition 2.2 of G p Λ A p Λ ´ ¨qq . It also fulfills condition (iv) since BB x µ λ p x, y q ˇˇ y “ x “ BB x µ λ p Λ ´ x, Λ ´ y q ˇˇ y “ x “ p Λ ´ q νµ BB z ν λ p z, Λ ´ y q ˇˇ z “ Λ ´ x,y “ x “ Λ µν A ν p Λ ´ x q (53)and similarly B yµ λ p x, y q ˇˇ x “ y “ ´ Λ µν A ν p Λ ´ x q , where we have used p Λ ´ q νµ “ Λ µν . Thisshows L p S, Λ q Σ P λ,ǫu Σ p L p S, Λ q Σ q ´ “ P λ,ǫ Λ u Σ , which implies L p S, Λ q Σ P λ Σ p L p S, Λ q Σ q ´ “ P λ Σ in the limit as ǫ Ó
0; recall from Lemma 2.3 that the limit does not depend on the choice of u, Λ u P Past.Again by Lemma 2.9, there is a W P C ΛΣ p Λ A p Λ ´ ¨qq such that P W ΛΣ ´ P λ ΛΣ P I p H ΛΣ q . Weconclude L p S, Λ q Σ P V Σ ´ L p S, Λ q Σ ¯ ´ ´ P W ΛΣ “ L p S, Λ q Σ ` P V Σ ´ P λ Σ ˘ ´ L p S, Λ q Σ ¯ ´ ´ ´ P W ΛΣ ´ P λ ΛΣ ¯ P I p H ΛΣ q . (54)Claim (ii): The integral kernel of e ´ i Ω P λ,ǫu Σ e i Ω for λ P G p A q , ǫ ą u P Past equals e ´ i Ω p x q p λ,ǫu Σ p x, y q e i Ω p y q “ e ´ i Ω p x q e ´ iλ p x,y q p ´ p y ´ x ` iǫu q e i Ω p y q “ p λ,ǫu Σ p x, y q , (55)where λ p x, y q “ Ω p x q ` λ p x, y q ´ Ω p y q , which clearly fulfills λ P G p A ` B Ω q ; cf. Definition 2.2.Taking the limit as ǫ Ó
0, the claim follows from the same kind of reasoning as in part (i).Finally, one can also use the self-adjoint operator P A Σ from (39) to construct a unitaryoperator e Q A Σ : H Σ ý which adapts the standard polarization H ´ Σ to one corresponding to A | T Σ , more precisely, e Q A Σ H ´ Σ P C Σ p A q . It is defined as follows:15 efinition 2.10. We set Q A Σ : “ r P A Σ , P ´ Σ s “ P ` Σ p P A Σ ´ P ´ Σ q P ´ Σ ´ P ´ Σ p P A Σ ´ P ´ Σ q P ` Σ . (56) Proof of Theorem 1.7.
In this proof, we use a 2 ˆ H Σ ý . This matrix notation always refers to the splitting H Σ “ H ` Σ ‘ H ´ Σ . Inparticular, we set ˆ ∆ `` ∆ `´ ∆ ´` ∆ ´´ ˙ “ ∆ P λ A Σ “ P A Σ ´ P ´ Σ , (57)cf. (36) for λ “ λ A . Using this matrix notation, we write Q A Σ “ ˆ `´ ´ ∆ ´` ˙ . (58)In the following we use the notation X “ Y mod I p H Σ q to mean X ´ Y P I p H Σ q . By (iii)of Lemma 2.3 we know that p ∆ P λ A Σ q P I p H Σ q , and therefore p P A Σ q “ p P ´ Σ ` ∆ P λ A Σ q “ P A Σ ` ˆ ´ ∆ ``
00 ∆ ´´ ˙ mod I p H Σ q . (59)Furthermore, Lemma 2.9 implies for all V P C Σ p A q that the corresponding orthogonal pro-jector P V Σ fulfills P A Σ ´ P V Σ P I p H Σ q . However, this means also that p P A Σ q ´ P A Σ P I p H Σ q ,and therefore, ∆ `` , ∆ ´´ P I p H Σ q ; see (59). In conclusion, we obtain P A Σ “ P ´ Σ ` ∆ P λ A Σ “ ˆ `´ ∆ ´` id H ´ Σ ˙ mod I p H Σ q . (60)Since p ∆ P λ A Σ q P I p H Σ q we have ∆ ´` ∆ `´ , ∆ ´` ∆ `´ P I p H Σ q and hence p Q A Σ q P I p H Σ q ;cf. (58). Defining Π A Σ : “ e Q A Σ P ´ Σ e ´ Q A Σ , (61)we concludeΠ A Σ “ p id H Σ ` Q A Σ q P ´ Σ p id H Σ ´ Q A Σ q “ ˆ `´ ∆ ´` id H ´ Σ ˙ “ P A Σ “ P V Σ mod I p H Σ q . (62)Furthermore, we observe that e Q A Σ is unitary because Q A Σ is skew-adjoint, so that Π A Σ is anorthogonal projector. Summarizing, we have shown e Q A Σ H ´ Σ “ Π A Σ H Σ P C Σ p A q , which provesthe claim of Theorem 1.7.As an addendum we prove the refinement of Theorem 1.7 described in Remark 1.8. Forthis it is left to show that charge p U A ΣΣ in H ´ Σ in , Π A Σ H Σ q “
0. We choose a future orientedfoliation p Σ t q t P R of space-time such that Σ “ Σ in and Σ “ Σ. Recall the choice of Σ in described in (7). The operators Q A Σ t are compact because they are skew-adjoint and p Q A Σ t q P p H Σ t q . Hence, the operators e ´ Q A Σ t are compact perturbations of the identity operatorsid H Σ t . Translating this fact to an interaction picture, the operators Q t : “ U in Σ t e ´ Q A Σ t U t Σ in (63)are as well compact perturbations of the identity operator id H Σin . We define the evolutionoperators in the interaction picture U t : “ U in Σ t U A Σ t Σ in , (64)which are continuous in t P R w.r.t. the operator norm; this follows from Lemma 3.9 in [3].Moreover, using V « W ô P V Σ P W K Σ , P V K Σ P W Σ P I p H Σ q , the just proven Theorem 1.7 implies e Q A Σ H ´ Σ « U A ΣΣ in H ´ Σ in ñ P ˘ Σ e ´ Q A Σ U A ΣΣ in P ¯ Σ in P I p H Σ q (65) ñ U in Σ P ˘ Σ e ´ Q A Σ U A ΣΣ in P ¯ Σ in P I p H Σ q (66) ñ P ˘ Σ in Q t U t P ¯ Σ in “ P ˘ Σ U in Σ e ´ Q A Σ U A ΣΣ in P ¯ Σ in P I p H Σ in q . (67)Since Q t ´ id H Σin is compact, the operator P ˘ Σ in p Q t ´ id H Σin q U t P ¯ Σ in is compact as well. Takingthe difference with the compact operator in (67) yields that P ˘ Σ in U t P ¯ Σ in is compact so that ˆ P ` Σ in U t P ` Σ in P ´ Σ in U t P ´ Σ in ˙ “ U t ´ ˆ P ` Σ in U t P ´ Σ in P ´ Σ in U t P ` Σ in ˙ (68)deviates from the unitary operator U t by a compact perturbation, and hence, is a Fredholmoperator. This implies that P ´ Σ in U t P ´ Σ in ˇˇ H ´ Σin ý is a Fredholm operator. We note that theFredholm index of P ´ Σ in U t “ P ´ Σ in ˇˇ H ´ Σin ý “ id H ´ Σin equals zero. The map t ÞÑ P ´ Σ in U t P ´ Σ in iscontinuous in the operator norm which implies that the Fredholm index is constant, andhence,0 “ index P ´ Σ in U t “ ˇˇ H ´ Σin ý “ index P ´ Σ in U in Σ U A ΣΣ in ˇˇ H ´ Σin ý “ index P ´ Σ U A ΣΣ in ˇˇ H ´ Σin Ñ H ´ Σ “ index P ´ Σ ˇˇ U A ΣΣin H ´ Σin Ñ H ´ Σ “ index P ´ Σ e ´ Q A Σ ˇˇ U A ΣΣin H ´ Σin Ñ H ´ Σ “ index e Q A Σ P ´ Σ e ´ Q A Σ ˇˇ U A ΣΣin H ´ Σin Ñ Π A Σ H Σ “ charge p U A ΣΣ in H ´ Σ in , Π A Σ H Σ q , (69)where in the fifth equality we have used that e ´ Q A Σ is a compact perturbation of the identity.This concludes the proofs of the main results under the condition that the claims inSection 2.1 are true. The proofs of these claims will be provided in the next two sections. Proof of Lemma 2.1.
