Dynamical C*-algebras and kinetic perturbations
aa r X i v : . [ m a t h - ph ] A ug Dynamical C*-algebras and kinetic perturbations
Detlev Buchholz (1) and Klaus Fredenhagen (2) (1)
Mathematisches Institut, Universit¨at G¨ottingen,Bunsenstr. 3-5, 37073 G¨ottingen, Germany (2)
II. Institut f¨ur Theoretische Physik, Universit¨at HamburgLuruper Chaussee 149, 22761 Hamburg, Germany
Abstract
The framework of dynamical C*-algebras for scalar fields in Mink-owski space, based on local scattering operators, is extended totheories with locally perturbed kinetic terms. These terms en-code information about the underlying spacetime metric, so thecausality relations between the scattering operators have to be ad-justed accordingly. It is shown that the extended algebra describesscalar quantum fields, propagating in locally deformed Minkowskispaces. Concrete representations of the abstract scattering opera-tors, inducing this motion, are known to exist on Fock space. Theproof that these representers also satisfy the generalized causal-ity relations requires, however, novel arguments of a cohomolog-ical nature. They imply that Fock space representations of theextended dynamical C*-algebra exist, involving linear as well askinetic and pointlike quadratic perturbations of the field.
We continue here our construction of dynamical C*-algebras for scalarquantum fields in Minkowski space [5]. These algebras are generatedby unitary operators S ( F ), where F denotes some real functional actingon the underlying classical field. The classical field is described by real,smooth functions x φ ( x ) on d -dimensional Minkowski space M ≃ R d ,1nd the functionals considered in [5] were of the specific form F [ φ ] . = − k X j =0 (1 /j !) Z dx g j ( x ) φ ( x ) j . (1.1)Here g j ∈ D ( M ) are real test functions on M with compact supports.The term for j = 0 denotes the constant functional. These functionals areinterpreted as perturbations of the underlying Lagrangean by point likeself interactions of the field. Their support (in the sense of functionals)is defined as union of the supports of the underlying test functions g j for j >
0; the constant functional (corresponding to j = 0) has emptysupport and hence can be placed everywhere. The unitaries S ( F ) are thescattering operators corresponding to the perturbations F . As was shownin [5], they satisfy for a given Lagrangean a dynamical relation, based onthe Schwinger-Dyson equation, as well as the causal factorization rule S ( F + G ) S ( G ) − S ( G + H ) = S ( F + G + H ) . (1.2)This relation holds whenever the spacetime support of F succeeds thesupport of H with regard to the Minkowski metric. The support of thefunctional G , having the preceding special form, is completely arbitrary.In the present article we consider also localized perturbations of thekinetic part of the underlying Lagrangeans. This is of interest if onethinks of perturbations of the theory by gravitational forces. But italso provides a basis for the discussion of symmetry properties of thetheory, related to Noether’s theorem. The corresponding functionals arequadratic in the partial derivatives of the underlying field, P [ φ ] . = (1 / Z dx ∂ µ φ ( x ) p µν ( x ) ∂ ν φ ( x ) . (1.3)Here x p · · ( x ) are smooth functions with compact support, whichhave values in the space of real, symmetric d × d matrices. As we shallsee, these functions have to comply with further constraints in order toadmit a meaningful interpretation as kinetic perturbations. To avoid thediscussion of the special situation in two dimensions, we assume d > P of this kind, we consider the corre-sponding scattering operators S ( P ). Whereas the respective dynamicalrelations remain unaffected, the causal factorization rule needs to be2dapted to the particular choice of P . This can be understood if onetakes into account that the unitary operators F S ( P ) − S ( F + P )describe scattering processes, induced by functionals F of the preced-ing types, which evolve under the perturbed dynamics with perturbationgiven by P . Thus if the functional P is of kinetic type, this scatter-ing process effectively takes place in a locally distorted Minkowski spacewhose causal structure, fixed by P , enters in the factorization rules. Yetoperators S ( P ) , S ( Q ), assigned to functionals having their supports inspacelike separated regions of Minkowski space, still commute. By argu-ments given in [5], this extended family of operators therefore generateslocal nets of C*-algebras in Minkowski space, complying with all Haag-Kastler axioms [9].We will study in more detail the subalgebra of the dynamical C*-algebra, which is generated by scattering operators assigned to function-als of the classical field as well as its kinetic and quadratic point likeperturbations. This algebra describes quantum fields in locally distortedMinkowski spaces, which satisfy corresponding field equations and com-mutation relations. We will also exhibit some algebraic relations betweenthe field and the underlying scattering operators.These results enter in our construction of representations of this al-gebra on Fock space. In this construction we make use of the knownfact that the unitary scattering operators associated with kinetic pertur-bations can be represented on Fock space [17]. Yet the phase factors ofthese operators remained ambiguous in that analysis. They matter, how-ever, for the proof that there is a choice such that the resulting operatorssatisfy the causal factorization relations. In order to establish this fact,we develop arguments akin to cohomology theory. The existence of Fockrepresentations of the dynamical C*-algebra, generated by the field andits quadratic perturbations, is thereby established.Our article is organized as follows. In the subsequent section we intro-duce notions from classical field theory and discuss the form of admissiblekinetic perturbations. Section 3 contains the definition of the extendeddynamical C*-algebra and remarks on some of its general properties. InSec. 4 we study the subalgebra generated by the field and its quadratickinetic as well as point like perturbations and determine its algebraicstructure. These results are used in Sec. 5 in an analysis of represen-tations of the scattering operators and of their products on Fock space.The ambiguities left open in the phase factors are discussed in Sec. 6;3here it is shown that, for some coherent choice of these factors, the scat-tering operators satisfy the causal factorization rules and thus define arepresentation of the C*-algebra on Fock space. The article concludeswith a brief outlook and a technical appendix. We adopt the notation used in [5] and adjust it to the more general set-ting, considered here. As already mentioned, we proceed from a classicalscalar field on d -dimensional Minkowski space M ≃ R d with its stan-dard metric η ( x, x ) = x − x , where x , x denote the time and spacecomponents of x ∈ R d . The field is described by real, smooth functions x φ ( x ), which constitute its configuration space E . The Lagrangeandensity of a non-interacting field with mass m ≥ x
7→ L ( x )[ φ ] = 1 / ∂ µ φ ( x ) η µν ∂ ν φ ( x ) − m φ ( x ) ) . (2.1)Its spacetime integral (if defined) is the corresponding Lagrangean action.The passage to fields which are subject to interaction, as given in (1.1)or (1.3), is accomplished by adding to this Lagrangean the respectivedensities.On the configuration space E of the field acts the additive group E ofdeformations, described by test functions φ ∈ D ( M ). Their action onthe affine space E is given by local shifts of the field, φ φ + φ . Withtheir help one defines variations of the Lagrangean action functionals,given by δ L ( φ )[ φ ] . = Z dx (cid:0) L ( x )[ φ + φ ] − L ( x )[ φ ] (cid:1) . (2.2)These variations are well defined for local Lagrangeans and arbitraryfields φ in view of the compact support of φ . Their stationary pointsdefine the solutions of the classical field equation for the given Lagrangean(“on shell fields”).In case of the non-interacting Lagrangean (2.1), the correspondingon shell field satisfies the Klein-Gordon equation. If one adds to thisLagrangean the densities of a kinetic perturbation P as in (1.3) and of aquadratic perturbation F with potential g = q as in (1.1), the resultingfield equation reads ∂ µ ( η µν + p µν ( x )) ∂ ν φ ( x ) + ( m + q ( x )) φ ( x ) = 0 . (2.3)4e restrict our attention here to perturbations P for which this equationdescribes the propagation of the field φ on a globally hyperbolic spacetimewith metric g P . This metric is, up to a factor, the inverse of the principalsymbol [10] of the underlying differential operator, x ∈ M , | det g P ( x ) | − / g P ( x ) . = ( η + p ( x )) − . (2.4)In order to simplify the discussion of the causal factorization relations,we restrict our attention to metrics g P for which (a) the constant timeplanes of Minkowski space for some fixed time coordinate are Cauchysurfaces and (b) the time coordinate is positive timelike with regard toall of these metrics. As is shown in the appendix, it amounts to thefollowing condition. Standing assumption:
