Deformations of quantum eld theories on spacetimes with ... - Dappiaggi, Lechner, Morfa-Mor… · The recent construction and analysis of deformations of quantum eld theories by warped

Deformations of quantum field theorieson spacetimes with Killing vector fields

Claudio Dappiaggi1a Gandalf Lechner2b Eric Morfa-Morales3c

1 II Institut fur Theoretische Physik D-22763 Hamburg Deutschland

2 Faculty of Physics University of Vienna A-1090 Vienna Austria

3 Erwin Schrodinger Institute for Mathematical Physics Vienna A-1090 Vienna Austria

aclaudiodappiaggiesiacat bgandalflechnerunivieacat cemorfamoesiacat

Abstract The recent construction and analysis of deformations of quantum field theoriesby warped convolutions is extended to a class of curved spacetimes These spacetimes carrya family of wedge-like regions which share the essential causal properties of the Poincaretransforms of the Rindler wedge in Minkowski space In the setting of deformed quantum fieldtheories they play the role of typical localization regions of quantum fields and observablesAs a concrete example of such a procedure the deformation of the free Dirac field is studied

1 Introduction

Deformations of quantum field theories arise in different contexts and have been studiedfrom different points of view in recent years One motivation for considering such modelsis a possible noncommutative structure of spacetime at small scales as suggested bycombining classical gravity and the uncertainty principle of quantum physics [DFR95]Quantum field theories on such noncommutative spaces can then be seen as deformationsof usual quantum field theories and it is hoped that they might capture some aspects ofa still elusive theory of quantum gravity (cf [Sza03] for a review) By now there existseveral different types of deformed quantum field theories see [GW05 BPQV08 Sol08GL08 BGK+08 BDFP10] for some recent papers and references cited therein

Certain deformation techniques arising from such considerations can also be used asa device for the construction of new models in the framework of usual quantum fieldtheory on commutative spaces [GL07 BS08 GL08 BLS10 LW10] independent of theirconnection to the idea of noncommutative spaces From this point of view the deforma-tion parameter plays the role of a coupling constant which changes the interaction of themodel under consideration but leaves the classical structure of spacetime unchanged

Deformations designed for either describing noncommutative spacetimes or for con-structing new models on ordinary spacetimes have been studied mostly in the case of aflat manifold either with a Euclidean or Lorentzian signature In fact many approachesrely on a preferred choice of Cartesian coordinates in their very formulation and do notgeneralize directly to curved spacetimes The analysis of the interplay between space-time curvature and deformations involving noncommutative structures thus presents a

challenging problem As a first step in this direction we study in the present paper howcertain deformed quantum field theories can be formulated in the presence of externalgravitational fields (ie on curved spacetime manifolds) see also [ABD+05 OS09] forother approaches to this question We will not address here the fundamental questionof dynamically coupling the matter fields with a possible noncommutative geometry ofspacetime [PV04 Ste07] but rather consider as an intermediate step deformed quantumfield theories on a fixed Lorentzian background manifold M

A deformation technique which is well suited for our purposes is that of warped convo-lutions see [BS08] and [GL07 GL08] for precursors and related work Starting from aHilbert space H carrying a representation U of IRn the warped convolution AQ of anoperator A on H is defined as

AQ = (2π)minusnintdnx dny eminusixy U(Qx)AU(y minusQx) (11)

Here Q is an antisymmetric (ntimes n)-matrix playing the role of deformation parameterand the integral can be defined in an oscillatory sense if A and U meet certain regularityrequirements For deformations of a single algebra the mapping A rarr AQ has manyfeatures in common with deformation quantization and the Weyl-Moyal product and infact was recently shown [BLS10] to be equivalent to specific representations of Rieffelrsquosdeformed Clowast-algebras with IRn-action [Rie92] In application to field theory modelshowever one has to deform a whole family of algebras corresponding to subsystemslocalized in spacetime and the parameter Q has to be replaced by a family of matricesQ adapted to the geometry of the underlying spacetime

To apply this scheme to quantum field theories on curved manifolds we will considerspacetimes with a sufficiently large isometry group containing two commuting Killingfields which give rise to a representation of IR2 as required in (11) This setting is wideenough to encompass a number of cosmologically relevant manifolds such as Friedmann-Robertson-Walker spacetimes or Bianchi models Making use of the algebraic frame-work of quantum field theory [Haa96 Ara99] we can then formulate quantum fieldtheories in an operator-algebraic language and study their deformations Despite thefact that the warped convolution was invented for the deformation of Minkowski spacequantum field theories it turns out that all reference to the particular structure of flatspacetime such as Poincare transformations and a Poincare invariant vacuum state canbe avoided

We are interested in understanding to what extent the familiar structure of quantumfield theories on curved spacetimes is preserved under such deformations and investigatein particular covariance and localization properties Concerning locality it is known thatin warped models on Minkowski space point-like localization is weakened to localizationin certain infinitely extended wedge-shaped regions [GL07 BS08 GL08 BLS10] Theseregions are defined as Poincare transforms of the Rindler wedge

WR = (x0 x1 x2 x3) isin IR4 x1 gt |x0| (12)

Because of their intimate relation to the Poincare symmetry of Minkowski spacetime it isnot obvious what a good replacement for such a collection of regions is in the presence

of non-vanishing curvature In fact different definitions are possible and wedges onspecial manifolds have been studied by many authors in the literature [Kay85 BB99Reh00 BMS01 GLRV01 BS04 LR07 Str08 Bor09]

In Section 2 the first main part of our investigation we show that on those four-dimensional curved spacetimes which allow for the application of the deformation meth-ods in [BLS10] and thus carry two commuting Killing fields there also exists a familyof wedges with causal properties analogous to the Minkowski space wedges Because ofthe prominent role wedges play in many areas of Minkowski space quantum field the-ory [BW75 Bor92 Bor00 BDFS00 BGL02] this geometric and manifestly covariantconstruction is also of interest independently of its relation to deformations

In Section 3 we then consider quantum field theories on curved spacetimes anddeform them by warped convolution We first show that these deformations can becarried through in a model-independent operator-algebraic framework and that theemerging models share many structural properties with deformations of field theorieson flat spacetime (Section 31) In particular deformed quantum fields are localized inthe wedges of the considered spacetime These and further aspects of deformed quantumfield theories are also discussed in the concrete example of a Dirac field in Section 32Section 4 contains our conclusions

2 Geometric setup

To prepare the ground for our discussion of deformations of quantum field theories oncurved backgrounds we introduce in this section a suitable class of spacetimes andstudy their geometrical properties In particular we show how the concept of wedgesknown from Minkowski space generalizes to these manifolds Recall in preparationthat a wedge in four-dimensional Minkowski space is a region which is bounded bytwo non-parallel characteristic hyperplanes [TW97] or equivalently a region whichis a connected component of the causal complement of a two-dimensional spacelikeplane The latter definition has a natural analogue in the curved setting Making use ofthis observation we construct corresponding wedge regions in Section 21 and analysetheir covariance causality and inclusion properties At the end of that section wecompare our notion of wedges to other definitions which have been made in the literature[BB99 BMS01 BS04 LR07 Bor09] and point out the similarities and differences

In Section 22 the abstract analysis of wedge regions is complemented by a numberof concrete examples of spacetimes fulfilling our assumptions

21 Edges and wedges in curved spacetimes

In the following a spacetime (M g) is understood to be a four-dimensional Hausdorff(arcwise) connected smooth manifold M endowed with a smooth Lorentzian metric gwhose signature is (+minusminusminus) Notice that it is automatically guaranteed that M isalso paracompact and second countable [Ger68 Ger70] The (open) causal complementof a set O subM is defined as

Oprime = M[J+(O) cup Jminus(O)

] (21)

where Jplusmn(O) is the causal future respectively past of O in M [Wal84 Section 81]To avoid pathological geometric situations such as closed causal curves and also to

define a full-fledged Cauchy problem for a free field theory whose dynamics is determinedby a second order hyperbolic partial differential equation we will restrict ourselves toglobally hyperbolic spacetimes So in particular M is orientable and time-orientableand we fix both orientations While this setting is standard in quantum field theoryon curved backgrounds we will make additional assumptions regarding the structure ofthe isometry group Iso(M g) of (M g) motivated by our desire to define wedges in Mwhich resemble those in Minkowski space

Our most important assumption on the structure of (M g) is that it admits twolinearly independent spacelike complete commuting smooth Killing fields ξ1 ξ2 whichwill later be essential in the context of deformed quantum field theories We refer hereand in the following always to pointwise linear independence which entails in particularthat these vector fields have no zeros Denoting the flows of ξ1 ξ2 by ϕξ1 ϕξ2 the orbitof a point p isinM is a smooth two-dimensional spacelike embedded submanifold of M

E = ϕξ1s1(ϕξ2s2(p)) isinM s1 s2 isin IR (22)

where s1 s2 are the flow parameters of ξ1 ξ2Since M is globally hyperbolic it is isometric to a smooth product manifold IRtimesΣ

where Σ is a smooth three-dimensional embedded Cauchy hypersurface It is knownthat the metric splits according to g = βdT 2minush with a temporal function T IRtimesΣrarr IRand a positive function β isin Cinfin(IRtimes Σ (0infin)) while h induces a smooth Riemannianmetric on Σ [BS05 Thm 21] We assume that with E as in (22) the Cauchy surfaceΣ is smoothly homeomorphic to a product manifold I timesE where I is an open intervalor the full real line Thus M sim= IRtimes I timesE and we require in addition that there existsa smooth embedding ι IRtimes I rarrM By our assumption on the topology of I it followsthat (IRtimes I ιlowastg) is a globally hyperbolic spacetime without null focal points a featurethat we will need in the subsequent construction of wedge regions

Definition 21 A spacetime (M g) is called admissible if it admits two linearly indepen-dent spacelike complete commuting smooth Killing fields ξ1 ξ2 and the correspondingsplit M sim= IRtimes I times E with E defined in (22) has the properties described above

The set of all ordered pairs ξ = (ξ1 ξ2) satisfying these conditions for a given ad-missible spacetime (M g) is denoted Ξ(M g) The elements of Ξ(M g) will be referredto as Killing pairs

For the remainder of this section we will work with an arbitrary but fixed admissiblespacetime (M g) and usually drop the (M g)-dependence of various objects in ournotation eg write Ξ instead of Ξ(M g) for the set of Killing pairs and Iso in placeof Iso(M g) for the isometry group Concrete examples of admissible spacetimes suchas Friedmann-Robertson-Walker- Kasner- and Bianchi-spacetimes will be discussed inSection 22

The flow of a Killing pair ξ isin Ξ is written as

ϕξs = ϕξ1s1 ϕξ2s2 = ϕξ2s2 ϕξ1s1 ξ = (ξ1 ξ2) isin Ξ s = (s1 s2) isin IR2 (23)

where s1 s2 isin IR are the parameters of the (complete) flows ϕξ1 ϕξ2 of ξ1 ξ2 Byconstruction each ϕξ is an isometric IR2-action by diffeomorphisms on (M g) ieϕξs isin Iso and ϕξsϕξu = ϕξs+u for all s u isin IR2

On the set Ξ the isometry group Iso and the general linear group GL(2 IR) act in anatural manner

Lemma 22 Let h isin Iso N isin GL(2 IR) and define ξ = (ξ1 ξ2) isin Ξ

hlowastξ = (hlowastξ1 hlowastξ2) (24)

(Nξ)(p) = N(ξ1(p) ξ2(p)) p isinM (25)

These operations are commuting group actions of Iso and GL(2 IR) on Ξ respectivelyThe GL(2 IR)-action transforms the flow of ξ isin Ξ according to s isin IR2

ϕNξs = ϕξNT s (26)