Given φ, ψ P C Σ , we set p φ “ F M Σ φ and p ψ “ F M Σ ψ where F M Σ is thegeneralized Fourier transform p F M Σ ψ qp p q “ { p ` m m p π q ´ { ż Σ e ipx i γ p d x q ψ p x q for ψ P C Σ , p P M , (70)17ntroduced in Theorem 2.15 of [3]. This theorem ensures that p φ p p q p ψ p p q i p p d p q is integrableon M ´ . Let u P Past. With justifications given below, we compute the following. @ φ, P ´ Σ ψ D “ lim ǫ Ó ż p P M ´ e ´ ǫpu p φ p p q p ψ p p q i p p d p q m (71) “ p π q m lim ǫ Ó ż p P M ´ e ´ ǫpu ż x P Σ φ p x q i γ p d x q e ´ ipx ˆ { p ` m m ˙ ż y P Σ e ipy i γ p d y q ψ p y q i p p d p q (72) “ p π q m lim ǫ Ó ż p P M ´ ż x P Σ φ p x q i γ p d x q { p ` m m ż y P Σ e ip p y ´ x ` iǫu q i γ p d y q ψ p y q i p p d p q (73) “ p π q m lim ǫ Ó ż x P Σ φ p x q i γ p d x q ż y P Σ ż p P M ´ { p ` m m e ip p y ´ x ` iǫu q i p p d p q i γ p d y q ψ p y q (74) “ lim ǫ Ó ż x P Σ φ p x q i γ p d x q ż y P Σ p ´ p y ´ x ` iǫu q i γ p d y q ψ p y q . (75)The interchange of the p -integral and the limit ǫ Ó p φ p p q p ψ p p q i p p d p q is integrable on M ´ and by | e ´ ǫpu | ď ǫ ą p P M ´ .In the step from (71) to (72) we have used (70) and that γ p γ µ q ˚ γ “ γ µ , from (72) to(73) that { p “ p and that p “ m for p P M ´ . In the step from (73) to (74) we haveused Fubini’s theorem to interchange the integrals. This is justified because φ and ψ arebounded and compactly supported, and because for any given ǫ ą | e ip p y ´ x ` iǫu q | “ e ´ ǫpu tends exponentially fast to 0 as | p | Ñ 8 , p P M ´ . This proves the claim (29).Now we prove the claimed properties of D and p ´ . For any w P R ` i Past, the modulus | e ipw | “ e ´ p Im w tends exponentially fast to 0 as | p | Ñ 8 , p P M ´ . Consequently, exchangingdifferentiation and integration in the following calculation is justified: p ´ p w q “ p π q m ż M ´ ´ i {B w ` m m e ipw i p p d p q“ p π q m ´ i {B w ` m m ż M ´ e ipw i p p d p q “ ´ i {B ` m m D p w q . (76)To show the second equality in (31), we proceed as follows: First, we show that w P R ` i Pastimplies ´ w µ w µ P C z R ´ “ domain p?¨q . We take w “ z ` iu with z P R and u P Past,and assume ´ w µ w µ P R . Then 0 “ Im p w µ w µ q “ z µ u µ , i.e., z is orthogonal to u in theMinkowski sense. Because u is time-like, we conclude that z is space-like or zero. Weobtain w µ w µ “ Re p w µ w µ q “ z µ z µ ´ u µ u µ ă
0, i.e., ´ w µ w µ P domain p?¨q . It follows that ?´ w µ w µ P R ` ` i R “ domain p K q . In particular, r D : R ` i Past Q w ÞÑ ´ m π K p m ?´ w µ w µ q m ?´ w µ w µ (77)18s a well-defined holomorphic function. Because | e ipw | decays fast as | p | Ñ 8 , p P M ´ ,uniformly for w in any compact subset of R ` i Past, D : R ` i Past Q w ÞÑ p π q m ż M ´ e ipw i p p d p q (78)is also a holomorphic function. We need to show D “ r D . By the identity theorem forholomorphic functions, it suffices to show that the restrictions of D and r D to i Past coincide.Given w “ iu P i Past, we choose a proper, orthochronous Lorentz transform Λ P SO Ò p , q Ď R ˆ that maps u to the negative time axis:Λ u “ ´ te “ p´ t, , , q with t “ a u µ u µ “ a ´ w µ w µ ą . (79)By Lorentz invariance of the volume-form i p p d p q on M ´ , we know ż M ´ e ipw i p p d p q “ ż M ´ e ip Λ w i p p d p q (80)and ?´ w µ w µ “ a ´p Λ w q µ p Λ w q µ . Summarizing, we have reduced the claim D “ r D to itsspecial case D p w q “ r D p w q for w “ ´ ite , t “ ?´ w µ w µ ą
0. This special case is proven asfollows. Using i p p d p q “ m p d p on p T p M q , (81)rotational symmetry, and the substitution s “ ? k ` m m , k “ m ? s ´ , m s ds “ k dk, (82)we obtain with the abbreviation E p p q “ a p ` m : ż M ´ e ipw i p p d p q “ ´ m ż R e ´ E p p q t d p E p p q“ ´ πm ż exp ´ ´ t ? k ` m ¯ k dk ? k ` m “ ´ πm ż e ´ mts ? s ´ ds “ ´ πm K p mt q mt , (83)using the definition of K in (32), and hence, the claim D p´ ite q “ r D p´ ite q .The representation (77) of D shows also that D can be analytically extended to allarguments w P C with ´ w µ w µ P domain p?¨q “ C z R ´ . The same holds true for p ´ “p m q ´ p´ i {B ` m q D . To sum up, p ´ has an analytic continuation p ´ : domain p r q Ñ C ˆ ,which also concludes the proof of Lemma 2.1.19 roof of Lemma 2.3. We remark that most of the arguments in this proof are valid withoutregularization, i.e., also in the case ǫ “
0. This is in contrast to Section 2.4 below, where theregularization with ǫ ą A P C c p R , R q , λ P G p A q , and Σ be a Cauchy surface. Before proving the claim(34)-(35) it will be convenient to introduce the operators ∆ P λ,ǫu Σ , ǫ ě