The coefficients p µν ( x ), µ, ν = 0 , . . . , d − x ∈ M , of the kinetic perturbations P are smooth functions with com-pact support which satisfy(i) 1 + p ( x ) > δ ij + p ij ( x ) is positive definite, i, j = 1 , . . . , d − x ∈ M , convex and stable under scalings by positive numbers which arebounded by 1, cf. the appendix. It is also invariant under spacetimetranslations. In view of the choice of a distinguished time coordinateunderlying its definition, it is, however, not Lorentz invariant. We willreturn to this point in subsequent sections. The functionals F : E → R considered in this section contain, in additionto point like interactions as in equation (1.1), kinetic perturbations (1.3)with properties specified in the standing assumption. The family of thesefunctionals is denoted by F . Whereas F is, in general, not stable underaddition, we will deal with special pairs and triples of functionals in F for which all (partial) sums satisfy the standing assumption. Such tupleswill be termed admissible .Apart from the spacetime localization of the functionals, fixed by thesupports of the underlying test functions, we must also take into account5heir impact on the causal structure of spacetime. For P ∈ F , thisstructure is determined by the metric g P , which is fixed by the kineticpart of P according to equation (2.4). Given any region O ⊂ R d , wedenote by J P ± ( O ) the causal future, respectively past, of O with regardto g P . In case of the Minkowski metric, P = 0, we write J ± ( O ).Given an admissible triple P, Q, N ∈ F , we say that P succeeds Q with regard to (the metric induced by) N if supp P does not intersect thepast cone of supp Q , determined by g N , i.e. supp P ∩ J N − (supp Q ) = ∅ .We then write P ≻ N Q . In particular, P ≻ Q means that P succeeds Q inMinkowski space. Note that ≻ is not an ordering relation, in particularit is not transitive. Based on these notions, we can proceed now to anextension of the dynamical algebras, introduced in [5], by adding to themthe kinetic perturbations. As in [5], we begin by defining a dynamicalgroup, generated by symbols S ( P ), P ∈ F , which are subject to two re-lations. These relations involve a given Lagrangean L , the correspondingrelative action (2.2), and shifts of the functionals P by elements φ ∈ E ,denoted by P φ [ φ ] . = P [ φ + φ ], φ ∈ E . Compared to [5], we employ herea somewhat simplified “on shell” version of this group. Definition:
Given a local Lagrangean L on Minkowski space M , thecorresponding dynamical group G L is the free group generated by ele-ments S ( P ), P ∈ F , with S (0) = 1, modulo the relations(i) S ( P ) = S ( P φ + δ L ( φ )) for P ∈ F , φ ∈ E ,(ii) S ( P + N ) S ( N ) − S ( Q + N ) = S ( P + Q + N ) for any admissible triple P, Q, N ∈ F such that P succeeds Q with regard to N , P ≻ N Q . Remark:
If one puts N = 0 in the second condition, one obtains thecausality relation S ( P ) S ( Q ) = S ( P + Q ) if P succeeds Q with regard tothe Minkowski metric. Thus if P , Q have spacelike separated supportsin Minkowski space, then also S ( Q ) S ( P ) = S ( Q + P ) and the operatorscommute.A thorough discussion of the origin and interpretation of these rela-tions is given in [5, 6]. The only difference with regard to the presentframework appears in relation (ii), where the impact of the kinetic func-tionals on the causal structure of spacetime is taken into account.The passage from the dynamical group G L to a corresponding C*-algebra is accomplished by standard arguments, cf. [5]. One regards the6lements of G L as basis of some complex vector space A L ; the productin A L is inherited from G L by the distributive law, and the *-operationcan be defined such that the generating elements S ( P ) become unitaryoperators. The resulting *-algebra has faithful states and thus can beequipped with a (maximal) C*-norm. Its completion defines the dy-namical C*-algebra A L for given Lagrangean L and generating operators S ( P ), P ∈ F , describing local operations on the underlying system.A distinguished role is played by the constant functionals which, for c ∈ R , are given by c [ φ ] . = c , φ ∈ E . Their support is empty, hence S ( c ) S ( P ) = S ( c + P ) = S ( P ) S ( c ) by the causality condition (ii), so c S ( c ) defines a unitary group in the center of A L . As in [5], we fixits scale and put S ( c ) = e ic c ∈ R .In a similar manner, one can define extended dynamical algebras fortheories on arbitrary globally hyperbolic spacetimes. There the admissi-ble kinetic perturbations need to be adjusted to the underlying metric.We restrict our attention here to Minkowski space and its local deforma-tions, inherited from functionals in F . For given M ∈ F , these pertur-bations can still be described by unitary operators in the algebra A L . Asbrought to light by Bogoliubov [3, 4], they are defined by S M ( P ) . = S ( M ) − S ( M + P ) , P ∈ F . (3.1)One easily verifies that these operators also satisfy the two defining rela-tions of some dynamical algebra. In the first relation, the Lagrangean L is to be replaced by L M , i.e. the Lagrangean obtained from L by addingto it the density inherent in M . The factorization equation in the secondrelation is satisfied for admissible quadruples P, Q, N, M ∈ F , provided P succeeds Q with regard to ( M + N ), i.e. P ≻ ( M + N ) Q . We take from now on as dynamical input the algebra A for the La-grangian L , cf. (2.2), omitting in the following the subscript L . Infact, we are primarily interested in its subalgebra A ⊂ A , which isgenerated by unitaries S ( P ) with functionals P ∈ P , where P ⊂ F denotes the family of functionals which are at most quadratic in theunderlying field and satisfy our standing assumption; its subset of gen-uine quadratic functionals is denoted by Q . As we shall see, the algebra7 comprises non-interacting quantum fields, propagating in locally de-formed Minkowski spaces.We adopt the notation used in [5]. Thus K . = − ( ∂ µ η µν ∂ ν + m ) isthe negative Klein-Gordon operator, ∆ R and ∆ A are the correspondingretarded and advanced propagators, their difference ∆ = (∆ R − ∆ A )is the commutator function, and ∆ D = (1 / R + ∆ A ) is the Diracpropagator. Further below, we will also introduce perturbed versions ofthese entities.As in [5], we consider perturbations involving linear functionals of thefields φ ∈ E , given by F f [ φ ] = L f [ φ ] + (1 / h f, ∆ D f i , f ∈ D ( M ) . (4.1)Here L f [ φ ] . = R dx f ( x ) φ ( x ) and h f, g i . = R dx f ( x ) g ( x ) are constant func-tionals, where f, g are smooth functions whose product f g is compactlysupported. It was shown in [5] that the unitary operators W ( f ) . = S ( F f ) = S ( L f ) e ( i/ h f, ∆ D f i ∈ A , f ∈ D ( M ) , (4.2)have the algebraic properties of Weyl operators on Minkowski space. Inparticular, W ( Kf ) = 1 , W ( f ) W ( g ) = e − ( i/ h f, ∆ g i W ( f + g ) , f, g ∈ D ( M ) . (4.3)So these operators can be interpreted as exponential functions of a quan-tum field, which satisfies the Klein-Gordon equation and has c-numbercommutation relations given by the commutator function ∆.Next, we compute the product of Weyl operators with arbitrary ele-ments of the full algebra A . The result is stated in the following lemma.There we make use again of the shift of functionals by elements of E .As a matter of fact, taking advantage of the support properties of thefunctionals, these shifts are canonically extended in the lemma to a largerfamily of smooth functions. Lemma 4.1.
Let P ∈ F and let f ∈ D ( M ) . Then(i) W ( f ) S ( P ) = S ( F f + P ∆ R f ) , S ( P ) W ( f ) = S ( F f + P ∆ A f ) (ii) W ( f ) S ( P ) W ( f ) − = S ( P ∆ f ) . he condition of associativity does not entail further relations for multipleproducts of Weyl operators with operators S ( P ) .Proof. To compute W ( f ) S ( P ), we decompose f into f = f P + Kg P ,where f P , g P are test functions and the support of f P succeeds that of P with regard to the Minkowski metric, cf. [5, Sec. 4]. Thus W ( f ) = W ( f P ),hence, making use of the causal factorization condition as well as thedynamical relation underlying A , we obtain W ( f ) S ( P ) = W ( f P ) S ( P ) = S ( F f P + P )= S (cid:0) F g P f P + P g P + δ L ( g P ) (cid:1) . (4.4)By an elementary computation one finds that F g P f P + δ L ( g P ) = F f .Since the support of f P , whence that of ∆ R f P , succeeds that of P and g P = ∆ R Kg P = ∆ R ( f − f P ), one has P g P = P ∆ R f . Thus we arrive at W ( f ) S ( P ) = S ( F f + P ∆ R f ). In an analogous manner one obtains thesecond equality in the first part of the statement.As to the second part, we make use of W ( f ) − = W ( − f ), giving (cid:0) W ( f ) S ( P ) (cid:1) W ( − f ) = S ( F f + P ∆ R f ) W ( − f )= S ( F − f + F − ∆ A ff + P (∆ R − ∆ A ) f ) . (4.5)Since the commutator function ∆ = ∆ R − ∆ A is antisymmetric, the firsttwo functionals in the latter operator compensate each other, viz. F − f + F − ∆ A ff = h f, ∆ D f i − h f, ∆ A f i = (1 / h f, ∆ f i = 0 , (4.6)proving statement (ii).It remains to establish the assertion about multiple products. Pickingany f, g ∈ D ( M ), it follows from the Weyl relations and the precedingstep that (cid:0) W ( f ) W ( g ) (cid:1) S ( P ) = e − ( i/ h f, ∆ g i S ( F f + g + P ∆ R ( f + g ) )= S ( F f + g − (1 / h f, ∆ g i + P ∆ R ( f + g ) )) . (4.7)On the other hand, interchanging brackets, one obtains W ( f ) (cid:0) W ( g ) S ( P ) (cid:1) = W ( f ) S ( F g + P ∆ R g )= S ( F f + F ∆ R fg + P ∆ R ( g + f ) ) . (4.8)9y another elementary computation, one verifies that F f + g − (1 / h f, ∆ g i = F f + F ∆ R fg , (4.9)hence the operators in the preceding two relations coincide. In a similarmanner one sees that also all other products do not produce any newrelations.We turn now to the analysis of the subalgebra A ⊂ A . Its generatingelements S ( P ) are given by functionals of the form P . = ( P + P + P ) ∈ P , (4.10)where P is constant, P is linear, and P is quadratic in the underlyingfield.Given a functional P ∈ Q , we consider perturbations of the La-grangean L by adding to it the density P of P [ φ ] = (1 / h φ, P φ i , φ ∈ E . The perturbed Lagrangean is denoted by L P and the resultingclassical field equation (2.3) involves the differential operator − ( K + P ).As is well known, cf. for example [2], there exist corresponding retardedand advanced propagators ∆ PR and ∆ PA , fixing the commutator function∆ P . = (∆ PR − ∆ PA ), and the Dirac propagator ∆ PD . = (1 / PR + ∆ PA ). Inview of the regularity properties of P , these distributions map test func-tions into smooth functions. We will frequently make use of the basicrelation ( K + P ) ∆ PA,R = ∆