If hlowastξ = Nξ for some ξ isin Ξ h isin Iso N isin GL(2 IR) then detN = plusmn1

Proof Due to the standard properties of isometries Iso acts on the Lie algebra ofKilling fields by the push-forward isomorphisms ξ1 7rarr hlowastξ1 [OrsquoN83] Therefore forany (ξ1 ξ2) isin Ξ also the vector fields hlowastξ1 hlowastξ2 are spacelike complete commutinglinearly independent smooth Killing fields The demanded properties of the splittingM sim= IR times I times E directly carry over to the corresponding split with respect to hlowastξ Sohlowast maps Ξ onto Ξ and since hlowast(klowastξ1) = (hk)lowastξ1 for h k isin Iso we have an action of Iso

The second map ξ 7rarr Nξ amounts to taking linear combinations of the Killing fieldsξ1 ξ2 The relation (26) holds because ξ1 ξ2 commute and are complete which entailsthat the respective flows can be constructed via the exponential map Since detN 6= 0the two components of Nξ are still linearly independent and since E (22) is invariantunder ξ 7rarr Nξ the splitting M sim= IRtimes I timesE is the same for ξ and Nξ Hence Nξ isin Ξie ξ 7rarr Nξ is a GL(2 IR)-action on Ξ and since the push-forward is linear it is clearthat the two actions commute

To prove the last statement we consider the submanifold E (22) together with itsinduced metric Since the Killing fields ξ1 ξ2 are tangent to E their flows are isometriesof E Since hlowastξ = Nξ and E is two-dimensional it follows that N acts as an isometryon the tangent space TpE p isin E But as E is spacelike and two-dimensional we canassume without loss of generality that the metric of TpE is the Euclidean metric andtherefore has the two-dimensional Euclidean group E(2) as its isometry group ThusN isin GL(2 IR) cap E(2) = O(2) ie detN = plusmn1

The GL(2 IR)-transformation given by the flip matrix Π =(

0 11 0

)will play a special

role later on We therefore reserve the name inverted Killing pair of ξ = (ξ1 ξ2) isin Ξ for

ξprime = Πξ = (ξ2 ξ1) (27)

Note that since we consider ordered tuples (ξ1 ξ2) the Killing pairs ξ and ξprime are notidentical Clearly the map ξ 7rarr ξprime is an involution on Ξ ie (ξprime)prime = ξ

After these preparations we turn to the construction of wedge regions in M and beginby specifying their edges

Definition 23 An edge is a subset of M which has the form

Eξp = ϕξs(p) isinM s isin IR2 (28)

for some ξ isin Ξ p isinM Any spacelike vector nξp isin TpM which completes the gradient ofthe chosen temporal function and the Killing vectors ξ1(p) ξ2(p) to a positively orientedbasis (nablaT (p) ξ1(p) ξ2(p) nξp) of TpM is called an oriented normal of Eξp

It is clear from this definition that each edge is a two-dimensional spacelike smoothsubmanifold of M Our definition of admissible spacetimes M sim= IR times I times E explicitlyrestricts the topology of I but not of the edge (22) which can be homeomorphic to aplane cylinder or torus

Note also that the description of the edge Eξp in terms of ξ and p is somewhat redun-dant Replacing the Killing fields ξ1 ξ2 by linear combinations ξ = Nξ N isin GL(2 IR)or replacing p by p = ϕξu(p) with some u isin IR2 results in the same manifold Eξp = Eξp

Before we define wedges as connected components of causal complements of edges wehave to prove the following key lemma from which the relevant properties of wedges fol-low For its proof it might be helpful to visualize the geometrical situation as sketchedin Figure 1

Figure 1 Three-dimensional sketch of the wedge Wξp and its edge Eξp

Lemma 24 The causal complement E primeξp of an edge Eξp is the disjoint union of twoconnected components which are causal complements of each other

Proof We first show that any point q isin E primeξp is connected to the base point p by a smoothspacelike curve Since M is globally hyperbolic there exist Cauchy surfaces ΣpΣq

passing through p and q respectively We pick two compact subsets Kq sub Σq containingq and Kp sub Σp containing p If Kp Kq are chosen sufficiently small their union KpcupKq

is an acausal compact codimension one submanifold of M It thus fulfils the hypothesisof Thm 11 in [BS06] which guarantees that there exists a spacelike Cauchy surfaceΣ containing the said union In particular there exists a smooth spacelike curve γconnecting p = γ(0) and q = γ(1) Picking spacelike vectors v isin TpΣ and w isin TqΣ we

have the freedom of choosing γ in such a way that γ(0) = v and γ(1) = w If v andw are chosen linearly independent from ξ1(p) ξ2(p) and ξ1(q) ξ2(q) respectively thesevectors are oriented normals of Eξp respectively Eξq and we can select γ such that itintersects the edge Eξp only in p

Let us define the region

Wξp = q isin E primeξp exist γ isin C1([0 1]M) with γ(0) = p γ(1) = q Eξp cap γ = pγ(0) is an oriented normal of Eξp γ(1) is an oriented normal of Eξq (29)

and exchanging ξ with the inverted Killing pair ξprime we correspondingly define the regionWξprimep It is clear from the above argument that Wξp cupWξprimep = E primeξp and that we canprescribe arbitrary normals nm of Eξp Eξq as initial respectively final tangent vectorsof the curve γ connecting p to q isin Wξp

The proof of the lemma consists in establishing that Wξp and Wξprimep are disjoint andcausal complements of each other To prove disjointness of WξpWξprimep assume thereexists a point q isin Wξp capWξprimep Then q can be connected with the base point p by twospacelike curves whose tangent vectors satisfy the conditions in (29) with ξ respectivelyξprime By joining these two curves we have identified a continuous loop λ in E primeξp As an

oriented normal the tangent vector λ(0) at p is linearly independent of ξ1(p) ξ2(p) sothat λ intersects Eξp only in p

Recall that according to Definition 21 M splits as the product M sim= IRtimes I times Eξpwith an open interval I which is smoothly embedded in M Hence we can considerthe projection π(λ) of the loop λ onto I which is a closed interval π(λ) sub I becausethe simple connectedness of I rules out the possibility that π(λ) forms a loop and onaccount of the linear independence of ξ1(p) ξ2(p) nξp the projection cannot be justa single point Yet as λ is a loop there exists pprime isin λ such that π(pprime) = π(p) We alsoknow that πminus1(π(p)) = IRtimes π(p) times Eξp is contained in J+(Eξp) cup Eξp cup Jminus(Eξp)and since p and pprime are causally separated the only possibility left is that they both lieon the same edge Yet per construction we know that the loop intersects the edge onlyonce at p and thus p and pprime must coincide which is the sought contradiction

To verify the claim about causal complements assume there exist points q isin Wξpqprime isin Wξprimep and a causal curve γ connecting them γ(0) = q γ(1) = qprime By definition of thecausal complement it is clear that γ does not intersect Eξp In view of our restrictionon the topology of M it follows that γ intersects either J+(Eξp) or Jminus(Eξp) Thesetwo cases are completely analogous and we consider the latter one where there existsa point qprimeprime isin γ cap Jminus(Eξp) In this situation we have a causal curve connecting q isin Wξp

with qprimeprime isin Jminus(Eξp) and since q isin Jminus(qprimeprime) sub Jminus(Eξp) it follows that γ must be pastdirected As the time orientation of γ is the same for the whole curve it follows thatalso the part of γ connecting qprimeprime and qprime is past directed Hence qprime isin Jminus(qprimeprime) sub Jminus(Eξp)which is a contradiction to qprime isin Wξprimep Thus Wξprimep sub Wξp

To show that Wξprimep coincides with Wξpprime let q isin Wξp

prime = Wξpprime sub E primeξp = Wξp tWξprimep

Yet q isin Wξp is not possible since q isin Wξpprime and Wξp is open So q isin Wξprimep ie we have

shown Wξpprime sub Wξprimep and the claimed identity Wξprimep = Wξp

prime follows

Eprimeξp in a Lorentz cylinder

Lemma 24 does not hold if the topological require-ments on M are dropped As an example considera cylinder universe IR times S1 times IR2 the product of theLorentz cylinder IR times S1 [OrsquoN83] and the Euclideanplane IR2 The translations in the last factor IR2 de-fine spacelike complete commuting linearly indepen-dent Killing fields ξ Yet the causal complement of theedge Eξp = 0 times 1 times IR2 has only a single con-nected component which has empty causal complementIn this situation wedges lose many of the useful prop-erties which we establish below for admissible space-times

In view of Lemma 24 wedges in M can be defined as follows

Definition 25 (Wedges)A wedge is a subset of M which is a connected component of the causal complement ofan edge in M Given ξ isin Ξ p isin M we denote by Wξp the component of E primeξp whichintersects the curves γ(t) = expp(t nξp) t gt 0 for any oriented normal nξp of EξpThe family of all wedges is denoted

W = Wξp ξ isin Ξ p isinM (210)

As explained in the proof of Lemma 24 the condition that the curve IR+ 3 t 7rarrexpp(t nξp) intersects a connected component of E primeξp is independent of the chosen normalnξp and each such curve intersects precisely one of the two components of E primeξp

Some properties of wedges which immediately follow from the construction carriedout in the proof of Lemma 24 are listed in the following proposition

Proposition 26 (Properties of wedges)Let W = Wξp be a wedge Then

a) W is causally complete ie W primeprime = W and hence globally hyperbolic

b) The causal complement of a wedge is given by inverting its Killing pair

(Wξp)prime = Wξprimep (211)

c) A wedge is invariant under the Killing flow generating its edge

ϕξs(Wξp) = Wξp s isin IR2 (212)

Proof a) By Lemma 24 W is the causal complement of another wedge V and thereforecausally complete W primeprime = V primeprimeprime = V prime = W Since M is globally hyperbolic this impliesthat W is globally hyperbolic too [Key96 Prop 125]

b) This statement has already been checked in the proof of Lemma 24

c) By definition of the edge Eξp (28) we have ϕξs(Eξp) = Eξp for any s isin IR2and since the ϕξs are isometries it follows that ϕξs(E

primeξp) = E primeξp Continuity of the flow

implies that also the two connected components of this set are invariant

Corollary 27 (Properties of the family of wedge regions)The family W of wedge regions is invariant under the isometry group Iso and undertaking causal complements For h isin Iso it holds

h(Wξp) = Whlowastξh(p) (213)

Proof Since isometries preserve the causal structure of a spacetime we only need tolook at the action of isometries on edges We find

hEξp = h ϕξs hminus1(h(p)) s isin IR2 = ϕhlowastξs(h(p)) s isin IR2 = Ehlowastξh(p) (214)

by using the well-known fact that conjugation of flows by isometries amounts to thepush-forward by the isometry of the associated vector field Since hlowastξ isin Ξ for anyξ isin Ξ h isin Iso (Lemma 22) the family W is invariant under the action of the isometrygroup Closedness of W under causal complementation is clear from Prop 26 b)

In contrast to the situation in flat spacetime the isometry group Iso does not acttransitively on W(M g) for generic admissible M and there is no isometry mapping agiven wedge onto its causal complement This can be seen explicitly in the examplesdiscussed in Section 22 To keep track of this structure of W(M g) we decomposeΞ(M g) into orbits under the Iso- and GL(2 IR)-actions

Definition 28 Two Killing pairs ξ ξ isin Ξ are equivalent written ξ sim ξ if there existh isin Iso and N isin GL(2 IR) such that ξ = Nhlowastξ

As ξ 7rarr Nξ and ξ 7rarr hlowastξ are commuting group actions sim is an equivalence relationAccording to Lemma 22 and Prop 26 b) c) acting with N isin GL(2 IR) on ξ eitherleaves WNξp = Wξp invariant (if detN gt 0) or exchanges this wedge with its causalcomplement WNξp = W prime