0, which shall act onany ψ P H Σ as ´ ∆ P λ,ǫu Σ ψ ¯ p x q “ ż y P Σ p e ´ iλ p x,y q ´ q p ´ p y ´ x ` iǫu q i γ p d y q ψ p y q , (84)where the fixed vector u P R is past-directed time-like. We remark that the special case ǫ “ P λ, “ ∆ P λ Σ ; cf. (36).We show now that ∆ P λ,ǫu Σ : H Σ ý is well-defined. Recall the parametrization π Σ p x q ofΣ as stated in (13) and the identity i γ p d x q “ Γ p x q d x on p T x Σ q given in (19). We use theabbreviation x “ π Σ p x q , y “ π Σ p y q in the following. Line (84) can be recast into ´ ∆ P λ,ǫu Σ ψ ¯ p x q “ ż R ∆ p λ,ǫu Σ p x , y q Γ p y q ψ p y q d y for (85)∆ p λ,ǫu Σ p x , y q : “ ` e ´ iλ p x,y q ´ ˘ p ´ p y ´ x ` iǫu q . (86)To show at the same time that the right-hand side of (85), i.e., (84), is well-defined for ψ P H Σ and almost every x P Σ, and that ∆ P λ,ǫ Σ ψ P H Σ , it suffices to prove that for every φ P H Σ , we have ż x P R ż y P R ˇˇˇ φ p x q Γ p x q ∆ p λ,ǫu Σ p x , y q Γ p y q ψ p y q ˇˇˇ d y ď C } φ }} ψ } (87)with some constant C p u, V max q . We collect the necessary ingredients: • As λ is smooth and vanishes on the diagonal, there is a positive constant C p λ q suchthat | e ´ iλ p x,y q ´ | ď C | x ´ y |r K p x q _ K p y qs for x, y P R . (88)Note that this bound holds globally, not only locally close to the diagonal, because e ´ iλ ´ K ˆ R Y R ˆ K for some compact set K . • The bounds (28) from the appendix, cf. (15), show that for all x, y P Σ and p z , z q “ z “ y ´ x we find | z | ď | z | ď a ` V max2 | z | . • Formula (238) in Corollary A.1 of the Appendix ensures for all ǫ ě z “ p z , z q such that z “ y ´ x for x, y P Σ and z ‰ } p ´ p z ` iǫu q} ď O u,V max ˆ e ´ C D | z | | z | ˙ . (89)20hanks to these ingredients we find the estimate } ∆ p λ,ǫu Σ p x , y q} ď C e ´ C D | y ´ x | | y ´ x | r K p x q _ K p y qs (90)for all x, y P Σ such that y ´ x ‰ ǫ ě C p u, V max , λ q . Conse-quently, using the bound for Γ from (20), we have the dominating functionsup ǫ ě ˇˇˇ φ p x q Γ p x q ∆ p λ,ǫu Σ p x , y q Γ p y q ψ p y q ˇˇˇ ď C Γ max2 | φ p x q| e ´ C D | y ´ x | | y ´ x | | ψ p y q| , (91)which is integrable, as the following calculation shows: C Γ max2 ż x P R ż y P R | φ p x q| e ´ C D | y ´ x | | y ´ x | | ψ p y q| d y d x (92) “ C Γ max2 ż z P R e ´ C D | z | | z | ż x P R | φ p π Σ p x qq|| ψ p π Σ p x ` z qq| d x d z (93) ď πC Γ max2 ż e ´ C D s ds } φ ˝ π Σ } } ψ ˝ π Σ } (94) ď C } φ } } ψ } , (95)for a constant C p u, V max , λ q . In the step from (93) to (94) we use the Cauchy-Schwarzinequality, and in the step from (94) to (95), we use that the norms }¨ ˝ π Σ } and }¨} areequivalent. On the one hand, this proves claim (87), which implies that the operators∆ P λ,ǫu Σ : H Σ ý described in (85) and (86) are well-defined for all ǫ ě ǫ ě } ∆ P λ,ǫu Σ } H Σ ý ď C . (96)On the other hand, we use again the integrable domination from (91) together with thepoint-wise convergence lim ǫ Ó p ´ p y ´ x ` iǫu q “ p ´ p y ´ x q (97)for x, y P Σ with x ‰ y ; cf. the analytic continuation of p ´ described in Lemma 2.1. Usingthese ingredients, the dominated convergence theorem yields the following convergence inthe weak operator topology: A φ, ∆ P λ,ǫu Σ ψ E ǫ Ó ÝÑ @ φ, ∆ P λ Σ ψ D for φ, ψ P H Σ . (98)The next argument needs this fact only restricted to φ, ψ P C Σ . Using the notation (35) andLemma 2.1, we get for φ, ψ P C Σ A φ, P λ,ǫu Σ ψ E “ @ φ, P ,ǫu Σ ψ D ` A φ, ∆ P λ,ǫu Σ ψ E ǫ Ó ÝÑ @ φ, P ´ Σ ψ D ` @ φ, ∆ P λ Σ ψ D . (99)Because P ´ Σ , ∆ P λ Σ : H Σ ý are bounded operators and C Σ is dense in H Σ , this implies that P λ Σ : “ P ´ Σ ` ∆ P λ Σ : H Σ ý (100)21s the unique bounded operator that satisfies (34), together with the bound } P λ Σ } H Σ ý ď } P ´ Σ } H Σ ý ` } ∆ P λ Σ } H Σ ý ď ` C p u, V max , λ q (101)coming from (96). Note that we may take any fixed u P Past, e.g., u “ p´ , , , q , in thisbound and in the bounds below.Next, we show that K λ : “ | ∆ P λ Σ | is a Hilbert-Schmidt operator. It is the integraloperator (here written in 3-vector notation) K λ ψ p x q “ ż R k λ p x , y q Γ p y q ψ p y q d y (102)for ψ P H Σ and almost all x P Σ with the integral kernel k λ p x , y q “ ż R γ ∆ p λ, p x , z q ˚ γ Γ p z q ∆ p λ, p z , y q d z . (103)We remark that under the symmetry assumption λ p x, y q “ ´ λ p y, x q , we have γ ∆ p λ, p x , z q ˚ γ “ ∆ p λ, p z , x q ; (104)cf. formula (110) below. Thanks to the estimate (90) we find ›› k λ p x , y q ›› ď Γ max C ż R e ´ C D | x ´ z | | x ´ z | e ´ C D | z ´ y | | z ´ y | p K p x q _ K p z qqp K p z q _ K p y qq d z . (105)Next, we use the bound e ´ C D | x ´ z | e ´ C D | z ´ y | p K p x q _ K p z qqp K p z q _ K p y qq ď C e ´ C D p| y ´ x |`| x |q{ (106)with the constant C p λ, V max q “ sup z P K e C D | z |{ . Substituting this bound in (105) and carry-ing out the integration yields } k λ p x , y q} ď Γ max C C e ´ C D p| y ´ x |`| x |q{ ż R d z | x ´ z | | z ´ y | “ C e ´ C D p| y ´ x |`| x |q{ | y ´ x | (107)for a finite constant C p λ, V max q . We can therefore bound the Hilbert-Schmidt norm of K λ as follows: } K λ } I p H Σ q “ ż R ż R trace r γ k λ p x , y q ˚ γ Γ p x q k λ p x , y q Γ p y qs d x d y ď max2 ż R ż R } k λ p x , y q} d x d y ď max2 C ż R ż R e ´ C D p| y ´ x |`| x |q | y ´ x | d x d y ă 8 . (108)This proves that K λ “ | ∆ P λ Σ | is a Hilbert-Schmidt operator, and therefore, ∆ P λ Σ is compact.22o prove part (iv) of Lemma 2.3, we assume λ p x, y q “ ´ λ p y, x q for all x, y P Σ. From thesymmetries D p w ˚ q “ D p w q ˚ and D p´ w q “ D p w q for all w P domain p r q and p γ µ q ˚ “ γ γ µ γ ,we conclude p ´ p´ w ˚ q “ γ p ´ p w q γ , (109)and hence, using the assumed symmetry of λ , γ ` e ´ iλ p y,x q p ´ p y ´ x ` iǫu q ˘ ˚ γ “ e ´ iλ p x,y q p ´ p x ´ y ` iǫu q (110)for x, y P Σ, ǫ ą u P Past. Substituting this in the specification (34)-(35) of P λ Σ , itfollows that P λ Σ is self-adjoint and concludes the proof. Proof of Theorem 2.4.