PA,R ( K + P ) = 1 (4.11)and the resolvent equation∆ PA,R − ∆ A,R = − ∆ PA,R ( P ∆ A,R ) = ∆
A,R ( P ∆ PA,R ) . (4.12)These relations hold on the test functions D ( M ). Note that P ∆ A,R and P ∆ PA,R = (1 − K ∆ PA,R ) map test functions into test functions.The analysis of the properties of the operators S ( P ), P ∈ P , sim-plifies by making use of the fact that the contributions coming from theconstant and linear functionals P and P can be factored out from S ( P ).For constant functionals, this was already shown in the preceding section.For the linear functionals, introduced above, this is a consequence of thepreceding lemma. Namely, making use of the quadratic dependence of P on the field, one obtains F f + P ∆ A f = L ( K + P )∆ A f + (1 / h ∆ A f, ( K + P )∆ A f i + P . (4.13)10hus, by the preceding lemma and the definition of Weyl operators, S ( P ) S ( L f ) e ( i/ h f, ∆ D f i = S ( L ( K + P )∆ A f + P ) e ( i/ h ∆ A f, ( K + P )∆ A f i . (4.14)Noticing that the inverse of ( K + P )∆ A is given by K ∆ PA , one sees thatthe linear functionals can be extracted from the operators S ( P ), as well.We may therefore restrict our attention in the following to quadraticperturbations P ∈ Q and omit the index 2. Without danger of confusion,we will also equate these perturbations with their respective densities.Given a perturbation P ∈ Q , the perturbed algebra A L P ⊂ A for theLagrangean L P is generated by the unitary operators, cf. equation (3.1), S P ( Q ) . = S ( P ) − S ( P + Q ) , Q ∈ Q . (4.15)Defining, in analogy to (4.1), functionals F Pf [ φ ] . = L f [ φ ] + (1 / h f, ∆ PD f i on E , it turns out that the corresponding perturbed operators W P ( f ) . = S P ( F Pf ) , f ∈ D ( M ) , (4.16)coincide with the Weyl operators for perturbed test functions. In fact,according to relation (4.13) we have F f + P ∆ A f = F P ( K + P )∆ A f + P . Hence,making use of the lemma and the fact that (cid:0) ( K + P )∆ A (cid:1) − = K ∆ PA , wearrive at W P ( f ) = W ( K ∆ PA f ) , f ∈ D ( M ) . (4.17)The perturbed operators W P ( f ), f ∈ D ( M ), describe the exponentialfunction of a quantum field which satisfies a linear field equation withregard to K + P . This follows from W P (( K + P ) f ) = W ( K ∆ PA ( K + P ) f ) = W ( Kf ) = 1 . (4.18)Moreover, they satisfy the Weyl relations with respect to the commutatorfunction ∆ P fixed by ( K + P ). In order to verify this we need to computethe symplectic form h ( K ∆ PA f ) , ∆ ( K ∆ PA g ) i for f, g ∈ D ( M ). Bearing inmind the properties of propagators, mentioned above, we have h ∆ A (1 − P ∆ PA ) f, (1 − P ∆ PA ) g i = h ∆ PA f, g i − h ∆ PA f , P ∆ PA g i , h (1 − P ∆ PA ) f, ∆ A (1 − P ∆ PA ) g i = h f, ∆ PA g i − h P ∆ PA f, ∆ PA g i . (4.19)11ince P is compactly supported, it acts as a symmetric operator onsmooth functions, so the last terms in the preceding two equalities coin-cide. We therefore obtain h ( K ∆ PA f ) , ∆ ( K ∆ PA g ) i = h (1 − P ∆ PA ) f, ∆ (1 − P ∆ PA ) g i = h ∆ A (1 − P ∆ PA ) f, (1 − P ∆ PA ) g i − h (1 − P ∆ PA ) f, ∆ A (1 − P ∆ PA ) g i = h ∆ PA f, g i − h f, ∆ PA g i = h f, ∆ P g i . (4.20)Thus we arrive at the Weyl relations for the perturbed operators, W P ( f ) W P ( g ) = e − ( i/ h f, ∆ P g i W P ( f + g ) , f, g ∈ D ( M ) . (4.21)It follows from this equality that the commutation relations of the op-erators in A L P , P ∈ Q , depend on the causal structure induced by theprincipal symbol of ( K + P ). This implies in particular that a pertur-bative expansion of the operators S P ( Q ), Q ∈ P , based on the quantumfield on Minkowski space, will in general not converge. Whereas for Weyl operators the existence of Fock representations is awell known fact, the question of whether these representations can beextended to the full dynamical algebras involving arbitrary local inter-actions is an open problem. As a matter of fact, this question may beregarded as the remaining fundamental problem of constructive quan-tum field theory [5]. We therefore restrict our attention here to the alge-bra A , involving perturbations of the non-interacting Lagrangean whichare at most quadratic in the underlying field. Even there, the questionof whether this algebra is represented on Fock space has remained opento date, to the best of our knowledge.In order to discuss this problem, we adopt the following strategy:proceeding from a representation of the Weyl algebra on Fock space, wemake use of the fact that the quadratic perturbations induce automor-phisms of this algebra. It then follows from a result by Wald [17] thatthese automorphisms can be unitarily implemented on Fock space. Inthe present section we complement this result by the observation thatthe automorphisms satisfy an automorphic version of the causal factor-ization condition. Since the Weyl algebra is irreducibly represented on12ock space, this implies that the implementing unitary operators satisfythe factorization condition, up to phase factors. In the subsequent sec-tion we will then show that the phase of the unitary operators can beadjusted such that they fully comply with causal factorization.The computation of the adjoint action of the quadratic perturbations S ( P ) ∈ A on the Weyl operators, P ∈ Q , is accomplished with the helpof Lemma 4.1. It yields, f ∈ D ( M ), S ( P ) − W ( f ) S ( P ) = S ( P ) − S ( F f + P ∆ R f )= S ( P ) − S ( F P ( K + P )∆ R f + P ) = W P (( K + P )∆ R f ) . (5.1)In the second equality, we made use of equation (4.13), where ∆ A f hasbeen replaced by ∆ R f and, in the last equality, we employed definition(4.16) of the perturbed Weyl operators. According to relation (4.17), thelatter operator coincides with W (( K ∆ PA )(( K + P )∆ R ) f ). Noticing that( K ∆ PA )(( K + P )∆ R ) has an inverse given by T P . = ( K ∆ PR )(( K + P )∆ A ) = (1 − P ∆ PR )(1 + P ∆ A ) , (5.2)we arrive at S ( P ) W ( f ) S ( P ) − = W ( T P f ) , f ∈ D ( M ) . (5.3)One easily verifies that T P acts as the identity on K D ( M ), hence it de-fines a real linear operator on the quotient space D ( M ) /K D ( M ). It alsofollows from the preceding equality that it preserves the symplectic form,entering in the Weyl relations, which is given by the commutator func-tion ∆. So it is an invertible symplectic transformation on the symplecticspace D ( M ) /K D ( M ).This quotient space is canonically associated with the Fock space of aparticle. We denote by H the symmetric Fock space, based on the singleparticle space H of a particle with mass m ≥
0. The scalar product in H is fixed by (cid:0) f, g (cid:1) . = Z dp θ ( p ) δ ( p − m ) e f ( p ) e g ( p ) , f, g ∈ D ( M ) . (5.4)So the quotient D ( M ) /K D ( M ) can be identified with the dense sub-space of H , given by the restrictions of the Fourier transforms e f of the13est functions to the mass shell p = m , p ≥
0. Moreover, the imag-inary part of the scalar product in (5.4) coincides with the symplecticform h f, ∆ g i , f, g ∈ D ( M ).It follows that the operator on D ( M ) /K D ( M ), fixed by T P , acts asa real linear, symplectic, and invertible operator T P on a dense domainin the single particle space H . In fact, as was shown by Wald, theoperator T P is bounded [17, Sec. 2.1]. Denoting by T † P the adjoint of T P with regarded to the scalar product given by the real part of (5.4), Waldalso showed that the difference ( T † P T P −
1) lies in the Hilbert-Schmidtclass [17, Sec. 3]. This is a consequence of the fact that its kernel D P canbe represented as difference of two Hadamard bi-solutions of the KleinGordon equation, i.e. as a smooth bi-solution,Re (cid:0) ( T P f, T P g ) − ( f, g ) (cid:1) = Z Z dxdy f ( x ) D P ( x, y ) g ( y ) . (5.5)Moreover, since ( T P − f , f ∈ D ( M ), are test functions, having theirsupports in the support of P , the kernel D P vanishes rapidly in spatialdirections if m >
0. Hence it determines a Hilbert-Schmidt operatoron H . If m = 0, this still holds true in spacetime dimensions d ≥ S ( P ) for the concrete Fock space repre-sentations of the abstractly defined operators. In the next step we showthat the symplectic operators T P , underlying their definition, satisfy acausal factorization relation.Let Q ∈ Q and let g . = ( K + Q )∆ A f with f ∈ D ( M ). Since ( K + Q )is a normally hyperbolic differential operator, there exist test functions g Q , h Q ∈ D ( M ) such that g = g Q + ( K + Q ) h Q (5.6)and supp g Q ∩ J − (supp Q ) = ∅ . In fact, one can put g Q = ( K + Q ) χ ∆ QR g , h Q = (1 − χ )∆ QR g , where χ is a smooth function which vanishes in aneighborhood of J − (supp Q ) and is equal to 1 in the complement of a14lightly larger neighborhood. Because of the support properties of g Q ,one has (∆ QR − ∆ R ) g Q = − ∆ QR ( Q ∆ R ) g Q = 0, hence T Q f = ( K ∆ QR )( g Q + ( K + Q ) h Q ) = g Q + Kh Q . (5.7)If supp g ∩ J − (supp Q ) = ∅ , there exists by the preceding argument adecomposition such that also supp h Q ∩ J − (supp Q ) = ∅ .Let us assume now that the pair P, Q ∈ Q is admissible and that thesupport of P succeeds that of Q in Minkowski space, i.e. P ≻ Q . Wechoose an open neighborhood C of some Cauchy surface in M which liesbetween P and Q , i.e. J (supp P ) ∩ C = J − (supp Q ) ∩ C = ∅ . (5.8)Let f ∈ D ( M ) with supp f ⊂ C . Then supp T Q f ⊂ J − ( C ) and there is adecomposition (5.7) such that supp g Q ⊂ C and supp h Q ∩ supp P = ∅ .Thus P ∆ A g Q = P h Q = 0. Since ∆ PR g Q has support in the complementof J − (supp Q ), whence (∆ P + QR − ∆ PR ) g Q = − ∆ P + QR ( Q ∆ PR ) g Q = 0 , itfollows that T P T Q f = ( K ∆ PR )(( K + P )∆ A ) ( g Q + Kh Q )= ( K ∆ PR ) ( g Q + ( K + P ) h Q ) = K ∆ P + QR g Q + Kh Q . (5.9)According to relation (5.6) g Q = g − ( K + Q ) h Q = ( K + Q )(∆ A f − h Q ) = ( K + P + Q )(∆ A f − h Q ) , (5.10)so we obtain T P T Q f = K ∆ P + QR (( K + P + Q )(∆ A f − h Q )) + Kh Q = T P + Q f . (5.11)Since any test function f can be represented in the form f = f C + Kg C with supp f C ⊂ C and the operators T P , T Q and T P + Q act on the image of K as the identity, the preceding relation holds for all f ∈ D ( M ). Thuswe have arrived at the causal factorization relation in Minkowski space T P T Q = T P + Q , P ≻ Q . (5.12)15e turn now to the general case. Let