ξp (if detN lt 0) Therefore the ldquocoherentrdquo1 subfamilies arisingin the decomposition of the family of all wedges along the equivalence classes [ξ] isin Ξsim

W =⊔[ξ]

W[ξ] W[ξ] = Wξp ξ sim ξ p isinM (215)

take the form

W[ξ] = Whlowastξp Wprimehlowastξp h isin Iso p isinM (216)

In particular each subfamily W[ξ] is invariant under the action of the isometry groupand causal complementation

1See [BS07] for a related notion on Minkowski space

In our later applications to quantum field theory it will be important to have controlover causal configurations W1 sub W prime

2 and inclusions W1 sub W2 of wedges W1W2 isin W SinceW is closed under taking causal complements it is sufficient to consider inclusionsNote that the following proposition states in particular that inclusions can only occurbetween wedges from the same coherent subfamily W[ξ]

Proposition 29 (Inclusions of wedges)Let ξ ξ isin Ξ p p isin M The wedges Wξp and Wξp form an inclusion Wξp sub Wξp if

and only if p isin Wξp and there exists N isin GL(2 IR) with detN gt 0 such that ξ = Nξ

Proof (lArr) Let us assume that ξ = Nξ holds for some N isin GL(2 IR) with detN gt 0and p isin Wξp In this case the Killing fields in ξ are linear combinations of those in ξand consequently the edges Eξp and Eξp intersect if and only if they coincide ie ifp isin Eξp If the edges coincide we clearly have Wξp = Wξp If they do not coincide it

follows from p isin Wξp that Eξp and Eξp are either spacelike separated or they can beconnected by a null geodesic

Consider now the case that Eξp and Eξp are spacelike separated ie p isin Wξp Picka point q isin Wξp and recall that Wξp can be characterized by equation (29) Sincep isin Wξp and q isin Wξp there exist curves γp and γq which connect the pairs of points(p p) and (p q) respectively and comply with the conditions in (29) By joining γpand γq we obtain a curve which connects p and q The tangent vectors γp(1) and γq(0)are oriented normals of Eξp and we choose γp and γq in such a way that these tangentvectors coincide Due to the properties of γp and γq the joint curve also complies withthe conditions in (29) from which we conclude q isin Wξp and thus Wξp sub Wξp

Consider now the case that Eξp and Eξp are connected by null geodesics ie p isinpartWξp Let r be the point in Eξp which can be connected by a null geodesic with p and

pick a point q isin Wξp The intersection Jminus(r) cap partWξp yields another null curve say microand the intersection micro cap Jminus(q) = pprime is non-empty since r and q are spacelike separatedand q isin Wξp The null curve micro is chosen future directed and parametrized in such a waythat micro(0) = pprime and micro(1) = r By taking ε isin (0 1) we find q isin Wξmicro(ε) and micro(ε) isin Wξp

which entails q isin Wξp(rArr) Let us assume that we have an inclusion of wedges Wξp sub Wξp Then clearly

p isin Wξp Since M is four-dimensional and ξ1 ξ2 ξ1 ξ2 are all spacelike they cannot belinearly independent Let us first assume that three of them are linearly independentand without loss of generality let ξ = (ξ1 ξ2) and ξ = (ξ2 ξ3) with three linearlyindependent spacelike Killing fields ξ1 ξ2 ξ3 Picking points q isin Eξp q isin Eξp these can

be written as q = (t x1 x2 x3) and q = (t x1 x2 x3) in the global coordinate system offlow parameters constructed from ξ1 ξ2 ξ3 and the gradient of the temporal function

For suitable flow parameters s1 s2 s3 we have ϕξ1s1(q) = (t x1 x2 x3) = qprime isin Eξpand ϕ(ξ2ξ3)(s2s3)(q) = (t x1 x2 x3) = qprime isin Eξp Clearly the points qprime and qprime areconnected by a timelike curve eg the curve whose tangent vector field is given bythe gradient of the temporal function But a timelike curve connecting the edges ofWξpWξp is a contradiction to these wedges forming an inclusion So no three of the

vector fields ξ1 ξ2 ξ1 ξ2 can be linearly independent

Hence ξ = Nξ with an invertible matrix N It remains to establish the correct signof detN and to this end we assume detN lt 0 Then we have (Wξp)

prime = Wξprimep sub Wξp

by (Prop 26 b)) and the (lArr) statement in this proof since ξ and ξprime are related by apositive determinant transformation and p isin Wξp This yields that both Wξp and itscausal complement must be contained in Wξp a contradiction Hence detN gt 0 andthe proof is finished

Having derived the structural properties of the setW of wedges needed later we nowcompare our wedge regions to the Minkowski wedges and to other definitions proposedin the literature

The flat Minkowski spacetime (IR4 η) clearly belongs to the class of admissible space-times with translations along spacelike directions and rotations in the standard timezero Cauchy surface as its complete spacelike Killing fields However as Killing pairsconsist of non-vanishing vector fields and each rotation leaves its rotation axis invariantthe set Ξ(IR4 η) consists precisely of all pairs (ξ1 ξ2) such that the flows ϕξ1 ϕξ2 aretranslations along two linearly independent spacelike directions Hence the set of alledges in Minkowski space coincides with the set of all two-dimensional spacelike planesConsequently each wedge W isin W(IR4 η) is bounded by two non-parallel characteristicthree-dimensional planes This is precisely the family of wedges usually considered inMinkowski space2 (see for example [TW97])

Besides the features we established above in the general admissible setting the familyof Minkowski wedges has the following well-known properties

a) Each wedge W isin W(IR4 η) is the causal completion of the world line of a uniformlyaccelerated observer

b) Each wedge W isin W(IR4 η) is the union of a family of double cones whose tips lieon two fixed lightrays

c) The isometry group (the Poincare group) acts transitively on W(IR4 η)

d) W(IR4 η) is causally separating in the sense that given any two spacelike separateddouble cones O1 O2 sub IR4 then there exists a wedge W such that O1 sub W sub Oprime2[TW97] W(IR4 η) is a subbase for the topology of IR4

All these properties a)ndashd) do not hold for the class W(M g) of wedges on a generaladmissible spacetime but some hold for certain subclasses as can be seen from theexplicit examples in the subsequent section

There exist a number of different constructions for wedges in curved spacetimes in theliterature mostly for special manifolds On de Sitter respectively anti de Sitter spaceBorchers and Buchholz [BB99] respectively Buchholz and Summers [BS04] construct

2Note that we would get a ldquotoo largerdquo family of wedges in Minkowski space if we would drop therequirement that the vector fields generating edges are Killing However the assumption that edgesare generated by commuting Killing fields is motivated by the application to deformations of quantumfield theories and one could generalize our framework to spacetimes with edges generated by completelinearly independent smooth Killing fields

wedges by taking property a) as their defining feature see also the generalization byStrich [Str08] In the de Sitter case this definition is equivalent to our definition of awedge as a connected component of the causal complement of an edge [BMS01] Butas two-dimensional spheres the de Sitter edges do not admit two linearly independentcommuting Killing fields Apart from this difference due to our restriction to commutinglinearly independent Killing fields the de Sitter wedges can be constructed in the sameway as presented here Thanks to the maximal symmetry of the de Sitter and anti deSitter spacetimes the respective isometry groups act transitively on the correspondingwedge families (c) and causally separate in the sense of d)

A definition related to the previous examples has been given by Lauridsen-Ribeirofor wedges in asymptotically anti de Sitter spacetimes (see Def 15 in [LR07]) Notethat these spacetimes are not admissible in our sense since anti de Sitter space is notglobally hyperbolic

Property b) has recently been taken by Borchers [Bor09] as a definition of wedgesin a quite general class of curved spacetimes which is closely related to the structure ofdouble cones In that setting wedges do not exhibit in general all of the features wederived in our framework and can for example have compact closure

Wedges in a class of Friedmann-Robertson-Walker spacetimes with spherical spatialsections have been constructed with the help of conformal embeddings into de Sitterspace [BMS01] This construction also yields wedges defined as connected componentsof causal complements of edges Here a) does not but c) and d) do hold see also ourdiscussion of Friedmann-Robertson-Walker spacetimes with flat spatial sections in thenext section

The idea of constructing wedges as connected components of causal complementsof specific two-dimensional submanifolds has also been used in the context of globallyhyperbolic spacetimes with a bifurcate Killing horizon [GLRV01] building on earlierwork in [Kay85] Here the edge is given as the fixed point manifold of the Killing flowassociated with the horizon

22 Concrete examples

In the previous section we provided a complete but abstract characterization of thegeometric structures of the class of spacetimes we are interested in This analysis is nowcomplemented by presenting a number of explicit examples of admissible spacetimes

The easiest way to construct an admissble spacetime is to take the warped product[OrsquoN83 Chap 7] of an edge with another manifold Let (E gE) be a two-dimensionalRiemannian manifold endowed with two complete commuting linearly independentsmooth Killing fields and let (X gX) be a two-dimensional globally hyperbolic space-time diffeomorphic to IRtimes I with I an open interval or the full real line Then given apositive smooth function f on X consider the warped product M = X timesf E ie theproduct manifold X times E endowed with the metric tensor field

g = πlowastX(gX) + (f πX) middot πlowastE(gE)

where πX M rarr X and πE M rarr E are the projections on X and E It readily followsthat (M g) is admissible in the sense of Definition 21

The following proposition describes an explicit class of admissible spacetimes in termsof their metrics

Proposition 210 Let (M g) be a spacetime diffeomorphic to IRtimes I times IR2 where I subeIR is open and simply connected endowed with a global coordinate system (t x y z)according to which the metric reads

ds2 = e2f0dt2 minus e2f1dx2 minus e2f2dy2 minus e2f3(dz minus q dy)2 (217)

Here t runs over the whole IR fi q isin Cinfin(M) for i = 0 3 and fi q do not depend ony and z Then (M g) is an admissible spacetime in the sense of Definition 21

Proof Per direct inspection of (217) M is isometric to IRtimes Σ with Σ sim= I times IR2 withds2 = β dt2minushijdxidxj where β is smooth and positive and h is a metric which dependssmoothly on t Furthermore on the hypersurfaces at constant t deth = e2(f1+f2+f3) gt 0and h is block-diagonal If we consider the sub-matrix with i j = y z this has a positivedeterminant and a positive trace Hence we can conclude that the induced metric on Σ isRiemannian or in other words Σ is a spacelike smooth three-dimensional Riemannianhypersurface Therefore we can apply Theorem 11 in [BS05] to conclude that M isglobally hyperbolic

Since the metric coefficients are independent from y and z the vector fields ξ1 = partyand ξ2 = partz are smooth Killing fields which commute and as they lie tangent to theRiemannian hypersurfaces at constant time they are also spacelike Furthermore sinceper definition of spacetime M and thus also Σ is connected we can invoke the Hopf-Rinow-Theorem [OrsquoN83 sect 5 Thm 21] to conclude that Σ is complete and thus allits Killing fields are complete As I is simply connected by assumption it follows that(M g) is admissible

Under an additional assumption also a partial converse of Proposition 210 is trueNamely let (M g) be a globally hyperbolic spacetime with two complete spacelike com-muting smooth Killing fields and pick a local coordinate system (t x y z) where y andz are the flow parameters of the Killing fields Then if the reflection map r M rarrM r(t x y z) = (t xminusyminusz) is an isometry the metric is locally of the form (217) aswas proven in [Cha83 CF84] The reflection r is used to guarantee the vanishing ofthe unwanted off-diagonal metric coefficients namely those associated to ldquodx dyrdquo andldquodx dzrdquo Notice that the cited papers allow only to establish a result on the local struc-ture of M and no a priori condition is imposed on the topology of I in distinction toProposition 210