To show the equivalence we need to control of the kernel of P λ Σ ´ P r λ Σ from above and from below. Let ∆ A : R Ñ R be the vector field on R with∆ A p x q ¨ z “ p A µ p x q ´ r A µ p x qq z µ (111)for any x “ p x , x q P Σ and z “ p z , z q P T x Σ. Then for any x “ p x , x q P Σ, A p x q| T x Σ “ r A p x q| T x Σ holds if and only if ∆ A p x q “
0. From λ P G p A q and λ P G p r A q , see Definition 2.2,we get the Taylor expansions e ´ iλ p x,y q “ ` iA µ p x qp y µ ´ x µ q ` O λ p| x ´ y | qp K p x q _ K p y qq , (112) e ´ i r λ p x,y q “ ` i r A µ p x qp y µ ´ x µ q ` O r λ p| x ´ y | qp K p x q _ K p y qq , (113) y ´ x “ ∇ t Σ p x q ¨ p y ´ x q ` O Σ p| x ´ y | q (114)for y, x P Σ from which we conclude e ´ iλ p x,y q ´ e ´ i r λ p x,y q “ i ∆ A p x q ¨ p y ´ x q ` r p x , y q (115)with an error term r that fulfills for any x, y P Σ | r p x , y q| ď O λ, r λ,V max p| x ´ y | q p K p x q _ K p y qq , (116)where we used | x ´ y | “ O V max p| x ´ y |q due to (15). Note that the bound (116) holds notonly locally near the diagonal but also globally for x, y P Σ because e ´ iλ ´ e ´ i r λ is boundedand λ and r λ vanish outside K ˆ R Y R ˆ K for some compact set K Ă R . For φ, ψ P H Σ formula (36) from Lemma 2.3 implies A φ, p P λ Σ ´ P r λ Σ q ψ E “ ż x P Σ φ p x q i γ p d x q ż y P Σ p e ´ iλ p x,y q ´ e ´ i r λ p x,y q q p ´ p y ´ x q i γ p d y q ψ p y q“ ż x P R ż y P R φ p x q ˚ γ Γ p x qr t p x, y q ` t p x, y qs γ Γ p y q ψ p y q d y d x (117)23ith t p x, y q “ i ∆ A p x q ¨ p y ´ x q p ´ p y ´ x q γ , (118) t p x, y q “ r p x , y q p ´ p y ´ x q γ , (119)where we use the abbreviations x “ π Σ p x q , y “ π Σ p y q again, and Γ is defined in (19). Wehave introduced two extra factors γ in (117) in order to have a positive-definite weight γ Γ.We claim that the kernel t p x, y q γ gives rise to a Hilbert-Schmidt-operator T . Indeed,using the bound (20) for Γ, the bound (238) from Corollary A.1 in the appendix for p ´ , andthe bound (116) for r , we have } T } I p H Σ q “ ż x P R ż y P R trace r t p x, y q ˚ γ Γ p x q t p x, y q γ Γ p y qs d y d x ď C ż x P R ż y P R ˇˇˇˇˇ e ´ C D | y ´ x | | y ´ x | ˇˇˇˇˇ p K p x q ` K p y qq d y d x ď C ă 8 (120)for some constants C and C that depend on Σ , λ, ˜ λ .If A | T Σ “ r A | T Σ then ∆ A “
0. This implies t “ P λ Σ ´ P r λ Σ “ T is aHilbert-Schmidt operator. This proves the “ ð ” part of the claim (40).Conversely, let us assume that A | T Σ “ r A | T Σ does not hold. Then we can take some x P R with ∆ A p x q ‰
0. By continuity of ∆ A , we have inf x P U | ∆ A p x q| ą U of x . Furthermore there is a constant C p V max q such that γ Γ p x q ´ C ispositive-semidefinite for all x “ p x , x q P Σ. Consequently, we get the following bound forall x P U and y P R :trace ” t p x , y q ˚ γ Γ p x q t p x , y q γ Γ p y q ı ě C trace ” t p x , y q ˚ t p x , y q ı ě C | ∆ A p x q ¨ p y ´ x q| } p ´ p y ´ x q} ě C | ∆ A p x q ¨ p y ´ x q| ˆ e ´ m | y ´ x | | y ´ x | ˙ . (121)with two positive constants C and C depending on V max . In the last step, we have usedthe lower bound (239) for } p ´ } from Corollary A.2 in the appendix. Because the lower boundgiven in (121) is not integrable over p x , y q P U ˆ R , we conclude that T is not a Hilbert-Schmidt operator. Because T is a Hilbert-Schmidt operator, this implies that P λ Σ ´ P r λ Σ cannot be a Hilbert-Schmidt operator. Thus, we have proven part “ ñ ” of the Theorem. This section contains the centerpiece of this work. The proof of Theorem 2.8 will be givenat the end of this section. To show that the claimed equality (47) holds, we analyze thedifference of matrix elements A φ, p P A Σ t ` S A Σ t q ψ E ´ A φ, p P A Σ t ` S A Σ t q ψ E (122)24or ψ, φ P C A . This is done in two steps. First, using Stokes’ theorem, we provide a formulafor the derivative w.r.t. the flow parameter of the family of Cauchy surfaces p Σ t q t P T inLemma 2.11 and Corollary 2.12. Second, we give the relevant estimates on this derivative inLemmas 2.13-2.15 which are summarized in Corollary 2.16, and conclude with the proof ofTheorem 2.8.For the first step, the following notations for the Dirac operators acting from the left andfrom the right, respectively, are convenient: D A ψ p x q “ D Ax ψ p x q : “ p i {B x ´ { A p x q ´ m q ψ p x q , (123) φ p y q ÐÝ D A “ φ p y q ÐÝ D Ay : “ φ p y qp´ i ÐÝ{B y ´ { A p y q ´ m q “ D Ay φ p y q , (124)where f p y qÐÝ{B y “ f p y qÐÝ{B : “ B µ f p y q γ µ . Lemma 2.11.
Let k : R ˆ R Ñ C ˆ be a smooth function. Let φ, ψ P C A . Then for any t P T we have ddt ż x P Σ t ż y P Σ t φ p x q i γ p d x q k p x, y q i γ p d y q ψ p y q“ ´ i ż x P Σ t ż y P Σ t φ p x q i γ p d x q D At k p x, y q i γ p d y q ψ p y q (125) with D At k p x, y q : “ v t p x q{ n t p x q D Ax k p x, y q ´ k p x, y qÐÝ D Ay v t p y q{ n t p y q . (126) Proof.
Assume that φ , ψ : R Ñ C are smooth functions with supp φ X supp ψ Ď K ` Causal for some compact set K Ă R .We set Σ t t : “ tp x, t q P Σ | t ď t ď t u (127)for any real numbers t ď t . By Stokes’ theorem, we have: ˜ż Σ t ´ ż Σ t ¸ φ p x q i γ p d x q ψ p x q “ ż Σ t t d r φ p x q i γ p d x q ψ p x qs . (128)We calculate: d r φ p x q i γ p d x q ψ p x qs “ B µ p φ p x q γ µ ψ p x qq d x “ pB µ φ p x qq γ µ ψ p x q d x ` φ p x q γ µ B µ ψ p x q d x “ {B φ p x q ψ p x q d x ` φ p x q {B ψ p x q d x “ iD A φ p x q ψ p x q d x ´ iφ p x q D A ψ p x q d x, (129)25ee also the calculation from (17) to (20) in [3]. Integration yields ˜ż Σ t ´ ż Σ t ¸ φ p x q i γ p d x q ψ p x q“ i ż Σ t t r D A φ p x q ψ p x q ´ φ p x q D A ψ p x qs d x “ i ż t t ż Σ t r D A φ p x q ψ p x q ´ φ p x q D A ψ p x qs i v t n t p d x q dt. (130)Differentiating this with respect to the upper boundary t , we conclude ddt ż Σ t φ p x q i γ p d x q ψ p x q“ i ż Σ t r D A φ p x q ψ p x q ´ φ p x q D A ψ p x qs i v t n t p d x q“ i ż Σ t r φ p x q ÐÝ D A v t p x q{ n t p x q i γ p d x q ψ p x q ´ φ p x q i γ p d x q v t p x q{ n t p x q D A ψ p x qs , (131)using (19). In the special case φ P C A this boils down to ddt ż Σ t φ p x q i γ p d x q ψ p x q “ ´ i ż Σ t φ p x q i γ p d x q v t p x q{ n t p x q D A ψ p x q , (132)while in the special case ψ P C A it boils down to ddt ż Σ t φ p x q i γ p d x q ψ p x q “ i ż Σ t φ p x q ÐÝ D A v t p x q{ n t p x q i γ p d x q ψ p x q . (133)We consider the function F : T ˆ T Ñ C , F p s, t q : “ ż x P Σ s ż y P Σ t φ p x q i γ p d x q k p x, y q i γ p d y q ψ p y q . (134)We apply (132) to φ “ φ and ψ p x q “ ş y P Σ t k p x, y q i γ p d y q ψ p y q to get BB s F p s, t q “ ´ i ż x P Σ s ż y P Σ t φ p x q i γ p d x q v s p x q{ n s p x q D Ax k p x, y q i γ p d y q ψ p y q . (135)Similarly, we apply (133) to φ p y q “ ş y P Σ t φ p x q i γ p d x q k p x, y q and ψ “ ψ to get BB t F p s, t q “ i ż x P Σ s ż y P Σ t φ p x q i γ p d x q k p x, y qÐÝ D Ay v t p y q{ n t p y q i γ p d y q ψ p y q . (136)From the chain rule, claim (125) follows: ddt F p t, t q“ ´ i ż x P Σ s ż y P Σ t φ p x q i γ p d x qr v t p x q{ n t p x q D Ax k p x, y q ´ k p x, y qÐÝ D Ay v t p y q{ n t p y qs i γ p d y q ψ p y q . (137)26rom formula (125) and the chain rule, we immediately get the following corollary. Corollary 2.12.