P, Q, N be an admissible tripleof quadratic perturbations such that P succeeds Q with regard to N .Putting T NP . = T − N T P + N , we need to show that T NP T NQ = T NP + Q if P ≻ N Q . (5.13)For the metric g N , fixed by N , there exists an open neighborhood C ofsome Cauchy surface in M such that J N + (supp P ) ∩ C = J N − (supp Q ) ∩ C = ∅ . (5.14)Turning to the proof of the causality relation, we proceed from T NQ ( K ∆ NA )= ( K ∆ NA )(( K + N )∆ R ) | {z } T − N ( K ∆ N + QR )(( K + N + Q )∆ A ) | {z } T N + Q ( K ∆ NA ) (5.15)Now ∆ A,R ( K ∆ NA,R ) = ∆
NA,R , as a consequence of the resolvent equa-tion (4.12). Hence the preceding equality simplifies to T NQ ( K ∆ NA ) = ( K ∆ NA ) (cid:0) ( K + N )∆ N + QR (cid:1)(cid:0) ( K + N + Q )∆ NA (cid:1) . (5.16)We observe that after a similarity transformation with K ∆ NA , the oper-ator T NQ has the same form as T Q with the Klein Gordon operator K replaced by ( K + N ). Thus the argument for the product rule (5.13)is the same as for (5.12), noticing that all underlying propagators havesupport properties which are consistent with the causal order relativeto the chosen broadened Cauchy surface C . Multiplying equation (5.13)from the left by T N , we arrive at T P + N T − N T Q + N = T P + Q + N if P ≻ N Q . (5.17)This equality implies that the adjoint action of S ( P + N ) S ( N ) − S ( Q + N )on Weyl operators coincides with the action of S ( P + Q + N ). So thesetwo operators comply with the condition of causal factorization, up tosome undetermined phase factor.It also follows from equation (5.15), cf. also (4.17) and (5.3), that forany given N, Q ∈ Q the operators S N ( Q ) . = S ( N ) − S ( Q + N ) commutewith all perturbed Weyl operators W N ( f ) = W ( K ∆ NA f ) for test func-tions f having their support in the spacelike complement of supp Q with16egard to the metric g N . Note that under these circumstances Q ∆ NA f = 0and ∆ N + QR f = ∆ NR f , hence T NQ acts like the identity on ( K ∆ NA ) f . Thus,presuming that the perturbed Weyl operators satisfy the condition ofHaag duality [12], the operators S N ( Q ) are elements of the von Neumannalgebra generated by W N ( f ) for test functions f having their support inany causally closed region containing supp Q . Whence, pairs of opera-tors S N ( P ) , S N ( Q ) commute if the functionals P, Q ∈ Q have spacelikeseparated supports, denoted by P ⊥ N Q , relative to the metric g N .Let us mention as an aside that Haag duality has been established byAraki [1] in case of non-interacting scalar fields on Minkowski space, i.e. N = 0. Apparently, a fully satisfactory proof for perturbations N ∈ Q of this field has not yet appeared in the literature. Yet there exist un-published results to that effect [13], so we take it for granted here.We extend now the operators S ( P ), P ∈ Q , to arbitrary perturbations P ∈ P . This is accomplished by observations made in the preceding sec-tion. Namely, given any quadratic perturbation P , we put for arbitraryconstants c and linear functionals L f = ( F f − (1 / h f, ∆ D f i ), compareequation (4.14), S ( c + L f + P ) . = e i ( c − (1 / h f, ∆ PD f i ) S ( P ) W ( K ∆ PA f )= e i ( c − (1 / h f, ∆ PD f i ) W ( K ∆ PR f ) S ( P ) . (5.18)The second equality follows from the adjoint action of S ( P ) on Weyloperators, cf. (5.3), and T P K ∆ PA = K ∆ PR .The extended operators satisfy, for fixed P ∈ Q , the causal factoriza-tion relations. To give an example, the preceding relations imply aftersome elementary computation that, f, g ∈ D ( M ), S ( F f + P ) S ( P ) − S ( F g + P ) = e i h f, ∆ PA g i S ( F f + F g + P ) . (5.19)Thus if supp f ≻ P supp g , the phase factor is equal to 1, in accordance withthe condition of causal factorization. In a similar manner one verifies thecausal factorization for all products of Weyl operators and the extendedoperators involving a fixed quadratic perturbation. In other words, theambiguities in the phase factors appearing in the causal factorizationrelations of the unitaries S ( P ) depend only on the quadratic parts P ∈ Q of the functionals P ∈ P . 17elation (5.18) also implies that the extended operators satisfy thedynamical condition, involving the Lagrangean L . Since constant func-tionals factor out from this condition, it suffices to verify it for functionalsof the form ( F Pf + P ) for arbitrary f ∈ D ( M ). A by now routine com-putation shows that for perturbations P ∈ Q one obtains for the shiftedfunctionals the equality( F Pf + P ) φ + δ L ( φ ) = F Pf +( K + P ) φ + P , φ ∈ E . (5.20)Thus S ( P ) − S (( F Pf + P ) φ + δ L ( φ )) = S ( P ) − S ( F Pf +( K + P ) φ + P )= W P ( f + ( K + P ) φ ) = W P ( f ) = S ( P ) − S ( F Pf + P ) , (5.21)where in the second equality we made use of the definition (4.16) of theperturbed Weyl operators. The third equality is a consequence of theWeyl relations and the fact that W P (( K + P ) φ ) = 1. So we arrive, asclaimed, at S ( P φ + δ L ( φ )) = S ( P ) for P ∈ P , φ ∈ E . (5.22)We summarize the results obtained in this section in a proposition. Proposition 5.1.
Let P ∈ P . There exist unitary operators S ( P ) onFock space, inducing automorphisms of the Weyl algebra, which are deter-mined by equation (5.18) . These operators satisfy the dynamical equation S ( P φ + δ L ( φ )) = S ( P ) , φ ∈ E . (5.23) Moreover, for any admissible triple of functionals P , Q , N ∈ P satisfying P ≻ N Q , there exists a phase α ( N | P, Q ) ∈ T , depending only on thequadratic parts P, Q, N of the functionals, such that S ( P + N ) S ( N ) − S ( Q + N ) = α ( N | P, Q ) S ( P + Q + N ) . (5.24) If P , Q are spacelike separated, P ⊥ N Q , the product in (5.24) is sym-metric in P , Q , i.e. α ( N | P, Q ) = α ( N | Q, P ) . The family of functionals P is stable under translations, yet not un-der Lorentz transformations because of the choice of a time direction18n our standing assumption. Since there exists a unitary representa-tion λ U ( λ ) of the Poincar´e group on Fock space, one can proceedfrom the operators S ( P ), P ∈ P , to operators which are compatiblewith any other choice of the time direction. Namely, given λ , the uni-taries U ( λ ) S ( P ) U ( λ ) − induce automorphisms of the Weyl operatorswhose quadratic part is fixed by the transformed single particle opera-tors T P λ . = U ( λ ) T P U ( λ ) − . Thus these unitaries comply, for adequate λ ,with any choice of the time direction in the standing assumption and sat-isfy the proposition as well. In particular, the phase α in the propositioncan be chosen to be Poincar´e invariant. We turn now to the problem of fixing the phases of the operators S ( P ), P ∈ P , so that they fully comply with the causal factorization condition.A similar problem was treated by Scharf and Wreszinski [15] for the caseof a Fermi field, coupled to an external electromagnetic field, cf. also [8].The kinetic perturbations are more singular, however, and an analogouscomputational approach, based on explicit expressions for the factors α in (5.24) (see e.g. [14]) would require some coherent non-perturbativerenormalization scheme. We therefore adopt here a different strategy. Based on the results ofWald [17], we have established in the preceding section the existence ofunitary operators S ( P ) on Fock space, which determine a projective rep-resentation of the group Q , generated by the operators T P for quadraticperturbations P ∈ Q on the single particle space. The cohomology ofthis representation is known to be non-trivial due to the appearance ofSchwinger terms, cf. [11] and references quoted there. Yet these sin-gularities are expected not to affect the causal factorization, involvingperturbations with disjoint supports. We therefore focus on the projec-tive causal factorization equation, stated in Proposition 5.1, and look atit from a cohomological point of view.Let α ( N | P, Q ) ∈ T be the phase factors appearing in equation (5.24)for quadratic functionals P, Q, N ∈ Q . We begin by exhibiting two basic This is related to the problem of associating determinants to hyperbolic differ-ential operators. For recent progress in the case of elliptic operators see [7] where,however, the class of allowed perturbations is less singular. α ( N | P, Q ) is well defined if P, Q, N ∈ Q is an admissible triple,satisfying the causality condition P ≻ N Q . Lemma 6.1.
Let P , P , Q , Q , N ∈ Q . Putting P . = P + P and Q . = Q + Q , one has α ( N | P, Q ) = α ( N | P , Q ) α ( N + P | P , Q )= α ( N | P, Q ) α ( N + Q | P, Q ) , (6.1) provided all phases α are well defined. Remark:
These relations comprise within the present context the es-sential part of the information contained in the cocycle equations, deter-mined by the underlying projective representation of Q . Proof.