Some of the metrics (217) are used in cosmology For the description of a spatiallyhomogeneous but in general anisotropic universe M sim= J times IR3 where J sube IR (see sect5 in[Wal84] and [FPH74]) one puts f0 = q = 0 in (217) and takes f1 f2 f3 to depend onlyon t This yields the metric of Kasner spacetimes respectively Bianchi I models3

ds2 = dt2 minus e2f1dx2 minus e2f2dy2 minus e2f3dz2 (218)

3 The Bianchi models IndashIX [Ell06] arise from the classification of three-dimensional real Lie algebras

Clearly here the isometry group contains three smooth Killing fields locally given bypartx party partz which are everywhere linearly independent complete and commuting In par-ticular (partx party) (partx partz) and (party partz) are Killing pairs

A case of great physical relevance arises when specializing the metric further by takingall the functions fi in (218) to coincide In this case the metric assumes the so-calledFriedmann-Robertson-Walker form

ds2 = dt2 minus a(t)2 [dx2 + dy2 + dz2] = a(t(τ))2[dτ 2 minus dx2 minus dy2 minus dz2

] (219)

Here the scale factor a(t) = ef1(t) is defined on some interval J sube IR and in thesecond equality we have introduced the conformal time τ which is implicitely definedby dτ = aminus1(t)dt Notice that as in the Bianchi I model the manifold is M sim= J times IR3ie the variable t does not need to range over the whole real axis (This does not affectthe property of global hyperbolicity)

By inspection of (219) it is clear that the isometry group of this spacetime containsthe three-dimensional Euclidean group E(3) = O(3) o IR3 Disregarding the Minkowskicase where J = IR and a is constant the isometry group in fact coincides with E(3)Edges in such a Friedmann-Robertson-Walker universe are of the form τtimesS where S isa two-dimensional plane in IR3 and t(τ) isin J HereW consists of a single coherent familyand the Iso-orbits inW are labelled by the time parameter τ for the corresponding edgesAlso note that the family of Friedmann-Robertson-Walker wedges is causally separatingin the sense discussed on page 11

The second form of the metric in (219) is manifestly a conformal rescaling of theflat Minkowski metric Interpreting the coordinates (τ x y z) as coordinates of a pointin IR4 therefore gives rise to a conformal embedding ι M rarr IR4

Two wedges in FRW spacetime

In this situation it is interesting to note that theset of all images ι(E) of edges E in the Friedmann-Robertson-Walker spacetime coincides with the setof all Minkowski space edges which lie completelyin ι(M) = J times IR3 provided that J does not co-incide with IR These are just the edges parallelto the standard Cauchy surfaces of constant τ inIR4 So Friedmann-Robertson-Walker edges canalso be characterized in terms of Minkowski spaceedges and the conformal embedding ι analogousto the construction of wedges in Friedmann-Robertson-Walker spacetimes with sphericalspatial sections in [BMS01]

thought of as Lie subalgebras of the Lie algebra of Killing fields Only the cases Bianchi IndashVII in whichthe three-dimensional Lie algebra contains IR2 as a subalgebra are of direct interest here since only inthese cases Killing pairs exist

3 Quantum field theories on admissible spacetimes

Having discussed the relevant geometric structures we now fix an admissible spacetime(M g) and discuss warped convolution deformations of quantum field theories on it Formodels on flat Minkowski space it is known that this deformation procedure weakenspoint-like localization to localization in wedges [BLS10] and we will show here thatthe same holds true for admissible curved spacetimes For a convenient description ofthis weakened form of localization and for a straightforward application of the warpedconvolution technique we will work in the framework of local quantum physics [Haa96]

In this setting a model theory is defined by a net of field algebras and here weconsider algebras F(W ) of quantum fields supported in wedges W isin W(M g) Themain idea underlying the deformation is to apply the formalism developed in [BS08BLS10] but with the global translation symmetries of Minkowski space replaced bythe Killing flow ϕξ corresponding to the wedge W = Wξp under consideration In thecase of Minkowski spacetime these deformations reduce to the familiar structure of anoncommutative Minkowski space with commuting time

The details of the model under consideration will not be important in Section 31since our construction relies only on a few structural properties satisfied in any well-behaved quantum field theory In Section 32 the deformed Dirac quantum field ispresented as a particular example

31 Deformations of nets with Killing symmetries

Proceeding to the standard mathematical formalism [Haa96 Ara99] we consider a Clowast-algebra F whose elements are interpreted as (bounded functions of) quantum fields onthe spacetime M The field algebra F has a local structure and in the present contextwe focus on localization in wedges W isin W since this form of localization turns outto be stable under the deformation Therefore corresponding to each wedge W isin W we consider the Clowast-subalgebra F(W ) sub F of fields supported in W Furthermore weassume a strongly continuous action α of the isometry group Iso of (M g) on F anda BoseFermi automorphism γ whose square is the identity automorphism and whichcommutes with α This automorphism will be used to separate the BoseFermi parts offields F isin F in the model theory of a free Dirac field discussed later it can be chosenas a rotation by 2π in the Dirac bundle

To allow for a straightforward application of the results of [BLS10] we will alsoassume in the following that the field algebra is concretely realized on a separable Hilbertspace H which carries a unitary representation U of Iso implementing the action α ie

U(h)FU(h)minus1 = αh(F ) h isin Iso F isin F

We emphasize that despite working on a Hilbert space we do not select a state sincewe do not make any assumptions regarding U -invariant vectors in H or the spectrumof subgroups of the representation U 4 The subsequent analysis will be carried out in aClowast-setting without using the weak closures of the field algebras F(W ) in B(H)

4Note that every Clowast-dynamical system (A G α) where A sub B(H) is a concrete Clowast-algebra on aseparable Hilbert space H and α G rarr Aut(A) is a strongly continuous representation of the locally

For convenience we also require the BoseFermi automorphism γ to be unitarilyimplemented on H ie there exists a unitary V = V lowast = V minus1 isin B(H) such thatγ(F ) = V FV We will also use the associated unitary twist operator

Z =1radic2

(1minus iV ) (31)

Clearly the unitarily implemented α and γ can be continued to all of B(H) By a slightabuse of notation these extensions will be denoted by the same symbols

In terms of the data F(W )WisinW α γ the structural properties of a quantum fieldtheory on M can be summarized as follows [Haa96 Ara99]

a) Isotony F(W ) sub F(W ) whenever W sub W

b) Covariance under Iso

αh(F(W )) = F(hW ) h isin Iso W isin W (32)

c) Twisted Locality With the unitary Z (31) there holds

[ZFZlowast G] = 0 for F isin F(W ) G isin F(W prime) W isin W (33)

The twisted locality condition (33) is equivalent to normal commutation relationsbetween the BoseFermi parts Fplusmn = 1

2(Fplusmnγ(F )) of fields in spacelike separated wedges

[F+ Gplusmn] = [Fplusmn G+] = Fminus Gminus = 0 for F isin F(W ) G isin F(W prime) [DHR69]The covariance requirement (32) entails that for any Killing pair ξ isin Ξ the algebra

F carries a corresponding IR2-action τξ defined by

τξs = αϕξs = adUξ(s) s isin IR2

where Uξ(s) is shorthand for U(ϕξs) Since a wedge of the form Wξp with some p isinMis invariant under the flows ϕNξs for any N isin GL(2 IR) (see Prop 26 c) and Lemma22) we have in view of isotony

τNξs(F(Wξp)) = F(Wξp) N isin GL(2 IR) s isin IR2

In this setting all structural elements necessary for the application of warped con-volution deformations [BLS10] are present and we will use this technique to define adeformed net W 7minusrarr F(W )λ of Clowast-algebras on M depending on a deformation param-eter λ isin IR For λ = 0 we will recover the original theory F(W )0 = F(W ) and foreach λ isin IR the three basic properties a)ndashc) listed above will remain valid To achievethis the elements of F(W ) will be deformed with the help of the Killing flow leaving Winvariant We begin by recalling some definitions and results from [BLS10] adapted tothe situation at hand

compact group G has a covariant representation [Ped79 Prop 747 Lemma 749] build out of theleft-regular representation on the Hilbert space L2(G)otimesH

Similar to the Weyl product appearing in the quantization of classical systems thewarped convolution deformation is defined in terms of oscillatory integrals of F-valuedfunctions and we have to introduce the appropriate smooth elements first The action αis a strongly continuous representation of the Lie group Iso which acts automorphicallyand thus isometrically on the Clowast-algebra F In view of these properties the smoothelements Finfin = F isin F Iso 3 h 7rarr αh(F ) is middot F-smooth form a norm-dense lowast-subalgebra Finfin sub F (see for example [Tay86]) However the subalgebras F(Wξp) sub Fare in general only invariant under the IR2-action τξ and we therefore also introduce aweakened form of smoothness An operator F isin F will be called ξ-smooth if

IR2 3 s 7rarr τξs(F ) isin F (34)

is smooth in the norm topology of F On the Hilbert space level we have a dense domainHinfin = Ψ isin H Iso 3 h 7rarr U(h)Ψ is middot H-smooth of smooth vectors in H

As further ingredients for the definition of the oscillatory integrals we pick a smoothcompactly supported ldquocutoffrdquo function χ isin Cinfin0 (IR2 times IR2) with χ(0 0) = 1 and thestandard antisymmetric (2times 2)-matrix

(0 1minus1 0

) (35)

With these data we associate to a ξ-smooth F isin F the deformed operator (warpedconvolution) [BLS10]

Fξλ =1

4π2limεrarr0

intds dsprime eminusiss

primeχ(εs εsprime)Uξ(λQs)FUξ(s

prime minus λQs) (36)

where λ is a real parameter and ssprime denotes the standard Euclidean inner product ofs sprime isin IR2 The above limit exists in the strong operator topology of B(H) on the densesubspace Hinfin and is independent of the chosen cutoff function χ within the specifiedclass The thus (densely) defined operator Fξλ can be shown to extend to a boundedξ-smooth operator on all of H which we denote by the same symbol [BLS10] As canbe seen from the above formula setting λ = 0 yields the undeformed operator Fξ0 = F for any ξ isin Ξ

The deformation F rarr Fξλ is closely related to Rieffelrsquos deformation of Clowast-algebras[Rie92] where one introduces the deformed product

F timesξλ G =1

4π2limεrarr0

primeχ(εs εsprime) τξλQs(F )τξsprime(G) (37)

This limit exists in the norm topology of F for any ξ-smooth FG isin F and F timesξλ G isξ-smooth as well

As is well known this procedure applies in particular to the deformation of classicaltheories in terms of star products As field algebra one would then take a suitablecommutative lowast-algebra of functions on M endowed with the usual pointwise operationsThe isometry group acts on this algebra automorphically by pullback and in particularthe flow ϕξ of any Killing pair ξ isin Ξ induces automorphisms The Rieffel product

therefore defines a star product on the subalgebra of smooth elements f g for thisaction

(f ξλ g)(p) =1

4π2limεrarr0

intd2s d2sprimeeminusiss

primeχ(εs εsprime) f(ϕξλQs(p))g(ϕξsprime(p)) (38)

The function algebra endowed with this star product can be interpreted as a noncom-mutative version of the manifold M similar to the flat case [GGBI+04] Note that sincewe are using a two-dimensional spacelike flow on a four-dimensional spacetime the de-formation corresponds to a noncommutative Minkowski space with ldquocommuting timerdquoin the flat case

The properties of the deformation map F rarr Fξλ which will be relevant here are thefollowing