For any smooth function k : R ˆ R ˆ T Ñ C ˆ , p x, y, t q ÞÑ k t p x, y q , any φ, ψ P C A , and any t P T we have ddt ż x P Σ t ż y P Σ t φ p x q i γ p d x q k t p x, y q i γ p d y q ψ p y q“ ż x P Σ t ż y P Σ t φ p x q i γ p d x q „ ´ i D At k p x, y q ` B k t B t p x, y q i γ p d y q ψ p y q . (138)This completes step one, and next, we turn to the relevant estimates. In the followingcalculations for fixed t P T , we drop the index t in v “ v t and n “ n t . Also, the t –dependenceof the remainder terms r ... is suppressed in the notation below, as we have uniformity in t ofthe error bounds. Recall from equation (42) that E µ “ F µν n ν denotes the “electric field” ofthe electromagnetic field F µν “ B µ A ν ´ B ν A µ with respect to the local Cauchy surface Σ. Lemma 2.13.
For u P Past , ǫ ą , and x, y P R , let p A,ǫu p x, y q : “ e ´ iλ A p x,y q p ´ p y ´ x ` iǫu q (139) with λ A defined in (38) . Then for t P T , x, y P Σ t , z “ p z , z q “ y ´ x , and w “ z ` iǫu wehave D At p A,ǫu p x, y q“ v p x q{ n p x q γ ν F µν p x q z µ p A,ǫu p x, y q ` p A,ǫu p x, y q γ ν F µν p y q z µ v p y q{ n p y q ` r p x, y, ǫu q (140) “ ´ i m v p x q z µ E µ p x q {B D p w q ` r p x, y, ǫu q ` r p x, y, ǫu q (141) with error terms r “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ r K p x q _ K p y qs , (142) r “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ r K p x q _ K p y qs , (143) r “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ r K p x q _ K p y qs (144) for any compact set K containing the support of A . For any two different points x ‰ y in Σ t , the limit r p x, y, q : “ lim ǫ Ó r p x, y, ǫu q exists.Proof. We calculate for x, y P Σ t , u P Past, and ǫ ą D Ax r e ´ iλ A p x,y q p ´ p y ´ x ` iǫu qs“ r {B x λ A p x, y q ´ { A p x qs e ´ iλ A p x,y q p ´ p y ´ x ` iǫu q ` e ´ iλ A p x,y q p i {B x ´ m q p ´ p y ´ x ` iǫu q“ r {B x λ A p x, y q ´ { A p x qs p A,ǫu p x, y q , because p i {B x ´ m q p ´ p y ´ x ` iǫu q “ . (145)27sing the definition (38) of λ A , we get {B x λ A p x, y q ´ { A p x q “ γ ν r A ν p y q ´ A ν p x q ` p x µ ´ y µ qB xν A µ p x qs“ r γ ν F µν p x qp y µ ´ x µ q ` r p x, y qs (146)with the Taylor rest term r p x, y q “ γ ν r A ν p y q ´ A ν p x q ´ p y µ ´ x µ qB xµ A ν p x qs “ O A p| x ´ y | qr K p x q _ K p y qs“ O A p| z | qr K p x q _ K p y qs with z “ y ´ x ; (147)cf. formula (28) in the appendix, which compares | z | with | z | . Recall that K denotes acompact set containing the support of A . Similarly, we find r e ´ iλ A p x,y q p ´ p y ´ x ` iǫu qsÐÝ D Ay “ e ´ iλ A p x,y q p ´ p y ´ x ` iǫu qr´ {B y λ A p x, y q ´ { A p y qs ` p ´ p y ´ x ` iǫu qp´ i ÐÝ{B y ´ m q e ´ iλ A p x,y q “ p A,ǫu p x, y qr´ {B y λ A p x, y q ´ { A p y qs . (148)Using the symmetry λ A p x, y q “ ´ λ A p y, x q and interchanging x and y , equation (146) can berewritten in the form ´ {B y λ A p x, y q ´ { A p y q “ r´ γ ν F µν p y qp y µ ´ x µ q ` r p y, x qs . (149)Combining this with the definition (126) of D At , we find for x, y P Σ t , z “ y ´ x D At p A,ǫu p x, y q“ v p x q{ n p x qr γ ν F µν p x q z µ ` r p x, y qs p A,ǫu p x, y q` p A,ǫu p x, y qr γ ν F µν p y q z µ ´ r p y, x qs v p y q{ n p y q“ v p x q{ n p x q γ ν F µν p x q z µ p A,ǫu p x, y q ` p A,ǫu p x, y q γ ν F µν p y q z µ v p y q{ n p y q ` r p x, y, ǫu q (150)with the error term r p x, y, ǫu q “ v p x q{ n p x q r p x, y q p A,ǫu p x, y q ´ p A,ǫu p x, y q r p y, x q v p y q{ n p y q“ O A,u, Σ ˆ e ´ C D | z | | z | ˙ r K p x q _ K p y qs , (151)for t P T , x, y P Σ t , ǫ ą u P Past. Here we used the bound (238) in Lemma A.1 inthe appendix for p ´ , the quadratic bound (147) for r p x, y q , and the fact that | vn | , beingcontinuous, is bounded on compact sets. This proves the claim given in (140) with the errorbound (142). 28t remains to prove the claim given in (141) with the bounds (143) and (144). Recall thedefinitions of p A,ǫu and p ´ given in (139) and (30), respectively. We have p A,ǫu p x, y q “ ´ i m {B D p w q ` r p x, y, ǫu q (152)with the error term r p x, y, ǫu q “ e ´ iλ A p x,y q D p w q ` p e ´ iλ A p x,y q ´ q p ´ p z ` iǫu q “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ (153)using the bounds (232), (238) from the appendix and the Taylor bound | e ´ iλ A p x,y q ´ | “ O A p| z |q ď O A, Σ p| z |q , (154)which follows from λ A P G p A q , cf. Definition 2.2 and, once more, from the estimate (28) inthe appendix. Hence we get from (150) D At p A,ǫu p x, y q ´ r p x, y, ǫu q“ v p x q{ n p x q γ ν F µν p x q z µ p A,ǫu p x, y q ` p A,ǫu p x, y q γ ν F µν p y q z µ v p y q{ n p y q“ ´ i m v p x q{ n p x q γ ν F µν p x q z µ {B D p w q ´ i m {B D p w q γ ν F µν p y q z µ v p y q{ n p y q ` r p x, y, ǫu q (155)with the error term r “ v p x q{ n p x q γ ν F µν p x q z µ r ` r γ ν F µν p y q z µ v p y q{ n p y q “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ r K p x q _ K p y qs . (156)We employ estimate (236) for B D from the appendix and the fact supp F µν Ď K to find v p x q{ n p x q F µν p x q γ ν z µ {B D p w q “ v p x q{ n p x q F µν p x q γ ν w µ {B D p w q ` r p x, y, ǫu q (157) {B D p w q γ ν F µν p y q z µ v p y q{ n p y q “ {B D p w q γ ν F µν p y q w µ v p y q{ n p y q ` r p x, y, ǫu q (158)with the error terms r “ ´ v p x q{ n p x q F µν p x q γ ν iǫu µ {B D p w q “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ K p x q , (159) r “ ´ {B D p w q γ ν F µν p y q iǫu µ v p y q{ n p y q “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ K p y q . (160)Substituting this in (155), we conclude D At p A,ǫu p x, y q “ ´ i m v p x q{ n p x q γ ν F µν p x q w µ {B D p w q ´ i m {B D p w q γ ν F µν p y q w µ v p y q{ n p y q` p r ` r ` r qp x, y, ǫu q (161)29ith the additional error term r “ ´ i m p r ` r q “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ r K p x q _ K p y qs . (162)The following “Lorentz symmetry relation” will be used several times in the calculationsbelow. w ν B µ D p w q “ w µ B ν D p w q for w P domain p r q . (163)Equation (163) can be seen as follows. Using D “ f ˝ r with f p ξ q “ ´ m p π q ´ K p mξ q{p mξ q from (31) and B µ r p w q “ ´ w µ r p w q , we obtain w ν B µ D p w q “ ´ w ν w µ r p w q f p r p w qq “ w µ B ν D p w q .Using the anticommutator relation t γ µ , γ ν u “ g µν for the Dirac-matrices three timesand the Lorentz symmetry relation (163), we calculate v p x q{ n p x q F µν p x q γ ν w µ {B D p w q “ r { n p x q γ ν { w s v p x q F µν p x qB µ D p