We have α ( N | P , Q ) α ( N + P | P , Q ) S ( P + N + Q )= α ( N | P , Q ) α ( N + P | P , Q ) S ( P + ( P + N ) + Q )= α ( N | P , Q )) S ( P + ( P + N )) S ( P + N ) − S ( P + N + Q ) (6.2)= S ( P + N ) S ( P + N ) − S ( P + N ) | {z } =1 S ( N ) − S ( N + Q )= α ( N | P, Q ) S ( P + N + Q ) . So the first equality in the statement follows. The second equality isobtained in a similar manner.It is our goal to show that there exists a collection of phases β ( P ) ∈ T , P ∈ Q , such that for any admissible triple of functionals P, Q, N ∈ Q with P ≻ N Q , one has α ( N | P, Q ) = β ( P + N ) − β ( N ) β ( Q + N ) − β ( P + Q + N ) . (6.3)Note that for any choice of phases β , the expression on the right handside satisfies the equalities in the preceding lemma. So, in other words,we want to prove that these equalities admit only such trivial solutions,akin to the coboundaries solving cocycle equations in cohomology the-ory. Multiplying each operator S ( P ), P ∈ P , with the phase factor20 ( P ), corresponding to the quadratic part P of P , the resulting oper-ators satisfy the proper causal factorization relation (5.24), where thephase factor α is identical to 1. Moreover, since the quadratic part P of P is not affected in the dynamical relation (5.23), this relation still holdstrue for the modified operators β ( P ) S ( P ), P ∈ P . We thereby arrive atthe main result of this article. Theorem 6.2.
Let A be the dynamical C*-algebra generated by uni-taries S ( P ) , P ∈ P , which satisfy the dynamical condition (i) for theLagrangean L of a scalar field with mass m ≥ in d > spacetimedimensions, as well as the causal factorization equation (ii). If m > ,this algebra is represented by an extension of the Weyl algebra on the(positive energy) Fock space for any value of d ; if m = 0 , the dimensionmust satisfy d ≥ . Since the proof of relation 6.3 is cumbersome, consisting of severalsteps, we begin with an outline of our argument. The functionals P ∈ Q ,involving symmetric tensors and scalars, depend on test functions p on R d , having values in a real vector space of dimension n ( d ) = d ( d +1) / K ⊂ R n ( d ) which is contractible, i.e. it is mapped into itself by scaling it with factorsless than 1. This set can be covered by an increasing net of compact,convex and contractible subsets K c ⊂ K , c ≥
1, related to metrics ofMinkowski type, η c ( x, x ) = c x − x , x ∈ R d . The metric η c dominatesall metrics g P where p takes values in K c , i.e. the light cone fixed by η c contains the lightcone determined by the metric g P . (See the appendix.)The subset of functionals in Q involving test functions with values in K c is denoted by Q ( K c ).In our analysis of the phases α ( N | P, Q ), we need to consider limitednumbers of (at most six) admissible triples of functionals
P, Q, N ∈ Q .Any such collection of functionals is, together with the respective sums,contained in some Q ( K c ) for sufficiently large c . Making use of thisfact, we can simplify the discussion of the causal relations between thefunctionals appearing in the phases.Given any c ≥ P, Q, N ∈ Q ( K c ) satisfy-ing the causality condition P ≻ N Q , we restrict the corresponding phases α ( N | P, Q ) to the subset of triples satisfying the stronger causality condi-tion P c ≻ Q . The latter symbol indicates that the functional P does not21ntersect the past of the functional Q with regard to the metric η c , i.e. supp P ∩ J c − (supp Q ) = ∅ in an obvious notation. Thereby, the causalrelations between the restricted functionals in Q ( K c ) can be discussedin a simpler, unified manner. In order to mark this step, we denote therestricted phases by α c ( N | P, Q ) and introduce the following terminology.
Definition:
Let c ≥
1. A finite collection of phases α c ( N i | P i , Q i ) forgiven admissible triples P i , Q i , N i ∈ Q ( K c ) is said to be well defined if P i c ≻ Q i for i = 1 , . . . , N . In particular, the equalities (6.1) are satisfiedby such well defined collections of restricted phases.A major part of our argument consists of the proof that the restrictedphases α c ( N | P, Q ) can be extended to a considerably larger set of func-tionals. As we will see, they have unique extensions α c ( N | P, Q ), beingdefined for admissible triples
P, Q, N ∈ Q ( K c ) with supp P ∩ supp Q = ∅ .We will then show that these extensions are the restrictions to Q ( K c )of a global phase α ( N | P, Q ) which is defined for all admissible triples
P, Q, N ∈ Q satisfying supp P ∩ supp Q = ∅ . Moreover, α coincides withthe original phase α on its domain. The more transparent properties of α will enable us to prove that there exist phase factors β ( P ) ∈ T , P ∈ Q ,which trivialize it. That is, equation (6.3) is satisfied for all admissibletriples P, Q, N ∈ Q with supp P ∩ supp Q = ∅ . Thus, a fortiori , α canbe trivialized.We turn now to the proof that the restricted phases α c ( N | P, Q ) can beextended, as indicated. There we make use of the fact that the phases aresymmetric for spacelike separated
P, Q , cf. Proposition 5.1. In accordancewith our conventions, we will only consider pairs of functionals which arespacelike separated with regard to the metric η c . It is note worthy thatthe condition of Haag duality, entailing the symmetry of the phases,then follows already from the seminal results of Araki [1]. A crucial steptowards the extension of the phases is the following lemma. Lemma 6.3.
Let P , P , Q, N ∈ Q ( K c ) . Then α c ( N + P | Q, P ) α c ( N | P , Q ) = α c ( N + P | P , Q ) α c ( N | Q, P ) . if all occurring phases α c are well defined, cf. the preceding definition. Technical remark:
In the proof of this lemma, as well as in sub-sequent arguments, we will make use of the fact that any functional22 ∈ Q ( K c ) can be split within Q ( K c ) into “locally convex” combina-tions of functionals. This is accomplished by multiplying the (tensor-valued) test function p , underlying P , with some “pointwise convex”partition of unity, p k . = χ k p , where 0 ≤ χ k ≤ P nk =1 χ k = 1 on the support of p . Since K c is convex, the func-tionals P k , which are obtained by replacing p in P by p k , are containedin Q ( K c ), k = 1 , . . . , n . By some abuse of notation, we put P k . = χ k P ,giving P nk =1 P k = ( P nk =1 χ k ) P = P . Choosing suitable pointwise convexpartitions, we will split in this manner given functionals P into combi-nations of functionals with prescribed support properties, determined bythe supports of χ k . For the sake of shortness, we omit the phrase “point-wise convex” in the following. We will also use the notation J c ∩ . = J c + ∩ J c − and J c ∪ . = J c + ∪ J c − . Proof.
For the proof of the lemma, we proceed from the underlying con-dition P c ≻ Q c ≻ P . So there exists a decomposition P = P + + P suchthat supp P + ∩ J c − (supp Q ∪ supp P ) = ∅ and supp P ∩ J c ∪ (supp Q ) = ∅ .Making use Lemma 6.1, we then split the phases appearing in the state-ment: our underlying strategy consists of moving, whenever possible, P to the first entry and P + , P to the second, respectively third, entry. Forthe first factor, appearing on the left hand side of the equality in thestatement, we obtain α ( N + P + + P | Q, P ) = α ( N + P | Q + P + , P ) α ( N + P | P + , P ) − = α ( N + P + Q | P + , P ) α ( N + P | Q, P ) α ( N + P | P + , P ) − . (6.4)For the second factor, we get α ( N | P + + P , Q ) = α ( N + P | P + , Q ) α ( N | P , Q ) . (6.5)The factors appearing on the right hand side of the equality are treatedsimilarly. The first factor yields α ( N + P | P + + P , Q ) α ( N + P | P , Q ) − = α ( N + P + P | P + , Q )= α ( N + P | P + , Q + P ) α ( N + P | P + , P ) − = α ( N + P + Q | P + , P ) α ( N + P | P + , Q ) α ( N + P | P + , P ) − . (6.6)The second factor gives α ( N | Q, P ) α ( N + P | P , Q ) = α ( N | Q, P + P )= α ( N + P | Q, P ) α ( N | Q, P ) . (6.7)23oticing that α ( N | Q, P ) = α ( N | P , Q ) since supp P c ⊥ supp Q , weconclude that the products of the phase factors on the left and righthand side of the equality in the statement coincide, as claimed.Note that the conditions on the entries of the phase factors are metin each of the preceding steps; because the functionals appearing there,as well as their respective sums, are convex combinations of (sums of)the given functionals, and K c is convex and contractible.We are now in a position to extend the definition of the restrictedphases α c to more general entries. This is accomplished in several steps.Let P, Q, N ∈ Q ( K c ) be admissible and letsupp P ∩ J c ∩ (supp Q ) = ∅ . (6.8)There exists a partition χ + , χ − such that χ + + χ − = 1 on the support of P and P ± . = χ ± P satisfy supp P ± ∩ J c ∓ (supp Q ) = ∅ . Moreover, N + P ± are locally convex combinations of N and N + P . With these constraintson P, Q , we can define α c ( N | P, Q ) . = α c ( N | P + , Q ) α c ( N + P + | Q, P − ) . (6.9)This definition amounts to a symmetrization with regard to the causalorder of P, Q , viz. it implies α c ( N | P, Q ) = α c ( N | Q, P ) if P c ≻ Q or Q c ≻ P . We need to verify that α c , so defined, (i) extends α c and (ii)does not depend on the split P = P + + P − within the above limitations.As to (i), we note that if P c ≻ Q , then P − and Q have spacelikeseparated supports, P − c ⊥ Q , and we may interchange these functionalsin the second factor of the right hand side of the preceding equality. Itthen follows from Lemma 6.1 that α c ( N | P, Q ) = α c ( N | P, Q ). We alsonote that according to Lemma 6.3, one may interchange the role of P + and P − in the definition.Concerning (ii), we remark that the ambiguities involved in the split-ting of P pertain to the spacelike complement of the support of Q . So let P = ( P + + P ) + ( P − − P ) be another convex splitting, where P c ⊥ Q .Then, bearing in mind the symmetry of the phases in P , Q , we have α c ( N | P + + P , Q ) α c ( N + P + + P | Q, P − − P ) (6.10)= α c ( N | P + , Q ) α c ( N + P + | P , Q ) α c ( N + P + | Q, P ) − | {z } =1 α c ( N + P + | Q, P − ) , α c is well defined. The extended phase satisfiescocycle relations analogous to those established for α in Lemma (6.1). Lemma 6.4.