Lemma 31 [BLS10]Let ξ isin Ξ λ isin IR and consider ξ-smooth operators FG isin F Then

a) Fξλlowast = F lowastξλ

b) FξλGξλ = (F timesξλ G)ξλ

c) If 5 [τξs(F ) G] = 0 for all s isin IR2 then [Fξλ Gξminusλ] = 0

d) If a unitary Y isin B(H) commutes with Uξ(s) s isin IR2 then Y FξλYminus1 = (Y FY minus1)ξλ

and Y FξλYminus1 is ξ-smooth

Since we are dealing here with a field algebra obeying twisted locality we also pointout that statement c) of the above lemma carries over to the twisted local case

Lemma 32 Let ξ isin Ξ and FG isin F be ξ-smooth such that [Zτξs(F )Zlowast G] = 0 Then

[ZFξλZlowast Gξminusλ] = 0 (39)

Proof The BoseFermi operator V commutes with the representation of the isometrygroup and thus the same holds true for the twist Z (31) So in view of Lemma 31 d)the assumption implies that ZFZlowast is ξ-smooth and [τξs(ZFZ

lowast) G] = 0 for all s isin IR2In view of Lemma 31 c) we thus have [(ZFZlowast)ξλ Gξminusλ] = 0 But as Z and Uξ(s)commute (ZFZlowast)ξλ = ZFξλZ

lowast and the claim follows

The results summarized in Lemma 31 and Lemma 32 will be essential for estab-lishing the isotony and twisted locality properties of the deformed quantum field theoryTo also control the covariance properties relating different Killing pairs we need anadditional lemma closely related to [BLS10 Prop 29]

5In [BS08 BLS10] this statement is shown to hold under the weaker assumption that the commu-tator [τξs(F ) G] vanishes only for all s isin S+S where S is the joint spectrum of the generators of theIR2-representation Uξ implementing τξ But since usually S = IR2 in the present setting we refer hereonly to the weaker statement where S + S sub IR2 has been replaced by IR2

Lemma 33 Let ξ isin Ξ λ isin IR and F isin F be ξ-smooth

a) Let h isin Iso Then αh(F ) is hlowastξ-smooth and

αh(Fξλ) = αh(F )hlowastξλ (310)

b) For N isin GL(2 IR) it holds

FNξλ = FξdetN middotλ (311)

In particular

Fξprimeλ = Fξminusλ (312)

Proof a) The flow of ξ transforms under h according to hϕξs = ϕhlowastξsh so thatαh(τξs(F )) = τhlowastξs(αh(F )) Since F is ξ-smooth and αh is isometric the smoothnessof s 7rarr τhlowastξs(αh(F )) follows Using the strong convergence of the oscillatory integrals(36) we compute on a smooth vector Ψ isin Hinfin

αh(Fξλ)Ψ =1

4π2limεrarr0

primeχ(εs εsprime)U(hϕξλQsh

minus1)αh(F )U(hϕξsprimeminusλQshminus1)Ψ

4π2limεrarr0

primeχ(εs εsprime)U(ϕhlowastξλQs)αh(F )U(ϕhlowastξsprimeminusλQs)Ψ

= αh(F )hlowastξλΨ

which entails (310) since Hinfin sub H is denseb) In view of the transformation law ϕNξs = ϕξNT s (26) we get Ψ isin Hinfin

FNξλΨ =1

4π2limεrarr0

primeχ(εs εsprime)U(ϕNξλQs)FU(ϕNξsprimeminusλQs)Ψ

4π2| detN |limεrarr0

intds dsprime eminusi(N

minus1ssprime) χ(εs ε(NT )minus1sprime)Uξ(λNTQs)FUξ(s

prime minus λNTQs)Ψ

4π2limεrarr0

primeχ(εNs ε(NT )minus1sprime)Uξ(λN

TQNs)FUξ(sprime minus λNTQNs)Ψ

= FξdetN middotλΨ

In the last line we used the fact that the value of the oscillatory integral does notdepend on the choice of cutoff function χ or χN(s sprime) = χ(Ns (NT )minus1sprime) and theequation NTQN = detN middotQ which holds for any (2times 2)-matrix N

This proves (311) and since ξprime = Πξ with the flip matrix Π =(

0 11 0

)which has

det Π = minus1 also (312) follows

Having established these properties of individual deformed operators we now set outto deform the net W 7rarr F(W ) of wedge algebras In contrast to the Minkowski spacesetting [BLS10] we are here in a situation where the set Ξ of all Killing pairs is not asingle orbit of one reference pair under the isometry group Whereas the deformation

of a net of wedge algebras on Minkowski space amounts to deforming a single algebraassociated with a fixed reference wedge (Borchers triple) we have to specify here moredata related to the coherent subfamilies W[ξ] in the decomposition W =

⊔[ξ]W[ξ] of W

(215) For each equivalence class [ξ] we choose a representative ξ In case there existsonly a single equivalence class this simply amounts to fixing a reference wedge togetherwith a length scale for the Killing flow With this choice of representatives ξ isin [ξ] madewe introduce the sets p isinM

F(Wξp)λ = Fξλ F isin F(Wξp) ξ-smooth middot (313)

F(Wξprimep)λ = Fξprimeλ F isin F(W primeξp) ξ

prime-smooth middot (314)

Here λ isin IR is the deformation parameter and the superscript denotes norm closure inB(H) Note that the deformed operators in F(Wξprimep)λ have the form Fξprimeλ = Fξminusλ (312)ie the sign of the deformation parameter depends on the choice of reference Killingpair

The definitions (313 314) are extended to arbitrary wedges by setting

F(hWξp)λ = αh(F(Wξp)λ) F(hW primeξp)λ = αh(F(W prime

ξp)λ) (315)

Recall that as h p and [ξ] vary over Iso M and Ξsim respectively this defines F(W )λfor all W isin W (cf (216)) It has to be proven that this assignment is well-definedeg that (315) is independent of the way the wedge hWξp = hWξp is represented Thiswill be done below However note that the definition of F(W )λ does depend on ourchoice of representatives ξ isin [ξ] since rescaling ξ amounts to rescaling the deformationparameter (Lemma 33 b))

Before establishing the main properties of the assignment W rarr F(W )λ we check thatthe sets (313 314) are Clowast-algebras As the Clowast-algebra F(Wξp) is τξ-invariant and τξacts strongly continuously the ξ-smooth operators in F(Wξp) which appear in the defini-tion (313) form a norm-dense lowast-subalgebra Now the deformation F 7rarr Fξλ is evidentlylinear and commutes with taking adjoints (Lemma 31 a)) so the sets F(Wξp)λ arelowast-invariant norm-closed subspaces of B(H) To check that these spaces are also closedunder taking products we again use the invariance of F(Wξp) under τξ By inspectionof the Rieffel product (37) it follows that for any two ξ-smooth FG isin F(Wξp) alsothe product F timesξλ G lies in this algebra (and is ξ-smooth see [Rie92]) Hence themultiplication formula from Lemma 31 b) entails that the above defined F(W )λ areactually Clowast-algebras in B(H)

The map W 7rarr F(W )λ defines the wedge-local field algebras of the deformed quan-tum field theory Their basic properties are collected in the following theorem

Theorem 34 The above constructed map W 7minusrarr F(W )λ W isin W is a well-definedisotonous twisted wedge-local Iso-covariant net of Clowast-algebras on H ie W W isin W

F(W )λ sub F(W )λ for W sub W (316)

[ZFλZlowast Gλ] = 0 for Fλ isin F(W )λ Gλ isin F(W prime)λ (317)

αh(F(W )λ) = F(hW )λ h isin Iso (318)

For λ = 0 this net coincides with the original net F(W )0 = F(W ) W isin W

Proof It is important to note from the beginning that all claimed properties relate onlywedges in the same coherent subfamily W[ξ] This can be seen from the form (216) ofW[ξ] which is manifestly invariant under isometries and causal complementation andthe structure of the inclusions (Proposition 29) So in the following proof it is sufficientto consider a fixed but arbitrary equivalence class [ξ] with selected representative ξ

We begin with establishing the isotony of the deformed net and therefore considerinclusions of wedges of the form hWξp hW

primeξp with h isin Iso p isinM arbitrary and ξ isin Ξ

fixed Starting with the inclusions hWξp sube hWξp we note that according to (213) andProp 29 there exists N isin GL(2 IR) with positive determinant such that hlowastξ = NhlowastξEquivalently (hminus1h)lowastξ = Nξ which by Lemma 22 implies detN = 1 By definition ageneric ξ-smooth element of F(hWξp)λ is of the form αh(Fξλ) = αh(F )hlowastξλ with someξ-smooth F isin F(Wξp) But according to the above observation this can be rewrittenas

αh(Fξλ) = αh(F )hlowastξλ = αh(F )Nhlowastξλ = αh(F )hlowastξλ (319)

where in the last equation we used detN = 1 and Lemma 33 b) Taking into accountthat hWξp sube hWξp and that the undeformed net is covariant and isotonous we haveαh(F ) isin F(hWξp) sub F(hWξp) and so the very right hand side of (319) is an elementof F(hWξp)λ Going to the norm closures the inclusion F(hWξp)λ sub F(hWξp)λ ofClowast-algebras follows

Analogously an inclusion of causal complements hW primeξp sube hW prime

ξp leads to the inclu-

sion of Clowast-algebras F(hW primeξp)λ sub F(hW prime

ξp)λ the only difference to the previous argument

consisting in an exchange of hh and p pTo complete the investigation of inclusions of wedges in W[ξ] we must also consider

the case hWξp sube hW primeξp = Whlowastξprimep

By the same reasoning as before there exists a

matrix N with detN = 1 such that (hminus1h)lowastξ = Nξprime = NΠξ with the flip matrix Π SoN prime = NΠ has determinant detN prime = minus1 and hlowastξ = N primehlowastξ

prime Using (312) we find forξ-smooth F isin F(hWξp)

αh(Fξλ) = αh(F )hlowastξλ = αh(F )N primehlowastξprimeλ = αh(F )hlowastξprimeminusλ (320)

By isotony and covariance of the undeformed net this deformed operator is an elementof F(hW prime

ξp)λ (314) and taking the norm closure in (314) yields F(hWξp)λ sub F(hW primeξp)λ

So the isotony (316) of the net is established This implies in particular that the netFλ is well-defined since in case hWξp equals hWξp or its causal complement the samearguments yield the equality of F(hWξp)λ and F(hWξp)λ respectively F(hW prime

ξp)λThe covariance of W 7rarr F(W )λ is evident from the definition To check twisted

locality it is thus sufficient to consider the pair of wedges Wξp Wprimeξp In view of the

definition of the Clowast-algebras F(W )λ (313) as norm closures of algebras of deformedsmooth operators it suffices to show that any ξ-smooth F isin F(Wξp) G isin F(Wξprimep)fulfill the commutation relation

[ZFξλZlowast Gξprimeλ] = 0 (321)

But Gξprimeλ = Gξminusλ (312) and since the undeformed net is twisted local and covariantwe have [τξs(F ) G] = 0 for all s isin IR2 which implies [ZFξλZ

lowast Gξminusλ] = 0 by Lemma32

The fact that setting λ = 0 reproduces the undeformed net is a straightforwardconsequence of Fξ0 = F for any ξ-smooth operator ξ isin Ξ

Theorem 34 is our main result concerning the structure of deformed quantum fieldtheories on admissible spacetimes It states that the same covariance and localizationproperties as on flat spacetime can be maintained in the curved setting Whereas theaction of the isometry group and the chosen representation space of F are the same for allvalues of the deformation parameter λ the concrete Clowast-algebras F(W )λ depend in a non-trivial and continuous way on λ For a fixed wedge W the collection F(W )λ λ isin IRforms a continuous field of Clowast-algebras [Dix77] this follows from Rieffelrsquos results [Rie92]and the fact that F(W )λ forms a faithful representation of Rieffelrsquos deformed Clowast-algebra(F(W )timesλ) [BLS10]