w q“r n ν p x q { w ´ γ ν n σ p x q w σ ` w ν { n p x q ´ { wγ ν { n p x qs v p x q F µν p x qB µ D p w q“ n ν p x q { wv p x q F µν p x qB µ D p w q (164) ´ γ ν n σ p x q w σ v p x q F µν p x qB µ D p w q (165) ` w ν { n p x q v p x q F µν p x qB µ D p w q (166) ´ { wγ ν { n p x q v p x q F µν p x qB µ D p w q . (167)For the first term (164), using the Lorentz symmetry (163) again, we get(164) “ n ν p x q { wv p x q F µν p x qB µ D p w q “ v p x q w µ E µ p x q {B D p w q“ v p x q z µ E µ p x q {B D p w q ` r p x, y, ǫu q (168)with the error term r “ v p x q iǫu µ E µ p x q {B D p w q “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ K p x q , (169)where in the last step we have used estimate (236) once more. For the second term (165),we use n σ p x q z σ “ O Σ p| z | q , which holds because of x, y P Σ t and n p x q K T x Σ t , to get(165) “ ´ γ ν n σ p x q w σ v p x q F µν p x qB µ D p w q “ r p x, y, ǫu q ` r p x, y, ǫu q (170)with the error terms r “ ´ γ ν n σ p x q z σ v p x q F µν p x qB µ D p w q “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ K p x q , (171) r “ ´ γ ν n σ p x q iǫu σ v p x q F µν p x qB µ D p w q “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ K p x q . (172)We have used the estimates (234) and, once more, (236). The contribution of the third term(166) is zero, i.e. (166) “ w ν { n p x q v p x q F µν p x qB µ D p w q “ , (173)30ecause of symmetry w ν B µ D p w q “ w µ B ν D p w q , cf. (163), and antisymmetry F µν “ ´ F νµ . Toexpress the fourth term (167), we use the Lorentz symmetry relation (163) again and replace x by y up to the following error term: r p x, y q “ F µν p x q v p x q{ n p x q ´ F µν p y q v p y q{ n p y q “ O A, Σ p| z |qr K p x q _ K p y qs . (174)We obtain for the fourth term (167):(167) “ ´ { w B µ D p w q γ ν { n p x q v p x q F µν p x q “ ´ w µ {B D p w q γ ν F µν p x q v p x q{ n p x q“ ´ {B D p w q γ ν F µν p y q w µ v p y q{ n p y q ` r p x, y, ǫu q (175)with the error term r “ w µ {B D p w q γ ν r “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ r K p x q _ K p y qs . (176)We have used estimate (235) from the appendix and the bound (174). The expressions (168),(170), (173) and (175) of the four terms (164)-(167) give v p x q{ n p x q F µν p x q γ ν w µ {B D p w q “ (164) ` (165) ` (166) ` (167) (177) “r v p x q z µ E µ p x q {B D p w q ` r s ` r r ` r s ` ` r´ {B D p w q γ ν F µν p y q w µ v p y q{ n p y q ` r s , which can be rewritten in the form v p x q{ n p x q F µν p x q γ ν w µ {B D p w q ` {B D p w q γ ν F µν p y q w µ v p y q{ n p y q“ v p x q z µ E µ p x q {B D p w q ` r p x, y, ǫu q ` r p x, y, ǫu q (178)with the error terms r “ r ` r “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ r K p x q _ K p y qs , (179) r “ r ` r “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ r K p x q _ K p y qs . (180)We have used the estimates (171) and (176) to bound r and the estimates (169) and (172)to bound r . Substituting this result in equation (161) together with the error bounds (151),(156) and (162), we infer D At p A,ǫu p x, y q“ ´ i m v p x q{ n p x q F µν p x q γ ν w µ {B D p w q ´ i m {B D p w q γ ν F µν p y q w µ v p y q{ n p y q ` r ` r ` r “ ´ i m v p x q z µ E µ p x q {B D p w q ` r ` r (181)with the error terms r p x, y, ǫu q “ r ` r ´ i m r “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ r K p x q _ K p y qs , (182) r p x, y, ǫu q “ r ´ i m r “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ r K p x q _ K p y qs . (183)31his proves the claim given in (141) with the bounds (143), (144). Recall that despitethe uniformity in ǫ of the bound given in (182), r “ r p x, y, ǫu q depends on ǫ . To ensureexistence of the limit lim ǫ Ó r p x, y, ǫu q for two different points x, y P Σ t from the explicit formof r , we observe that z “ y ´ x is space-like, and hence z P domain r . As a consequence,the functions D and B µ D are continuous at z , cf. Lemma 2.1, which implies the claim.In the following, we abbreviate B µ “ B{B w µ . Recall the notation r p w q “ ?´ w µ w µ from(27). Lemma 2.14.
For w P domain p r q and µ “ , , , , one has B µ r r p w q {B D p w qs “ w µ {B D p w q ´ γ µ w ν B ν D p w q ` { ww µ m D p w q . (184) Proof.
The function D fulfills the Klein-Gordon equation p l ` m q D p w q “ , w P domain p r q . (185)Indeed, for w P R ` i Past, this can be seen from the definition (31) of D as follows:Because of the fast convergence of e ipw to 0 as | p | Ñ 8 , p P M ´ , we can interchange theKlein-Gordon-operator with the integral in the following calculation: p l ` m q D p w q “ p π q ´ m ´ ż M ´ p l w ` m q e ipw i p p d p q“ p π q ´ m ´ ż M ´ p´ p ` m q e ipw i p p d p q “ . (186)By analytic continuation, the Klein-Gordon equation (185) follows for all w P domain p r q .Equation (184) is proven by the following calculation: B µ r r p w q {B D p w qs “ ´B µ r w ν w ν {B D p w qs (163) “ ´B µ r w ν { w B ν D p w qs“ ´ { w B µ D p w q ´ w ν γ µ B ν D p w q ´ w ν { w B µ B ν D p w q“ ´ { w B µ D p w q ´ w ν γ µ B ν D p w q ´ { w B ν p w ν B µ D p w qq ` { w pB ν w ν qB µ D p w q“ { w B µ D p w q ´ w ν γ µ B ν D p w q ´ { w B ν p w ν B µ D p w qq (163) “ { w B µ D p w q ´ w ν γ µ B ν D p w q ´ { w B ν p w µ B ν D p w qq“ { w B µ D p w q ´ w ν γ µ B ν D p w q ´ { ww µ l D p w q (163) , (185) “ w µ {B D p w q ´ γ µ w ν B ν D p w q ` { ww µ m D p w q . (187)Recall the definition of the helper object s A,ǫu Σ p x, y q “ rp{ n { E qp x qsrp r {B D qp w qs{p m q intro-duced in Definition 2.6. The properties of s A,ǫu Σ p x, y q claimed in Lemma 2.7 follow analo-gously to the arguments used in (92)–(95), i.e., from the bound (233) given in Corollary A.1in the appendix, the compact support of E , boundedness of Bp{ n t { E t q{B t , and the dominatedconvergence theorem. 32 emma 2.15. For t P R , x, y P Σ t , z “ y ´ x , u P Past , and ǫ ą we have D At s A,ǫu Σ p x, y q “ i m v t p x q z µ E µ p x q {B D p w q ` r p x, y, ǫu q ` r p x, y, ǫu q , (188) D At p p A,ǫu Σ ` s A,ǫu Σ qp x, y q “ r p x, y, ǫu q ` r p x, y, ǫu q (189) with error terms that fulfill the bounds r “ O A,u, Σ ˆ e ´ C | z | | z | ˙ K p x q , r “ O A,u, Σ ˆ ? ǫ e ´ C | z | | z | { ˙ K p x q , (190) r “ O A,u, Σ ˆ e ´ C | z | | z | ˙ r K p x q _ K p y qs , r “ O A,u, Σ ˆ ? ǫ e ´ C | z | | z | { ˙ r K p x q _ K p y qs (191) with some positive constant C p Σ q . Furthermore, for x ‰ y the following limit exists: r p x, y, q : “ lim ǫ Ó r p x, y, ǫu q (192) Proof.