Let P , P , Q, Q , Q , N ∈ Q ( K c ) . Putting P = P + P ,one has α c ( N | P + P , Q ) = α c ( N | P , Q ) α c ( N + P | P , Q ) (6.11) α c ( N | P, Q ) α c ( N + Q | P, Q ) = α c ( N | P, Q ) α c ( N + Q | P, Q ) , (6.12) provided all terms are well defined. The latter condition now implies thatall phase factors contain admissible triples in Q ( K c ) , where the function-als in their second and third entry have disjoint supports, in agreementwith condition (6.8) .Proof. For the proof of the first equality in (6.11), we split P i = P i + + P i − ,where P i ± are functionals, i = 1 ,
2, with appropriate support propertiesrelative to Q , in accordance with definition (6.9) of the extended phases.The left hand side of (6.11) is then defined by α c ( N | P + P , Q ) α c ( N + P + P | Q, P − + P − ) . (6.13)Applying Lemma 6.1 to every factor, we obtain α c ( N | P + P , Q ) = α c ( N | P , Q ) α c ( N + P | P , Q ) ,α c ( N + P + P | Q, P − + P − ) = α c ( N + P + P | Q, P − ) · α c ( N + P + P + P − | Q, P − ) . (6.14)The factors appearing on the right hand side of (6.11) are given by α c ( N | P , Q ) = α c ( N | P , Q ) α c ( N + P | Q, P − ) , (6.15) α c ( N + P | P , Q ) = α c ( N + P | P , Q ) α c ( N + P + P | Q, P − ) . Comparing the four factors appearing on the right hand sides of theequalities in (6.14), respectively (6.15), we see that two of them coincide.For the products of the remaining pairs, we get α c ( N + P | P , Q ) α c ( N + P + P | Q, P − )= α c ( N + P | P − + P , Q )= α c ( N + P | Q, P − ) α c ( N + P + P − | P , Q ) , (6.16)completing the proof of relation (6.11).25urning to the proof of relation (6.12), we make use of the underlyingcondition supp P ∩ (cid:0) J c ∩ (supp Q ) ∪ J c ∩ (supp Q ) (cid:1) = ∅ . So there exists aconvex decomposition P = P ++ + P + − + P − + + P −− whose componentssatisfy supp P σσ ′ ∩ (cid:0) J c − σ (supp Q ) ∪ J c − σ ′ (supp Q ) (cid:1) = ∅ for σ, σ ′ = ± .We then apply relation (6.11) to the phases appearing on the left handside of equation (6.12) and obtain α c ( N | P, Q ) (6.17)= α c ( N | P ++ + P −− , Q ) α c ( N + P ++ + P −− | P + − + P − + , Q ) .α c ( N + Q | P, Q ) (6.18)= α c ( N + Q | P ++ + P −− , Q ) α c ( N + Q + P ++ + P −− | P + − + P − + , Q ) . The first factors on the right hand side of equations (6.17), respec-tively (6.18), are by definition equal to α c ( N | P ++ + P −− , Q ) = α c ( N | P ++ , Q ) α c ( N + P ++ | Q , P −− ) , (6.19) α c ( N + Q | P ++ + P −− , Q )= α c ( N + Q | P ++ , Q ) α c ( N + Q + P ++ | Q , P −− ) . (6.20)Hence, applying Lemma 6.1 twice, their product is given by α c ( N | P ++ , Q + Q ) α c ( N + P ++ | Q + Q , P −− ) . (6.21)It is thus symmetric in Q , Q .The second factors on the right hand side of (6.17) and (6.18) aretreated similarly. There we have α c ( N + P ++ + P −− | P + − + P − + , Q ) (6.22)= α c ( N + P ++ + P −− | Q , P − + ) α c ( N + P ++ + P −− + P − + | P + − , Q ) ,α c ( N + Q + P ++ + P −− | P + − + P − + , Q ) (6.23)= α c ( N + Q + P ++ + P −− | P − + , Q ) α c ( N + Q + P ++ + P −− + P − + | Q , P + − ) , We apply Lemma 6.3 to the product of the first factors on the right handside of (6.22), (6.23), changing the places of Q , Q , P − + with the result α c ( N + Q + P ++ + P −− | Q , P − + ) α c ( N + P ++ + P −− | P − + , Q ) . (6.24)For the product of the second factors we obtain α c ( N + Q + P ++ + P −− + P − + | P + − , Q ) α c ( N + P ++ + P −− + P − + | Q , P + − ) . (6.25)26ow the product of the first factors in (6.24) and (6.25) coincides bydefinition with the extended phase α c ( N + Q + P ++ + P −− | P + − + P − + , Q ) , (6.26)and the product of the second factors in (6.24) and (6.25) yields α c ( N + P ++ + P −− | P + − + P −− , Q ) . (6.27)Since the product of (6.26) and (6.27) coincides with the product of thesecond factors in (6.17) and (6.18), we conclude that this product is alsosymmetric in Q , Q . Noting once again that the phase factors, whichappeared in intermediate steps, were well defined for the respective triplesin Q ( K c ), the proof of equality (6.12) is complete.In a final step, we extend α c ( N | P, Q ) to triples
P, Q, N ∈ Q ( K c ),where P, Q have arbitrary disjoint supports, viz. we also admit function-als Q whose support is not causally closed. Let N , . . . , N n be an opencovering of supp Q such that J c ∩ ( N i ) ∩ supp P = ∅ , i = 1 , . . . , n . Pickinga corresponding partition of unity by test functions χ i , we proceed to thedecomposition Q = Q + · · · + Q n , where Q i . = χ i Q , i = 1 , . . . , n . Wethen put α c ( N | P, Q ) (6.28) . = α c ( N | P, Q ) α c ( N + Q | P, Q ) α c ( N + Q + · · · + Q n − | P, Q N ) . It follows from relation (6.12) that the right hand side of this equalitydoes not change if one exchanges the positions of Q i and Q i +1 . Hence itis stable under arbitrary permutations of the Q i , i = 1 , . . . , n −
1. It alsodoes not depend on the chosen partition of unity, as we will show next.Let ρ i , i = 1 , . . . , n , be another partition of unity for the chosencovering. We first consider the cases where ρ i + ρ j = χ i + χ j for somepair i = j and all other test functions coincide, ρ k = χ k , k = i, j .According to the preceding observation, we may reorder the indices andassume i = 1, j = 2. Putting R . = ( ρ − χ ) Q , we obtain ( Q + R ) = ρ Q ,( Q − R ) = ρ Q . Since supp P ∩ supp ρ i Q = ∅ , i = 1 ,
2, we can applyrelation (6.11), giving α c ( N | P, Q + R ) α c ( N + Q + R | P, Q − R ) (6.29)= α c ( N | P, Q ) α c ( N + Q | P, R ) α c ( N + Q | P, R ) − | {z } =1 α c ( N + Q | P, Q ) .
27e conclude that under these special changes of the partition of unity,the right hand side of definition (6.28) does not change. But any otherpartition of unity can be reached in a finite number of steps from parti-tions of this special type, so the definition does not depend on it either.Finally, the definition is also independent of the chosen covering. Tosee this we proceed to refinements of the given covering and correspond-ing refinements of the decompositions of the functionals. Let, for ex-ample, Q = Q + Q be such a refinement. Splitting P = P + + P − ,where supp P ± does not intersect J c ∓ (supp Q ), respectively, we have α c ( N | P, Q ) = α c ( N | P + , Q ) α c ( N + P + | Q , P − ). According to Lem-ma 6.1, the factors appearing on the right hand side can be split into α c ( N | P + , Q ) = α c ( N | P + , Q ) α c ( N + Q | P + , Q ) , (6.30) α c ( N + P + | Q , P − ) = α c ( N + P + | Q , P − ) α c ( N + Q + P + | Q , P − ) . The product of the first factors on the right hand sides of these equalitiesgives α c ( N | P, Q ) and that of the second factors α c ( N + Q | P, Q ).Thus we arrive at α c ( N | P, Q ) = α c ( N | P, Q ) α c ( N + Q | P, Q ) . (6.31)Iterating this argument, we see that definition (6.28) is invariant underfinite refinements of the covering. Since any two coverings have a jointrefinement, it follows that the extension of α c ( N | P, Q ) is well defined if
P, Q have disjoint supports and all (sums of the) functionals are containedin Q ( K c ). We summarize the preceding results. Proposition 6.5.
The phase factors α in Proposition 5.1 can be extendedto phases α which are defined for admissible triples P, Q, N ∈ Q withsupp P T supp Q = ∅ and satisfy α ( N | P, Q ) = α ( N | Q, P ) , (6.32) α ( N | P + P , Q ) = α ( N | P , Q ) α ( N + P | P , Q ) . (6.33) These equalities uniquely fix this extension.Proof.