For deformed nets on Minkowski space there also exist proofs showing that thenet W 7rarr F(W )λ depends on λ for example by working in a vacuum representationand calculating the corresponding collision operators [BS08] There one finds as astriking effect of the deformation that the interaction depends on λ ie that defor-mations of interaction-free models have non-trivial S-matrices However on genericcurved spacetimes a distinguished invariant state like the vacuum state with its posi-tive energy representation of the translations does not exist Consequently the resultconcerning scattering theory cannot be reproduced here Instead we will establish thenon-equivalence of the undeformed net W 7rarr F(W ) and the deformed net W 7rarr F(W )λλ 6= 0 in a concrete example model in Section 32

As mentioned earlier the family of wedge regionsW(M g) is causally separating in asubclass of admissible spacetimes including the Friedmann-Robertson-Walker universesIn this case the extension of the net Fλ to double cones or similar regions O subM via

F(O)λ =⋂WsupO

F(W )λ (322)

is still twisted local These algebras contain all operators localized in the region O inthe deformed theory On other spacetimes (M g) such an extension is possible only forspecial regions intersections of wedges whose shape and size depend on the structureof W(M g)

Because of the relation of warped convolution to noncommutative spaces wheresharp localization is impossible it is expected that F(O) contains only multiples of theidentity if O has compact closure We will study this question in the context of thedeformed Dirac field in Section 32

We conclude this section with a remark concerning the relation between the field andobservable net structure of deformed quantum field theories The field net F is com-posed of Bose and Fermi fields and therefore contains observable as well as unobserv-able quantities The former give rise to the observable net A which consists of the

subalgebras invariant under the grading automorphism γ In terms of the projectionv(F ) = 1

2(F + γ(F )) the observable wedge algebras are

A(W ) = F isin F(W ) F = γ(F ) = v(F(W )) W isin W (323)

so that A(W ) A(W ) commute (without twist) if W and W are spacelike separatedSince the observables are the physically relevant objects we could have considered

a deformation A(W ) rarr A(W )λ of the observable wedge algebras along the same linesas we did for the field algebras This approach would have resulted precisely in theγ-invariant subalgebras of the deformed field algebras F(W )λ ie the diagram

F(W )deformationminusminusminusminusminusminusminusrarr F(W )λ

y yvA(W )

deformationminusminusminusminusminusminusminusrarr A(W )λ

commutes This claim can quickly be verified by noting that the projection v commuteswith the deformation map F 7rarr Fξλ

32 The Dirac field and its deformation

After the model-independent description of deformed quantum field theories carried outin the previous section we now consider the theory of a free Dirac quantum field as aconcrete example model We first briefly recall the notion of Dirac (co)spinors and theclassical Dirac equation following largely [DHP09 San08] and partly [Dim82 FV02]where the proofs of all the statements below are presented and an extensive descriptionof the relevant concepts is available Afterwards we consider the quantum Dirac fieldusing Arakirsquos self-dual CAR algebra formulation [Ara71]

As before we work on a fixed but arbitrary admissible spacetime (M g) in the sense ofDefinition 21 and we fix its orientation Therefore as a four-dimensional time orientedand oriented globally hyperbolic spacetime M admits a spin structure (SM ρ) con-sisting of a principle bundle SM over M with structure group SL(2C) and a smoothbundle homomorphism ρ projecting SM onto the frame bundle FM which is a principalbundle over M with SO0(3 1) as structure group The map ρ preserves base points andis equivariant in the sense that it intertwines the natural right actions R of SL(2C) onSM and of SO0(3 1) on FM respectively

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

with the covering homomorphism Λ 7rarr Λ from SL(2C) to SO0(3 1)Although each spacetime of the type considered here has a spin structure [DHP09

Thm 21 Lemma 21] this is only unique if the underlying manifold is simply con-nected [Ger68 Ger70] ie if all edges are simply connected in the case of an admissiblespacetime In the following it is understood that a fixed choice of spin structure hasbeen made

The main object we shall be interested in is the Dirac bundle that is the associatedvector bundle

DM = SM timesT C4 (325)

with the representation T = D( 120) oplus D(0 1

2) of SL(2C) on C4 Dirac spinors ψ are

smooth global sections of DM and the space they span will be denoted E(DM) Thedual bundle DlowastM is called the dual Dirac bundle and its smooth global sections ψprime isinE(DlowastM) are referred to as Dirac cospinors

For the formulation of the Dirac equation we need two more ingredients Thefirst are the so-called gamma-matrices which are the coefficients of a global tensorγ isin E(T lowastM otimes DM otimes DlowastM) such that γ = γAaBe

a otimes EA otimes EB Here EA and EB

with AB = 1 4 are four global sections of DM and DlowastM respectively such that(EA E

B) = δBA with ( ) the natural pairing between dual elements Notice that EAdescends also from a global section E of SM since we can define EA(x) = [(E(x) zA)]where zA is the standard basis of C4 At the same time out of E we can construct eawith a = 1 4 as a set of four global sections of TM once we define e = ρ E as aglobal section of the frame bundle which in turn can be read as a vector bundle overTM with IR4 as typical fibre The set of all ea is often referred to as the non-holonomicbasis of the base manifold In this case upper indices are defined via the natural pairingover IR4 that is (eb ea) = δba Furthermore we choose the gamma-matrices to be of thefollowing form

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

where the σa are the Pauli matrices and In will henceforth denote the (ntimes n) identitymatrix These matrices fulfil the anticommutation relation γa γb = 2ηabI4 with theflat Minkowski metric η They therefore depend on the sign convention in the metricsignature and differ from those introduced in [DHP09] where a different conventionwas used

The last ingredient we need to specify is the covariant derivative (spin connection)on the space of smooth sections of the Dirac bundle that is

nabla E(DM)rarr E(T lowastM otimesDM) (327)

whose action on the sections EA is given as nablaEA = σBaAeaEB The connection coeffi-

cients can be expressed as σBaA = 14Γbadγ

dCA where Γbad are the coefficients arising from

the Levi-Civita connection expressed in terms of non-holonomic basis [DHP09 Lemma22]

We can now introduce the Dirac equation for spinors ψ isin E(DM) and cospinorsψprime isin E(DlowastM) as

Dψ = (minusiγmicronablamicro +mI)ψ = 0 (328)

Dprimeψprime = (+iγmicronablamicro +mI)ψprime = 0 (329)

where m ge 0 is a constant while I is the identity on the respective spaceThe Dirac equation has unique advanced and retarded fundamental solutions [DHP09]

Denoting the smooth and compactly supported sections of the Dirac bundle by D(DM)there exist two continuous linear maps

Splusmn D(DM)rarr E(DM)

such that SplusmnD = DSplusmn = I and supp (Splusmnf) sube Jplusmn(supp (f)) for all f isin D(DM) whereJplusmn stands for the causal futurepast In the case of cospinors we shall instead talkabout Splusmnlowast D(DlowastM)rarr E(DlowastM) and they have the same properties of Splusmn except thatSplusmnlowast D

prime = DprimeSplusmnlowast = I In analogy with the theory of real scalar fields the causal propagatorsfor Dirac spinors and cospinors are defined as S = S+ minus Sminus and Slowast = S+

lowast minus Sminuslowast respectively

For the formulation of a quantized Dirac field it is advantageous to collect spinorsand cospinors in a single object We therefore introduce the space

D = D(DM oplusDlowastM) (330)

on which we have the conjugation

Γ(f1 oplus f2) = f lowast1β oplus βminus1f lowast2 (331)

defined in terms of the adjoint f 7rarr f lowast on C4 and the Dirac conjugation matrix βThis matrix is the unique selfadjoint element of SL(4C) with the properties that γlowasta =minusβγaβminus1 a = 0 3 and iβnaγa is a positive definite matrix for any timelike future-directed vector n

Due to these properties the sesquilinear form on D defined as

(f1 oplus f2 h1 oplus h2) = minusi〈f lowast1β Sh1〉+ i〈Slowasth2 βminus1f lowast2 〉 (332)

where 〈〉 is the global pairing between E(DM) and D(DlowastM) or between E(DlowastM) andD(DM) is positive semi-definite Therefore the quotient

K = D(kerS oplus kerSlowast) (333)

has the structure of a pre-Hilbert space and we denote the corresponding scalar productand norm by 〈〉S and fS = 〈f f〉12S The conjugation Γ descends to the quotientK and we denote its action on K by the same symbol Moreover Γ is compatible withthe sesquilinear form (332) in such a way that it extends to an antiunitary involutionΓ = Γlowast = Γminus1 on the Hilbert space K [San08 Lemma 424]

Regarding covariance the isometry group Iso of (M g) naturally acts on the sectionsin D by pullback In view of the geometrical nature of the causal propagator this actiondescends to the quotientK and extends to a unitary representation u of Iso on the Hilbertspace K

Given the pre-Hilbert space K the conjugation Γ and the representation u as abovethe quantized Dirac field can be conveniently described as follows [Ara71] We considerthe Clowast-algebra CAR (KΓ) that is the unique unital Clowast-algebra generated by the sym-bols B(f) f isin K such that for all f h isin K

a) f 7minusrarr B(f) is complex linear

b) B(f)lowast = B(Γf)

c) B(f) B(h) = 〈Γf h〉S middot 1

The field equation is implicit here since we took the quotient with respect to the ker-nels of S Slowast The standard picture of spinors and cospinors can be recovered via theidentifications ψ(h) = B(0oplus h) and ψdagger(f) = B(f oplus 0)

As is well known the Dirac field satisfies the standard assumptions of quantum fieldtheory and we briefly point out how this model fits into the general framework usedin Section 31 The global field algebra F = CAR (KΓ) carries a natural Iso-action αby Bogoliubov transformations Since the unitaries u(h) h isin Iso commute with theconjugation Γ the maps

αh(B(f)) = B(u(h)f) f isin K

extend to automorphisms of F Similarly the grading automorphism γ is fixed by

γ(B(f)) = minusB(f)

and clearly commutes with αh The field algebra is faithfully represented on the FermiFock space H over K where the field operators take the form

B(f) =1radic2

(a(f)lowast + a(Γf)) (334)

with the usual Fermi Fock space creationannihilation operators a(f) f isin K Inthis representation the second quantization U of u implements the action α TheBoseFermi-grading can be implemented by V = (minus1)N where N isin B(H) is the Fockspace number operator [Foi83]

Regarding the regularity assumptions on the symmetries recall that the anticommu-tation relations of the CAR algebra imply that a(f) = fS and thus f 7rarr B(f) is alinear continuous map from K to F As s 7rarr uξ(s)f is smooth in the norm topology ofK for any ξ isin Ξ f isin K this implies that the field operators B(f) transform smoothlyunder the action α Furthermore the unitarity of u yields strong continuity of α on allof F as required in Section 31

The field algebra F(W ) sub F associated with a wedge W isin W is defined as the unitalClowast-algebra generated by all B(f) f isin K(W ) where K(W ) is the set of (equivalenceclasses of) smooth and compactly supported sections of DMoplusDlowastM with support in W Since 〈Γf g〉S = 0 for f isin K(W ) g isin K(W prime) we have B(f) B(g) = 0 for f isin K(W )g isin K(W prime) which implies the twisted locality condition (33) Isotony is clear from thedefinition and covariance under the isometry group follows from u(h)K(W ) = K(hW )