In this proof, we abbreviate w “ y ´ x ` iǫu “ z ` iuǫ . Moreover, we suppress the w dependence of r p w q , D p w q , B w and again also the t -dependence of v , n , and of the remainderterms r ... in the notation. Using the definition of D At given in (126) of Lemma 2.11, we get8 m D At s A,ǫu Σ t p x, y q“ v p x q{ n p x q D Ax r { n p x q { E p x q r {B D s ´ r { n p x q { E p x q r {B D sÐÝ D Ay { n p y q v p y q“ v p x q{ n p x q i {B x r { n p x q { E p x q r {B D s ´ r { n p x q { E p x q r {B D sÐÝ{B y p´ i q{ n p y q v p y q ` r p x, y, ǫu q“ iv p x q{ n p x q γ µ { n p x q { E p x qB xµ r r {B D s ` i { n p x q { E p x qB yµ r r {B D s γ µ { n p y q v p y q ` r p x, y, ǫu q“ ´ iv p x q{ n p x q γ µ { n p x q { E p x qB µ r r {B D s ` i { n p x q { E p x qB µ r r {B D s γ µ { n p x q v p x q ` r p x, y, ǫu q , (193)where the remainder terms are defined and estimated as follows:(i) Recalling the definitions (123) and (124) of the Dirac operators D A and ÐÝ D A and thefact that A is compactly supported, the estimate (233) of Corollary A.1 in the appendixensures r “ v p x q{ n p x qp´ m ´ { A p x qqr { n p x q { E p x q r {B D s ´ r { n p x q { E p x q r {B D sp´ m ´ { A p y qq{ n p y q v p y q“ O A,u, Σ ˆ e ´ C D | z | | z | ˙ K p x q (194)for some compact set K containing the support of E .(ii) Using once more that E has compact support and using the bound (233) again we havethe analogous estimate r “ r ` iv p x q{ n p x q γ µ ` B xµ r { n p x q { E p x qs ˘ r {B D ` i ` B yµ r { n p x q { E p x qs ˘ r {B Dγ µ { n p y q v p y q“ O A,u, Σ ˆ e ´ C D | z | | z | ˙ K p x q (195)33iii) Using the signs coming from inner derivatives: ´B x D p w q “ B y D p w q “ B D p w q and theTaylor expansion { n p y q v p y q “ { n p x q v p x q ` r p x, y q with r “ O Σ p| z |q (196)for x, y P Σ t with x P K we find with the help of bound (237) in the appendix: r “ r ` i { n p x q { E p x qB µ r r {B D s γ µ r “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ K p x q . (197)In the following calculations, we drop the argument x ; thus, v , n , and E stand for v p x q , n p x q , and E p x q , respectively, but r “ r p w q and D “ D p w q . Using Lemma 2.14, we get ´ i ´ m D At s A,ǫu Σ p x, y q ´ r ¯ “ ´ v { nγ µ { n { E B µ r r {B D s ` { n { E B µ r r {B D s γ µ { nv “ ´ v { nγ µ { n { E r w µ {B D ´ γ µ w ν B ν D s ` v { n { E r w µ {B D ´ γ µ w ν B ν D s γ µ { n ` r p x, y, ǫu q“ T ` T ` T ` T ` r (198)with the four terms T “ ´ v { n { w { n { E {B D,T “ v { n { E {B D { w { n, T “ v { nγ µ { n { Eγ µ w ν B ν D,T “ ´ v { n { Eγ µ γ µ { nw ν B ν D, (199)and the remainder term r “ ´ v { nγ µ { n { E { ww µ m D ` v { n { E { ww µ m Dγ µ { n “ O A,u, Σ ` e ´ C D | z | ˘ K p x q , (200)where the bound comes from (231) of Corollary A.1 in the appendix and from supp E Ď K .We evaluate the four terms T j separately. Using the anticommutation rules t γ µ , γ ν u “ g µν for the Dirac matrices and { n “
1, we get T “ ´ v { n r w ν n ν ´ { n { w s { E {B D “ ´ v { nw ν n ν { E {B D ` v r w µ E µ ´ { E { w s {B D “ ´ v { nw ν n ν { E {B D ` vw µ E µ {B D ´ v { Ew µ B µ D, (201)where in the last step we used the Lorentz symmetry (163) to compute { w {B D “ γ µ γ ν w µ B ν D “ p γ µ γ ν w µ B ν D ` γ µ γ ν w ν B µ D q“ p γ µ γ ν ` γ ν γ µ q w µ B ν D “ w µ B µ D. (202)Using the anticommutation rules again, the fact γ µ γ µ “
4, the definition E µ “ F µν n ν givenin (42), and the antisymmetry F µν “ ´ F νµ , we get γ µ { n { Eγ µ “ p n µ ´ { nγ µ qp E µ ´ γ µ { E q “ n µ E µ ´ { n { E ` { nγ µ γ µ { E “ n µ E µ “ n µ F µν n ν “ T “
0. Using the same argument that was used to derive (202) we also find {B D p w q { w “ w µ B µ D , and hence, T “ v { n { Ew µ B µ D { n. (204)Finally, we have T “ ´ v { n { E { nw ν B ν D, (205)which yields T ` T “ ´ v { n { E { nw µ B µ D “ v { n { Ew µ B µ D “ v { Ew µ B µ D. (206)We have used that { n and { E anticommute because of n µ E µ “ n µ F µν n ν “
0. Together withthe expression (201) for T and T “
0, we conclude T ` T ` T ` T “ vw µ E µ {B D ` r p x, y, ǫu q . (207)with the error terms r “ ´ v { nw ν n ν { E {B D “ r p x, y, ǫu q ` r p x, y, ǫu q , (208)where using w “ z ` iǫur “ ´ v { niǫu ν n ν { E {B D, r “ ´ v { nz ν n ν { E {B D. (209)Inequality (236) from the appendix and the fact supp E Ď K provide the bound r “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ K p x q . (210)For the next estimate, we observe p ∇ t Σ t p x q ¨ z , z q P T x Σ t K n p x q ; recall the parametrization(13) of Σ t . We obtain the Taylor expansion z ν n ν “ n p x qr t Σ t p y q ´ t Σ t p x qs ´ n p x q ¨ z “ n p x q ∇ t Σ t p x q ¨ z ´ n p x q ¨ z ` O Σ p| z | q “ O Σ p| z | q (211)uniformly for x in the compact set K . Using (234) from the appendix and the supportproperty of E again, this implies r “ O A,u, Σ ˆ e ´ C D | z | | z | ˙ K p x q . (212)Finally, we have from equation (207) T ` T ` T ` T ´ r “ vw µ E µ {B D “ vz µ E µ {B D ` r p x, y, ǫu q (213)with the error term r “ viǫu µ E µ {B D “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ K p x q , (214)35here once again we have used the bound (236) from the appendix and the fact supp E Ď K .Let us summarize: We use the equations (198), (213), and (208) to get the claimed formula D At s A,ǫu Σ t p x, y q “ i m vz µ E µ {B D ` r ` r (188)with the remainder terms r : “ r m ` i m p r ` r q “ O A,u, Σ ˆ e ´ C D | z | | z | ` e ´ C D | z | ˙ K p x q “ O A,u, Σ ˆ e ´ C | z | | z | ˙ K p x q (215) r : “ i m p r ` r q “ O A,u, Σ ˆ ? ǫ e ´ C D | z | | z | { ˙ K p x q “ O A,u, Σ ˆ ? ǫ e ´ C | z | | z | { ˙ K p x q (216)with any positive constant C p Σ q ă C D p Σ q . We have applied the error bounds (197), (200),and (212) for the first remainder term r , and the bounds (210) and (214) for the secondremainder term r . Finally, we have weakened the bounds slightly to get a simpler notation.This shows the claimed error bounds in (190).Combining this with Lemma 2.13 and setting r “ r ` r , r “ r ` r , equation(189) together with the corresponding error bounds (191) are immediate consequences.To ensure existence of the limit of r p x, y, ǫu q as ǫ Ó x, y P Σ t with x ‰ y , weuse the existence of the limits lim ǫ Ó r p x, y, ǫt q and lim ǫ Ó r p x, y, ǫu q . The existence of theformer limit was proven in Lemma 2.13, and existence of the latter limit follows by the sameargument, i.e., from the fact that the functions D and B µ D are continuous at z , and that r is explicitly given in terms of D and its derivative. This yields the claim. Corollary 2.16.