Let
P, Q, N be any admissible triple with supp P T supp Q = ∅ .There exists some c ≥ P, Q, N ∈ K c . As shown above, therestriction α c of α to admissible triples P ′ , Q ′ , N ′ ∈ K c satisfying P ′ c ≻ Q ′ can be uniquely extended to phases α c which are defined on all admissible28riples in K c for which supp P ′ T supp Q ′ = ∅ ; moreover, they satisfy thepreceding two equalities on their domain. The extension α c is uniquebecause these equalities comprise the defining equation for α c in termsof the restricted phases α c .Next, we show that the extended phase α c coincides with the originalphase α , restricted to Q ( K c ). So let P, Q, N ∈ K c with P ≻ N Q . Anypair of points ( x, y ) ∈ supp P × supp Q satisfies either x c ≻ y , or y c ≻ x .In the latter case, the point x is spacelike separated from y with regardto the metric g N induced by N , x ⊥ N y . Thus, since the supports of P , Q are compact, we can split these functionals with the help of suitablepartitions of unity into finite sums P = P i P i , Q = P j Q j , such thateither P i c ≻ Q j , or Q j c ≻ P i and P i ⊥ N Q j . By repeated application of thebasic Lemma 6.1, we obtain a corresponding decomposition of the phase α ( N | P, Q ), given by α ( N | P, Q ) = Π i,j α ( N + X k
1, hence K b c ⊃ K c . Forpairs P, Q ∈ K c the relation P b c ≻ Q implies P c ≻ Q . Hence α b c coincideswith α c on all admissible triples in K c satisfying the stronger causalitycondition. We proceed now as in the preceding step and decompose P = P i P i , Q = P j Q j such that for each pair ( i, j ) at least one ofthe relations P i b c ≻ Q j or Q j b c ≻ P i holds. Adopting the notation in thepreceding step, we find in the first case α c ( N ij | P i , Q j ) = α c ( N ij | P i , Q j )= α b c ( N ij | P i , Q j ) = α b c ( N ij | P i , Q j ) . (6.35)In the second case we obtain, bearing in mind the symmetry propertiesof the extended phases in their second and third argument, α c ( N ij | P i , Q j ) = α c ( N ij | Q j , P i ) = α c ( N ij | Q j , P i ) (6.36)= α b c ( N ij | Q j , P i ) = α b c ( N ij | Q j , P i ) = α b c ( N ij | P i , Q j ) . Thus, by another decomposition based on Lemma 6.1, we arrive at α c ( N | P, Q ) = Π ij α c ( N ij | P i , Q j )= Π ij α b c ( N ij | P i , Q j ) = α b c ( N | P, Q ) , (6.37)completing the proof.We will show now that the extended phases α can be trivialized. Triv-ial solutions of the equalities in the preceding proposition are obtainedby picking phases β ( P ) ∈ T , P ∈ Q , and putting δβ ( N | P, Q ) . = β ( P + N ) − β ( N ) β ( Q + N ) − β ( P + Q + N ) . (6.38)They correspond to coboundaries in cohomology theory. Thus we need toexhibit phase factors β for which α can be represented in this form. Theconstruction of these phase factors will be accomplished in successivesteps. Namely, we will adjust the phases β for increasing subsets offunctionals in Q such that the preceding equality is satisfied in each stepby α , restricted to the respective subsets of functionals. The desiredresult is then obtained by some limiting argument.30t will be convenient to describe this procedure by an iterative scheme.To this end we multiply α with the inverse of (6.38), involving the phasesdetermined in each step, α ( N | P, Q ) α ( N | P, Q ) δβ ( N | P, Q ) − . Theresulting phases still satisfy both equations in the preceding propositionand are equal to 1 on increasing subsets of functionals. From the pointof view of cohomology theory, we are staying by this procedure in thecohomology class of α . We will therefore denote the phase factors, whichare modified in this manner, again by α .Turning to the construction, we pick any two disjoint compact re-gions O , O ⊂ M , choose a partition of unity χ , χ , χ such that χ , χ have disjoint supports, are equal to 1 on O , respectively O , and put χ = 1 − χ − χ . Let P, Q, N ∈ Q be any admissible triple such thatsupp P ⋐ O , supp Q ⋐ O , where the symbol ⋐ indicates that the sup-ports are contained in the open interior of the given regions. Setting N j . = χ j N , j = 0 , ,
2, it follows from equations (6.33) and (6.32) that α ( N | P, Q ) = α ( N + N | P + N , Q ) α ( N + N | N , Q ) − α ( N + N | P + N , Q ) = α ( N | P + N , Q + N ) α ( N | P + N , N ) − α ( N + N | N , Q ) − = α ( N | N , Q + N ) − α ( N | N , N ) . (6.39)With this input, we define β ( R ) . = α ( χ R | χ R, χ R ) for R ∈ Q . Makinguse of the fact that χ P = χ P = 0 and χ Q = χ Q = 0, the equalities(6.39) imply that α ( N | P, Q ) = β ( P + Q + N ) β ( P + N ) − β ( Q + N ) − β ( N ) (6.40)for the restricted set of triples P, Q, N . Thus α is trivial for such triples.Multiplying α with the inverse of the right hand side, we obtain an im-proved phase α which still satisfies the equations in Proposition 6.5 and,in addition, is equal to 1 if supp P ⋐ O , supp Q ⋐ O .Given any α with these properties, we repeat the preceding procedure.So let O , O ⊂ M be another pair of disjoint compact regions andlet χ ′ , χ , χ be a corresponding partition of unity. We put as before β ( R ) . = α ( χ ′ R | χ R, χ R ) for R ∈ Q . Thus α satisfies equation (6.40)for the respective triples. Multiplying it with the inverse of the righthand side, we obtain a modified phase α which satisfies the equationsin Proposition 6.5 and is equal to 1 if supp P ⋐ O , supp Q ⋐ O . As amatter of fact, it turns out that this modified phase is still equal to 1 alsofor the original triples P, Q, N satisfying supp P ⋐ O , supp Q ⋐ O .31aking use of the properties of the improved phases α , establishedin the preceding step, and of Proposition 6.5, we have for admissible P, Q, R, N with supp P ⋐ O , supp Q ⋐ O α ( N + P | R, Q ) = α ( N | P + R, Q ) α ( N | P, Q ) − = α ( N | P + R, Q ) = α ( N | R, Q ) α ( N + R | P, Q )= α ( N | R, Q ) . (6.41)This equality will be used at several points in the proof of the followingimportant lemma. Lemma 6.6.
Let β be the phases, determined in the preceding step fordisjoint regions O , O from a given α , which is equal to on pairs offunctionals with support in disjoint regions O , O . Then β ( P + Q + N ) β ( P + N ) − β ( Q + N ) β ( N ) − = 1 (6.42) for admissible triples P, Q, N ∈ Q with supp P ⋐ O , supp Q ⋐ O .Proof. We put R = χ ′ R , R = χ R , R = χ R , thus R + R + R = R , R ∈ Q , and consider for admissible triples P, Q, N the phase β ( P + Q + N ) = α ( N + P + Q | N + P + Q , N + P + Q ) . (6.43)Making use of the given support properties of P, Q , we will split thisexpression with the help of Proposition 6.5 into a product of phases,where
P, Q do not appear, both, in the same factor. This is a somewhatlengthy procedure. We begin by applying relation (6.33) twice, giving α ( N + P + Q | N + P + Q , N + P + Q ) = α α α α , (6.44)where α = α ( N + N + N + P + Q | P + Q , P + Q ) ,α = α ( N + N + P + Q | N , P + Q ) α = α ( N + N + P + Q | P + Q , N ) ,α = α ( N + P + Q | N , N ) . (6.45)Turning to α , we apply again relation (6.33) twice and obtain α = α ( N + P + Q | P , P ) α ( N + P + Q + Q | P , Q ) · α ( N + P + Q + Q | Q , P ) α ( N + P + Q | Q , Q ) . (6.46)32ince the second and third entries in the two middle factors have supportin O , respectively O , these factors are equal to 1. By relation (6.41)we can omit in the first factor Q and in the fourth factor P , hence α = α ( N + P | P , P ) α ( N + Q | Q , Q ) . (6.47)To the second factor α we apply both equalities Proposition 6.5, giving α = α ( N + N + P + Q | N , P ) α ( N + N + P + Q + P | N , Q ) . (6.48)According to relation (6.41) we can omit Q in the first factor and P + P in the second factor with the result α = α ( N + N + P | N , P ) α ( N + N + Q | N , Q ) . (6.49)The third factor α is treated similarly and we find α = α ( N + N + P | P , N ) α ( N + N + Q | Q , N ) . (6.50)We turn now to the factor α . In order to analyze it, we need a finerresolution of the functional N . To this end we choose another partitionof unity ρ + ρ + ρ = 1 such that supp ρ ⊂ O , supp ρ ⊂ O andsupp ρ ∩ (supp P ∪ supp Q ) = ∅ . Since P, Q have supports in the interiorof the respective regions, such a partition exists and the supports of ρ N , ρ N are contained in O , respectively O . We then consider thefunctionals N ji . = ρ j χ i N for i = 3 , j = 0 , ,
2. Decomposing N , N in the second and third entry of α , we move the terms appearing in thecorresponding sums successively to the first entry with the help of thetwo equalities in Proposition 6.5. We thereby arrive at a product of ninefactors of the form α j,k . = α ( P + Q + M jk | N j , N k ) , j, k = 0 , , , (6.51)where M jk is a sum of N and certain specific terms in the decompositionof N , N . As a matter of fact, this assertion becomes more transparentby proceeding in reverse. Beginning with α ( P + Q + N | N , N ), onebuilds α by successive multiplication with appropriate factors α jk . Thefirst two steps are given in α ( P + Q + N | N , N ) α ( P + Q + N + N | N , N )= α ( P + Q + N | N + N , N ) ,α ( P + Q + N | N + N , N ) α ( P + Q + N + N + N | N , N )= α ( P + Q + N | N + N + N , N ) . (6.52)33ne then proceeds in the same manner with N in the third entries. Bythis procedure, one ensures in particular that M = N .Let us now have a closer look at the factors α j,k . Because of thesupport properties of the operators N j , N k for j, k = 1 ,
2, it follows fromrelation (6.41) that we can omit Q from α j,k for j = 1 as well as k = 1.Similarly, for j = 2 or k = 2 we can omit P . Thus the resulting termsare again products of phases depending only on N, P , respectively
N, Q .There remains the case j = k = 0. Recalling that M = N , we applyrelation (6.33) and get α = α ( N + P + Q | N , N ) (6.53)= α ( N + Q | P + N , N ) α ( N + Q | P , N ) − = α ( N + Q + N | P , N ) α ( N + Q | N , N ) α ( N + Q | P , N ) − . Again by relation (6.41), we can omit Q in the first and the third fac-tor of the latter product. So α also factors into a product of phasesdepending only on N, P , respectively
N, Q . So, to summarize, we suc-ceeded in proving that there exist phases β ( P, N ), β ( Q, N ), involvingdecompositions of their arguments depending only on the given regions O , . . . , O , such that β ( P + Q + N ) = β ( P, N ) β ( Q, N ) . (6.54)Making use of this equality also for the functional P = 0, respectively Q = 0, it is straight forward to verify relation (6.42), completing itsproof.By iteration of this argument, one can trivialize the extended phases α ( N | P, Q ) for admissible triples
P, Q, N ∈ Q , where
P, Q have theirsupports in any given finite number of pairs of disjoint compact regions.In order to cover all such triples, we make use of Tychonoff’s theorem.Let P be any finite collection of pairs O ′ × O ′′ of disjoint compact subsetsof M . We denote by B P the set of maps β : Q → T which trivialize α for the given subsets. Recalling the definition of δβ , cf. (6.38), one has α ( N | P, Q ) δβ ( N | P, Q ) − = 1 (6.55)if supp P × supp Q ⊂ O ′ × O ′′ ∈ P . We have shown that the sets B P arenot empty and it is also clear that B P ⊂ B P if P ⊃ P . Let B . = \ P B P . (6.56)34his set is non-empty. Because, otherwise, due to the compactness ofthe set of maps β : Q → T with respect to the topology of pointwiseconvergence (Tychonoff’s Theorem), already a finite intersection had tobe empty, which has been excluded. Every β ∈ B trivializes α . Wealso note that different elements differ by a local functional, i.e. a map γ : Q → T satisfying δγ = 1. We have thus established the followingproposition. Proposition 6.7.