The model of the Dirac field therefore fits into the framework of Section 31 and thewarped convolution deformation defines a one-parameter family of deformed nets FλBesides the properties which were established in the general setting in Section 31 wecan here consider the explicit deformed field B(f)ξλ A nice characterization of these

operators can be given in terms of their n-point functions associated with a quasifreestate ω on F

Let ω be an α-invariant quasifree state on F and let (Hω πωΩω) denote the as-sociated GNS triple As a consequence of invariance of ω the GNS space Hω carriesa unitary representation Uω of Iso which leaves Ωω invariant In this situation thewarping map

Fξλ 7rarr F ωξλ =

4π2limεrarr0

primeχ(εs εsprime)Uω

ξ (λQs)πω(F )Uωξ (sprime minus λQs) (335)

defined for ξ-smooth F isin F as before extends continuously to a representation of theRieffel-deformed Clowast-algebra (Ftimesξλ) on Hω [BLS10 Thm 28] Moreover the Uω-invariance of Ωω implies

F ωξλΩ

ω = πω(F )Ωω ξ isin Ξ λ isin IR F isin F ξ-smooth (336)

Since the CAR algebra is simple all its representations are faithful [BR97] We willtherefore identify F with its representation πω(F) in the following and drop the ω-dependence of HωΩω Uω and the warped convolutions F ω

ξλ from our notationTo characterize the deformed field operators B(f)ξλ we will consider the n-point

functions

ωn(f1 fn) = ω(B(f1) middot middot middotB(fn)) = 〈Ω B(f1) middot middot middotB(fn)Ω〉 f1 fn isin K

and the corresponding deformed expectation values of the deformed fields called de-formed n-point functions

ωξλn (f1 fn) = 〈Ω B(f1)ξλ middot middot middotB(fn)ξλΩ〉 f1 fn isin K

Of particular interest are the quasifree states where ωn vanishes if n is odd and ωn is alinear combination of products of two-point functions ω2 if n is even In particular theundeformed four-point function of a quasifree state reads

ω4(f1 f2 f3 f4) = ω2(f1 f2)ω2(f3 f4) + ω2(f1 f4)ω2(f2 f3)minus ω2(f1 f3)ω2(f2 f4) (337)

Proposition 35 The deformed n-point functions of a quasifree and Iso-invariant statevanish for odd n The lowest deformed even n-point functions are f1 f4 isin K

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

ωξλ4 (f1 f2 f3 f4) = ω2(f1 f2)ω2(f3 f4) + ω2(f1 f4)ω2(f2 f3) (339)

minus 1

4π2limεrarr0

primeχ(εs εsprime)ω2(f1 uξ(s)f3) middot ω2(f2 uξ(2λQs

prime)f4)

Proof The covariant transformation behaviour of the Dirac field Uξ(s)B(f)Uξ(s)minus1 =

B(uξ(s)f) the invariance of Ω and the form (335) of the warped convolution imply thatany deformed n-point function can be written as an integral over undeformed n-pointfunctions with transformed arguments As the latter functions vanish for odd n we alsohave ωξλn = 0 for odd n

Taking into account (336) we obtain for the deformed two-point function

ωξλ2 (f1 f2) = 〈Ω B(f1)ξλB(f2)ξλΩ〉= 〈(B(f1)lowast)ξλΩ B(f2)ξλΩ〉 = 〈B(f1)lowastΩ B(f2)Ω〉 = ω2(f1 f2)

proving (338)To compute the four-point function we use again B(f)ξλΩ = B(f)Ω and Uξ(s)Ω =

Ω Inserting the definition of the warped convolution (36) and using the transformationlaw Uξ(s)B(f)Uξ(s)

minus1 = B(uξ(s)f) and the shorthand f(s) = uξ(s)f we obtain

ωξλ4 (f1 f2 f3 f4) = 〈Ω B(f1)B(f2)ξλB(f3)ξλB(f4)Ω〉

= (2π)minus4 limε1ε2rarr0

intds eminusi(s1s

prime1+s2sprime2)χε(s)ω4(f1 f2(λQs1) f3(λQs2 + sprime1) f4(sprime1 + sprime2))

where ds = ds1 dsprime1 ds2 ds

prime2 and χε(s) = χ(ε1s1 ε1s

prime1)χ(ε2s2 ε2s

prime2) After the substitu-

tions sprime2 rarr sprime2minus sprime1 and sprime1 rarr sprime1minusλQs2 the integrations over s2 sprime2 and the limit ε2 rarr 0

can be carried out The result is

(2π)minus2 limε1rarr0

intds1 ds

prime1 eminusis1sprime1χ(ε1s1 ε1s

prime1)ω4(f1 f2(λQs1) f3(sprime1) f4(sprime1 minus λQs1))

with a smooth compactly supported cutoff function χ with χ(0 0) = 1We now use the fact that ω is quasi-free and write ω4 as a sum of products of two-

point functions (337) Considering the term where f1 f2 and f3 f4 are contracted inthe second factor ω2(f3(sprime1) f4(sprime1 minus λQs1)) the sprime1-dependence drops out because ω isinvariant under isometries So the integral over sprime1 can be performed and yields a factorδ(s1) in the limit ε1 rarr 0 Hence all λ-dependence drops out in this term as claimed in(339)

Similarly in the term where f1 f4 and f2 f3 are contracted all integrations disappearafter using the invariance of ω and making the substitution sprime1 rarr sprime1 + λQs1 Also thisterm does not depend on λ

Finally we compute the term containing the contractions f1 f3 and f2 f4

intds1 ds

prime1)ω2(f1 f3(sprime1)) middot ω2(f2(λQs1) f4(sprime1 minus λQs1))

= (2π)minus2 limε1rarr0

intds1 ds

prime1)ω2(f1 f3(sprime1)) middot ω2(f2 f4(sprime1 minus 2λQs1))

intds1 ds

prime1)ω2(f1 f3(sprime1)) middot ω2(f2 f4(2λQs1))

In the last step we substituted s1 rarr s1 + 12λQminus1sprime1 used the antisymmetry of Q and

absorbed the new variables in χ in a redefinition of this function Since the oscillatoryintegrals are independent of the particular choice of cutoff function comparison with(339) shows that the proof is finished

The structure of the deformed n-point functions build from a quasifree state ω is quitedifferent from the undeformed case In particular the two-point function is undeformed

but the four-point function depends on the deformation parameter For even n gt 4 astructure similar to the n-point functions on noncommutative Minkowski space [GL08]is expected which are all λ-dependent These features clearly show that the deformedfield B(f)ξλ λ 6= 0 differs from the undeformed field Considering the commutationrelations of the deformed field operators it is also straightforward to check that thedeformed field is not unitarily equivalent to the undeformed one

However this structure does not yet imply that the deformed and undeformed Diracquantum field theories are inequivalent For there could exist a unitary V onH satisfyingV U(h)V lowast = U(h) h isin Iso V Ω = Ω which does not interpolate the deformed andundeformed fields but the Clowast-algebras according to V F(W )λV

lowast = F(W ) W isin W Ifsuch a unitary exists the two theories would be physically indistinguishable

On flat spacetime an indirect way of ruling out the existence of such an intertwinerV and thus establishing the non-equivalence of deformed and undeformed theories isto compute their S-matrices and show that these depend on λ in a non-trivial mannerHowever on curved spacetimes collision theory is not available and we will thereforefollow the more direct non-equivalence proof of [BLS10 Lemma 46] adapted to oursetting This proof aims at showing that the local observable content of warped theoriesis restricted in comparison to the undeformed setting as one would expect becauseof the connection ot noncommutative spacetime However the argument requires acertain amount of symmetry and we therefore restrict here to the case of a Friedmann-Robertson-Walker spacetime M

As discussed in Section 22 M can then be viewed as J times IR3 sub IR4 via a conformalembedding where J sub IR is an interval Recall that in this case we have the Euclideangroup E(3) contained in Iso(M g) and can work in global coordinates (τ x y z) whereτ isin J and x y z isin IR are the flow parameters of Killing fields As reference Killingpair we pick ζ = (party partz) and as reference wedge the ldquoright wedgerdquo W 0 = Wζ0 =(τ x y z) τ isin J x gt |τ |

In this geometric context consider the rotation rϕ about angle ϕ in the x-y-planeand the cone

C = rϕW 0 cap rminusϕW 0 (340)

with some fixed angle |ϕ| lt π2 Clearly C sub W 0 and the reflected cone jxC where

jx(t x y z) = (tminusx y z) lies spacelike to W 0 and rϕW 0Moreover we will work in the GNS-representation of a particular state ω on F for

the subsequent proposition which besides the properties mentioned above also has theReeh-Schlieder property That is the von Neumann algebra F(C)primeprime sub B(Hω) has Ωω asa cyclic vector

Since the Dirac field theory is a locally covariant quantum field theory satisfyingthe time slice axiom [San10] the existence of such states can be deduced by spacetimedeformation arguments [San09 Thm 41] As M and the Minkowski space have uniquespin structures and M can be deformed to Minkowski spacetime in such a way that itsE(3) symmetry is preserved the state obtained from deforming the Poincare invariantvacuum state on IR4 is still invariant under the action of the Euclidean group

In the GNS representation of such a Reeh-Schlieder state on a Friedmann-Robertson-Walker spacetime we find the following non-equivalence result

Proposition 36 Consider the net Fλ generated by the deformed Dirac field on a Fried-mann-Robertson-Walker spacetime with flat spatial sections in the GNS representationof an invariant state with the Reeh-Schlieder property Then the implementing vector Ωis cyclic for the field algebra F(C)λprimeprime associated with the cone (340) if and only if λ = 0In particular the nets F0 and Fλ are inequivalent for λ 6= 0

Proof Let f isin K(C) so that f u(rminusϕ)f isin K(W 0) and both field operators B(f)ζλand B(u(rminusϕ)f)ζλ are contained in F(W 0)λ Taking into account the covariance of thedeformed net it follows that U(rϕ)B(u(rminusϕ)f)ζλU(rminusϕ) = B(f)rϕlowast ζλ lies in F(rϕW 0)λ

Now the cone C is defined in such a way that the two wedges W 0 and rϕW 0 liespacelike to jxC Let us assume that Ω is cyclic for F(C)λprimeprime which by the unitarityof U(jx) is equivalent to Ω being cyclic for F(jxC)λprimeprime Hence Ω is separating for thecommutant F(jxC)λprime which by locality contains F(W 0)λ and F(rϕW 0)λ But in view of(336) B(f)ζλ and B(f)rϕlowast ζλ coincide on Ω

B(f)rϕlowast ζλΩ = B(f)Ω = B(f)ζλΩ

so that the separation property implies B(f)ζλ = B(f)rϕlowast ζλTo produce a contradiction we now show that these two operators are actually

not equal To this end we consider a difference of four-point functions (339) withsmooth vectors f1 f2 = f f3 = f f4 and Killing pairs ζ respectively rϕlowast ζ With theabbreviations wϕij(s) = ω2(fi urϕlowast ζ(s)fj) we obtain

〈Ω B(f1)(B(f)ζλB(f)ζλ minusB(f)rϕlowast ζλB(f)rϕlowast ζλ

)B(f4)Ω〉

= ωζλ4 (f1 f f f4)minus ωrϕlowast ζλ

4 (f1 f f f4)

= (w013 λ w

024)(0)minus (wϕ13 λ w

ϕ24)(0)

where λ denotes the Weyl-Moyal star product on smooth bounded functions on IR2with the standard Poisson bracket given by the matrix (35) in the basis ζ1 ζ2 Nowthe asymptotic expansion of this expression for λrarr 0 gives in first order the differenceof Poisson brackets [EGBV89]