The error terms r p¨ , ¨ , ǫu q and r p¨ , ¨ , ǫu q in (189) give rise to boundedlinear operators R ǫu p t q , R ǫu p t q : H Σ t ý with matrix elements x φ, R ǫu p t q ψ y “ ż x P Σ t ż y P Σ t φ p x q i γ p d x q r p x, y, ǫu q i γ p d y q ψ p y q , ψ, φ P H Σ t (217) and similarly for r p x, y, ǫu q , R ǫu p t q . They fulfill:(i) The operators R ǫu p t q , ǫ ě , are Hilbert-Schmidt operators. There is a constant C p A, u, Σ q such that sup t P T,ǫ ą } R ǫu p t q} I p H Σ t q ď C . Furthermore, lim ǫ Ó } R ǫu p t q ´ R p t q} I p H Σ t q “ . (218) (ii) sup t P T } R ǫu p t q} H Σ t ý ď O A,u, Σ p? ǫ q .Proof. (i) For ψ, φ P H Σ t , using the bound (191) for r , we find uniformly for ǫ ą t P T that } R ǫu p t q} I p H Σ t q “ ż x P R ż y P R trace “ γ r p x, y, ǫu q ˚ γ Γ p x q r p x, y, ǫu q Γ p y q ‰ d y d x (219) ď C ż x P R ż y P R „ e ´ C D | y ´ x | | y ´ x | p K p x q _ K p y qq d y d x ă 8 . (220)36or some constant C p A, u, Σ q . The limit R ǫu p t q ǫ Ó ÝÑ R p t q in the I p H Σ t q norm is im-plied by the point-wise convergence (192) stated in Lemma 2.15 and the point-wise bound(191), using dominated convergence.(ii) For ψ, φ P H Σ t , using the bound in (191) for r and the Cauchy-Schwarz inequality,we find analogously to the calculation (92)–(95): | x φ, R ǫu p t q ψ y | ď O A,u, Σ p? ǫ q ż x P R ż y P R | φ p x q| | Γ p x q| d x e ´ C D | y ´ x | | y ´ x | { | Γ p y q| d y | ψ p y q| (221) ď O A,u, Σ p? ǫ q ż z P R e ´ C D | z | | z | { d z } φ } } ψ } , (222)which is finite and uniform in t .The existence of the bounded linear operators R ǫu p t q , R ǫu p t q : H Σ t ý follows.Finally, we prove the Theorem 2.8 with the collected ingredients. Proof of Theorem 2.8.
With justifications given below, we find that for φ, ψ P C A A φ | Σ t , p P A Σ t ` S A Σ t q ψ | Σ t E ´ A φ | Σ t , p P A Σ t ` S A Σ t q ψ | Σ t E (223) “ lim ǫ Ó ˜ż x P Σ t ż y P Σ t ´ ż x P Σ t ż y P Σ t ¸ φ p x q i γ p d x q p p A,ǫu ` s A,ǫu Σ t qp x, y q i γ p d y q ψ p y q (224) “ lim ǫ Ó ż t t ż x P Σ t ż y P Σ t φ p x q i γ p d x q « ´ i D At p p A,ǫu ` s A,ǫu Σ t q ` B s A,ǫu Σ t B t ff p x, y q i γ p d y q ψ p y q dt (225) “ lim ǫ Ó ż t t ż x P Σ t ż y P Σ t φ p x q i γ p d x q « ´ ir p x, y, ǫu q ´ ir p x, y, ǫu q ` B s A,ǫu Σ t B t p x, y q ff ¨ i γ p d y q ψ p y q dt (226) “ lim ǫ Ó ż t t ” ´ i x φ | Σ t , R ǫu p t q ψ | Σ t y ´ i x φ | Σ t , R ǫu p t q ψ | Σ t y ` A φ | Σ t , S A,ǫu Σ t , ψ | Σ t Eı dt (227)In the first step from (223) to (224) we expressed the matrix elements of the operators P λ A Σ and S A Σ in terms of the respective integral kernels p ǫu,λ A and s ǫu,A Σ given in Lemma 2.3 and part(i) of Lemma 2.7. The step from (224) to (225) follows from Corollary 2.12. The step from(225) to (226) is a consequence of equation (189) in Lemma 2.15. Finally, in the step from(226) to (227) we have used that the integral kernels r p¨ , ¨ , ǫu q , r p¨ , ¨ , ǫu q , and B s A,ǫu Σ t {B t give rise to bounded operators R ǫu p t q , R ǫu p t q , and S A,ǫu Σ t as ensured by Corollary 2.16 andpart (ii) of Lemma 2.7.Claim (ii) of Corollary 2.16 implies that R ǫu p t q converges to zero in operator normas ǫ Ó
0, uniformly in t P T . Furthermore, claim (i) of Corollary 2.16 and part (ii) ofLemma 2.7 guarantee that ´ iR ǫu p t q ` S A,ǫu Σ t converges in the I p H Σ t q norm to a Hilbert-Schmidt operator R p t q : “ ´ iR p t q ` S A, t such that sup t P T } R p t q} I p H Σ t q ă 8 . Calculation37223)–(227) can now be rewritten in the form of claim (47): A φ | Σ t , p P A Σ t ` S A Σ t q ψ | Σ t E ´ A φ | Σ t , p P A Σ t ` S A Σ t q ψ | Σ t E “ ż t t x φ | Σ t , R p t q ψ | Σ t y dt (228)at first for φ, ψ P C A , but then extended by a density argument to φ, ψ P H A . Since theoperators U A Σ are unitary, we get the estimate ››› U A Σ t p P A Σ t ` S A Σ t q U Σ t A ´ U A Σ t p P A Σ t ` S A Σ t q U Σ t A ››› I p H A q (229) ď ż t t } R p t q} I p H Σ t q dt ă 8 . (230)This proves the claim. Proof of Theorem 2.5.
As a consequence of Theorem 2.8 and Lemma 2.7 claim (41) holdsfor the special case λ “ λ A . For general λ P G p A q , Theorem 2.4 implies P A Σ ´ P λ Σ P I p H Σ q which concludes the proof for the general case. A Appendix
In this appendix we provide auxiliary estimates for the covariant functions D , its derivatives,and p ´ needed in the proofs of the main results. Lemma A.1 (Upper bounds) . Let u be a time-like four-vector. For all space-like z P R with | z | ď V max | z | and ǫ ě with w “ z ` iǫu ‰ we have the following bounds with theconstant C D p V max q “ m a ´ V max2 , reading { as `8 : | w µ w ν D p w q| ď O u,V max ` e ´ C D | z | ˘ , (231) | D p w q| ď O V max ˆ e ´ C D | z | | z | ˙ , (232) ˇˇ r p w q B µ D p w q ˇˇ ď O u,V max ˆ e ´ C D | z | | z | ˙ , (233) |B µ D p w q| ď O u,V max ˆ e ´ C D | z | | z | _ ǫ ˙ , (234) | w ν B µ D p w q| ď O u,V max ˆ e ´ C D | z | | z | _ ǫ ˙ , (235) | ǫu µ B ν D p w q| ď O u,V max ˆ ? ǫe ´ C D | z | | z | { ˙ , (236) ˇˇ B ν “ r p w q B µ D p w q ‰ˇˇ ď O u,V max ˆ e ´ C D | z | | z | ˙ , (237) ›› p ´ p w q ›› ď O u,V max ˆ e ´ C D | z | | z | ˙ . (238)38 or ǫ “ one may take, e.g., u “ p´ , , , q . In this case the u -dependence of the constantsin (231) - (238) drops out. Lemma A.2 (Lower Bound) . For all space-like z P R zt u one has the lower bound ›› p ´ p z q ›› ě C e ´ m | z | | z | (239) with a positive numerical constant C . The proofs have been carried out in [4]. However, they can also be inferred from theasymptotic behavior of the modified Bessel function K and its derivative given in [1, Chap-ters 9.6 and 9.7]. Acknowledgment
This work was partially funded by the Elite Network of Bavaria throughthe Junior Research Group “Interaction between Light and Matter”.
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