Let α ( N | P, Q ) be the extended phases for admissibletriples P, Q, N ∈ Q satisfying supp P ∩ supp Q = ∅ . There exists afunction β : Q → T such that α ( N | P, Q ) = β ( P + N ) − β ( N ) β ( Q + N ) − β ( P + Q + N ) . As was shown in Proposition 6.5, the phases α ( N | P, Q ) coincide withthe restriction of α ( N | P, Q ) to their domain, i.e. on admissible triples
P, Q, N ∈ Q satisfying P ≻ N Q . Thus they can be trivialized, so thefollowing corollary obtains. It completes the proof of Theorem 6.2. Corollary 6.8.
Let α ( N | P, Q ) be the causal phases, introduced in Propo-sition 5.1 for admissible triples P, Q, N ∈ Q satisfying P ≻ N Q . Thereexists a function β : Q → T such that α ( N | P, Q ) = β ( P + N ) − β ( N ) β ( Q + N ) − β ( P + Q + N ) . We conclude this section with a remark on the covariance proper-ties of our construction. As noted at the end of the preceding section,the unitaries U ( λ ) S ( P ) U ( λ ) − induce automorphisms of the Weyl op-erators, for any P ∈ P and Poincar´e transformation λ . They exhaustthe unitaries for perturbations P λ satisfying the standing assumption forany given time direction and they also satisfy the corresponding causalfactorization condition. A fully covariant description would require, how-ever, that the phase factors β in the preceding corollary can be chosen tobe Poincar´e invariant for the given (Poincar´e invariant) α . It is an openproblem whether such a choice exists. In this article we have extended the framework of dynamical C*-algebrasfor quantum field theories on Minkowski space [5], admitting also kinetic35erturbations. The novel feature appearing in this extended framework isthe influence of kinetic perturbations on the causal factorization relationsof the unitary operators, describing their impact on states. Whereasthese operators still generate a local, covariant net on Minkowski space,labelled by their support regions, the causal relations between them areaffected. This is due to the fact that they describe the propagation offields in distorted spacetimes. As a matter of fact, this feature imposesrestrictions on the admissible perturbations, put down in our standingassumption. They reflect the idea that the kinetic perturbations arecaused by gravitational effects on the fields. In accordance with thisidea, we have shown that the perturbed fields satisfy wave equations andcommutation relations on locally perturbed Minkowski spaces.The unitary operators describing these perturbations are well definedat the level of abstract C*-algebras, which admit an abundance of statesand corresponding Hilbert space representations. Yet it is not clear fromthe outset that there exist also states, describing situations of physicalinterest, such as a vacuum and its local excitations, or equilibrium states.As a matter of fact, a comprehensive representation theory of dynamicalC*-algebras is the missing corner stone in a rigorous proof that inter-acting quantum field theories exist in four spacetime dimensions [5]. Aswas already mentioned, perturbation theory is of little use in this contextsince it cannot converge in the presence of kinetic perturbations, due totheir impact on the underlying causal structure and resulting modifica-tions of commutation relations. Thus a non-perturbative approach tothis problem is needed.As a step into that direction, we have considered the subset of per-turbations, which are at most quadratic in the underlying field. Theseperturbations do not describe self-interactions of the field, but compriseits interaction with the spacetime background and perturbations of itsmass. Previous results by Wald [17] had settled the existence of cor-responding unitary operators and resulting local nets of C*-algebras onFock space. But a proof that by adjustment of their phase factors thereexist also operators which satisfy the causal factorization relations didnot yet exist. In fact, it turned out to be surprisingly involved.Since a direct construction of such unitary operators would have re-quired the development of a non-perturbative renormalization scheme fortime-ordered exponentials, we have taken here a different, still cumber-some path. Adopting methods from cohomology theory, we have shown36hat the ambiguous phase factors of the unitary operators can be fixedin a manner such that they satisfy the causal factorization equations, i.e. there are no cohomological obstructions in that respect. It completedour proof that the restricted dynamical algebra is represented on Fockspace in any number of spacetime dimensions. This observation providesfurther evidence to the effect that our novel algebraic approach to theconstruction of quantum field theories is viable.
Appendix
In this appendix we determine perturbations of the metric in Minkowskispace M which keep it globally hyperbolic, so that the hypersurfaces t = const (for a fixed time coordinate) are still Cauchy surfaces and thetime coordinate is positive timelike with regard to the perturbed metric.We also analyze in some detail their inverses, which enter in the corre-sponding hyperbolic differential operators. We will thereby justify ourstanding assumption and exhibit increasing families K c of perturbations,labelled by the velocity of light c ≥
1, which enter in our analysis.Let g be any such metric. We use the split into time and space anddescribe g by a block matrix g = (cid:18) g gg T − G (cid:19) , (A.1)where g is a ( d − G is a spatial ( d − × ( d − g , the chosen time coordinate is stillpositive timelike, thus g >
0, and spatial vectors are still spacelike, so G has to be positive definite. The lightcone V + ( g ) fixed by g at any givenpoint in M is determined by the equation for the corresponding lightlikedirections, v = (1 , v ) ∈ R d ,0 = g ( v, v ) = g + 2 h v , g i − h v , Gv i . (A.2)Since G ≥ k G − k − | v | k G − k − − | v || g | ≤ g . (A.3)It follows that the velocity of light, determined by g , satisfies the bound | v | ≤ c . = (cid:16)p g / k G − k + | g | + | g | (cid:17) k G − k . (A.4)37hus one has the inclusion of light cones V + ( g ) ⊂ V + ( η c ), where thelatter lightcone is of Minkowski type, V + ( η c ) = { ( t, x ) ∈ R d | t > , c t − x > } , c > . (A.5)Next, we determine the inverse metric. Using again the split intotime and space coordinates, we represent g − also as a block matrix g − = (cid:18) g hh T − H (cid:19) (A.6)and obtain by an elementary computation g = ( g + h g , G − g i ) − , h = g G − g , H = G − − ( g ) − | h ih h | . (A.7)The conditions on g can now also be formulated in terms of conditionson g − , namely g > H is to be positive definite.The kinetic perturbations P , considered in the main text, are de-scribed by differential operators with principal symbols p , which in thechosen coordinates are given by p = (cid:18) p pp T − P (cid:19) . (A.8)They fix the metric g P by the equation | det g P | − / g P = ( η + p ) − on M (for d > p ) > P ) is positive definite.These conditions agree with our standing assumption, characterizing theprincipal symbols of admissible perturbations.It is apparent that any convex combination of admissible principalsymbols p is again admissible. We restrict the admissible symbols tocompact, convex subsets in order to control the size of the lightconesdetermined by the corresponding metrics g P in Minkowski space. Given0 < ε ≤
1, we consider perturbations with principal symbols satisfying ε ≤ p ≤ ε − , ε ≤ P ≤ ε − . (A.9)38e also require that the length | p | is bounded by ε − . These conditionscharacterize compact convex subsets of principal symbols. Since p = 0 iscontained in any set, they are also contractible.In analogy to relation (A.2), one can determine now the momen-tum space light cone fixed by these data. By a similar computationas above one finds that the vectors (1 , k ) are contained in this cone if | k | ≤ ( √ − ε . Thus this cone contains the momentum space light-cone for the Minkowskian metric η c ( ε ) with velocity of light c ( ε ) = ( √ ε − . (A.10)For the dual lightcones in position space we get the opposite inclusion.Hence the metrics g P associated with the (for the given ε ) restrictedprincipal symbols p are dominated by the Minkowski metric η c ( ε ) . Inview of the significance of this parameter in the main text, we denote thecorresponding compact, convex and contractible sets of principal symbolsby K c for given c ≥
1. They increase with increasing c and exhaust theset of all admissible principal symbols. Acknowledgment
We are grateful to Valter Moretti and Rainer Verch for information onthe status of the problem of Haag duality in curved spacetimes. DB alsothanks Dorothea Bahns and the Mathematics Institute of the Universityof G¨ottingen for their generous hospitality.
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