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f4〉 minus 〈f1 P

ζ2 f〉〈f P

ζ1 f4〉

minus 〈f1 Prϕlowast ζ1 f〉〈f P rϕlowast ζ

2 f4〉+ 〈f1 Prϕlowast ζ2 f〉〈f P rϕlowast ζ

1 f4〉

where all scalar products are in K and P rϕlowast ζ1 P rϕlowast ζ

1 denote the generators of s 7rarr urϕlowast ζ(s)

By considering f4 orthogonal to P ζ1 f and P rϕlowast ζ

1 f we see that for B(f)ζλ = B(f)rϕlowast ζλit is necessary that 〈f1 (P

ζj minus P rϕlowast ζ

j )f〉 = 0 But varying f f1 within the specified

limitations gives dense subspaces in K ie we must have P ζj = P rϕlowast ζ

j This implies thattranslations in a spacelike direction are represented trivially on the Dirac field which isnot compatible with its locality and covariance properties

So we conclude that the deformed field operator B(f)rϕlowast ζλ is not independent of ϕfor λ 6= 0 and hence the cyclicity assumption is not valid for λ 6= 0 Since on theother hand Ω is cyclic for F(C)0

primeprime by the Reeh-Schlieder property of ω and a unitaryV leaving Ω invariant and mapping F(C)0 onto F(C)λ would preserve this property wehave established that the nets F0 and Fλ λ 6= 0 are not equivalent

4 Conclusions

In this paper we have shown how to apply the warped convolution deformation toquantum field theories on a large class of globally hyperbolic spacetimes Under therequirement that the group of isometries contains a two-dimensional Abelian subgroupgenerated by two commuting spacelike Killing fields many results known for Minkowskispace theories were shown to carry over to the curved setting by formulating conceptslike edges wedges and translations in a geometrical language In particular it has beendemonstrated in a model-independent framework that the basic covariance and wedge-localization properties we started from are preserved for all values of the deformationparameter λ As a concrete example we considered a warped Dirac field on a Friedmann-Robertson-Walker spacetime in the GNS representation of a quasifree and Iso-invariantstate with Reeh-Schlieder property It was shown that the deformed models depend ina non-trivial way on λ and violate the Reeh-Schlieder property for regions smaller thanwedges In view of the picture that the deformed models can be regarded as effectivetheories on a noncommutative spacetime where strictly local quantities do not existbecause of space-time uncertainty relations it is actually expected that they do notcontain operators sharply localized in bounded spacetime regions for λ 6= 0

At the current stage it is difficult to give a clear-cut physical interpretation to themodels constructed here since scattering theory is not available for quantum field theo-ries on generic curved spacetimes Nonetheless it is interesting to note that in a fieldtheoretic context the deformation often leaves invariant the two-point function (Prop35) the quantity which is most frequently used for deriving observable quantum effectsin cosmology (for the example of quantised cosmological perturbations see [MFB92])So when searching for concrete scenarios where deformed quantum field theories canbe matched to measurable effects one has to look for phenomena involving the highern-point functions

There exist a number of interesting directions in which this research could be extendedAs far as our geometrical construction of wedge regions in curved spacetimes is con-cerned we limited ourselves here to edges generated by commuting Killing fields Thisassumption rules out many physically interesting spacetimes such as de Sitter Kerr orFriedmann-Robertson-Walker spacetimes with compact spatial sections An extensionof the geometric construction of edges and wedges to such spaces seems to be straight-forward and is expected to coincide with the notions which are already available Forexample in the case of four-dimensional de Sitter spacetime edges have the topology ofa two-sphere Viewing the two-sphere as an SO(3) orbit a generalization of the warpedconvolution deformation formula could involve an integration over this group instead ofIR2 A deformation scheme of Clowast-algebras which is based on an action of this groupis currently not yet available (see however [Bie07] for certain non-Abelian group ac-tion) In the de Sitter case a different possibility is to base the deformation formula onthe flow of a spacelike and commuting timelike Killing field leaving a wedge invariantThe covariance properties of the nets which arise from such a procedure are somewhatdifferent and require an adaption of the kernel in the warped convolution [Mo10]

It is desireable to find more examples of adequate deformation formulas compatiblewith the basic covariance and (wedge-) localization properties of quantum field theoryA systematic exploration of the space of all deformations is expected to yield manynew models and a better understanding of the structure of interacting quantum fieldtheories

AcknowledgementsCD gratefully acknowledges financial support from the Junior Fellowship Programmeof the Erwin Schrodinger Institute and from the German Research Foundation DFGthrough the Emmy Noether Fellowship WO 14471-1 EMM would like to thank theEU-network MRTN-CT-2006-031962 EU-NCG for financial support We are gratefulto Valter Moretti and Helmut Rumpf for helpful discussions

References

[ABD+05] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schupp and J Wess A Grav-ity Theory on Noncommutative Spaces ClassQuantGrav 22 (2005) 3511ndash3532 [openaccess]

[Ara71] H Araki On quasifree states of CAR and Bogoliubov automorphisms Publ Res InstMath Sci Kyoto 6 (1971) 385ndash442 [open access]

[Ara99] H Araki Mathematical Theory of Quantum Fields Int Series of Monographs on PhysicsOxford University Press Oxford 1999

[BB99] H-J Borchers and D Buchholz Global properties of vacuum states in de Sitter spaceAnnales Poincare Phys Theor A70 (1999) 23ndash40 [open access]

[BDFP10] D Bahns S Doplicher K Fredenhagen and G Piacitelli Quantum Geometry on Quan-tum Spacetime Distance Area and Volume Operators Preprint (May 2010) [open

access]

[BDFS00] D Buchholz O Dreyer M Florig and S J Summers Geometric modular action andspacetime symmetry groups Rev Math Phys 12 (2000) 475ndash560 [open access]

[BGK+08] D Blaschke F Gieres E Kronberger M Schweda and M Wohlgenannt Translation-invariant models for non-commutative gauge fields JPhysA 41 (2008) 252002 [open

access]

[BGL02] R Brunetti D Guido and R Longo Modular localization and Wigner particles RevMath Phys 14 (2002) 759ndash786 [open access]

[Bie07] P Bieliavsky Deformation quantization for actions of the affine group Preprint (September2007) [open access]

[BLS10] D Buchholz G Lechner and S J Summers Warped Convolutions Rieffel Deformationsand the Construction of Quantum Field Theories Preprint (2010) [open access]

[BMS01] D Buchholz J Mund and S J Summers Transplantation of Local Nets and GeometricModular Action on Robertson-Walker Space-Times Fields Inst Commun 30 (2001) 65ndash81 [open access]

[Bor92] H-J Borchers The CPT theorem in two-dimensional theories of local observables Com-mun Math Phys 143 (1992) 315ndash332

[Bor00] H-J Borchers On revolutionizing quantum field theory with Tomitarsquos modular theoryJ Math Phys 41 (2000) 3604ndash3673

[Bor09] H-J Borchers On the Net of von Neumann algebras associated with a Wedge and Wedge-causal Manifolds Preprint (2009) [open access]

[BPQV08] A P Balachandran A Pinzul B A Qureshi and S Vaidya S-Matrix on the MoyalPlane Locality versus Lorentz Invariance PhysRevD 77 (2008) 025020 [open access]

[BR97] O Bratteli and D W Robinson Operator Algebras and Quantum Statistical MechanicsII Springer 1997

[BS04] D Buchholz and S J Summers Stable quantum systems in anti-de Sitter space Causal-ity independence and spectral properties J Math Phys 45 (2004) 4810ndash4831 [open

access]

[BS05] A N Bernal and M Sanchez Smoothness of time functions and the metric splitting ofglobally hyperbolic spacetimes CommunMathPhys 257 (2005) 43ndash50 [open access]

[BS06] A N Bernal and M Sanchez Further results on the smoothability of Cauchy hypersurfacesand Cauchy time functions LettMathPhys 77 (2006) 183ndash197 [open access]

[BS07] D Buchholz and S J Summers String- and brane-localized fields in a strongly nonlocalmodel J Phys A40 (2007) 2147ndash2163 [open access]

[BS08] D Buchholz and S J Summers Warped Convolutions A Novel Tool in the Constructionof Quantum Field Theories In E Seiler and K Sibold editors Quantum Field Theoryand Beyond Essays in Honor of Wolfhart Zimmermann pages 107ndash121 World Scientific2008 [open access]

[BW75] J J Bisognano and E H Wichmann On the Duality Condition for a Hermitian ScalarField J Math Phys 16 (1975) 985ndash1007

[CF84] S Chandrasekhar and V Ferrari On the Nutku-Halil solution for colliding impulsivegravitational waves Proc Roy Soc Lond A396 (1984) 55

[Cha83] S Chandrasekhar The mathematical theory of black holes Oxford University Press 1983

[DFR95] S Doplicher K Fredenhagen and J E Roberts The Quantum structure of space-time atthe Planck scale and quantum fields Commun Math Phys 172 (1995) 187ndash220 [openaccess]

[DHP09] C Dappiaggi T-P Hack and N Pinamonti The extended algebra of observables forDirac fields and the trace anomaly of their stress-energy tensor RevMathPhys 21 (2009)1241ndash1312 [open access]

[DHR69] S Doplicher R Haag and J E Roberts Fields observables and gauge transformationsI Commun Math Phys 13 (1969) 1ndash23 [open access]

[Dim82] J Dimock Dirac quantum fields on a manifold Trans Amer Math Soc 269 (1982)no 1 133ndash147

[Dix77] J Dixmier C-Algebras North-Holland-Publishing Company Amsterdam New YorkOxford 1977

[EGBV89] R Estrada J M Gracia-Bondia and J C Varilly On Asymptotic expansions of twistedproducts J Math Phys 30 (1989) 2789ndash2796

[Ell06] G F R Ellis The Bianchi models Then and now Gen Rel Grav 38 (2006) 1003ndash1015

[Foi83] J J Foit Abstract Twisted Duality for Quantum Free Fermi Fields Publ Res InstMath Sci Kyoto 19 (1983) 729ndash741 [open access]

[FPH74] S A Fulling L Parker and B L Hu Conformal energy-momentum tensor in curvedspacetime Adiabatic regularization and renormalization Phys Rev D10 (1974) 3905ndash3924

[FV02] C J Fewster and R Verch A quantum weak energy inequality for Dirac fields in curvedspacetime Commun Math Phys 225 (2002) 331ndash359 [open access]

[Ger68] R P Geroch Spinor structure of space-times in general relativity I J Math Phys 9(1968) 1739ndash1744

[Ger70] R P Geroch Spinor structure of space-times in general relativity II J Math Phys 11(1970) 343ndash348

[GGBI+04] V Gayral J M Gracia-Bondia B Iochum T Schucker and J C Varilly Moyal planesare spectral triples Commun Math Phys 246 (2004) 569ndash623 [open access]

[GL07] H Grosse and G Lechner Wedge-Local Quantum Fields and Noncommutative MinkowskiSpace JHEP 11 (2007) 012 [open access]

[GL08] H Grosse and G Lechner Noncommutative Deformations of Wightman Quantum FieldTheories JHEP 09 (2008) 131 [open access]

[GLRV01] D Guido R Longo J E Roberts and R Verch Charged sectors spin and statistics inquantum field theory on curved spacetimes Rev Math Phys 13 (2001) 125ndash198 [openaccess]

[GW05] H Grosse and R Wulkenhaar Renormalisation of phi4 theory on noncommutative R4in the matrix base Commun Math Phys 256 (2005) 305ndash374 [open access]

[Haa96] R Haag Local Quantum Physics - Fields Particles Algebras Springer 2 edition 1996

[Kay85] B S Kay The double-wedge algebra for quantum fields on Schwarzschild and Minkowskispace-times Commun Math Phys 100 (1985) 57 [open access]

[Key96] M Keyl Causal spaces causal complements and their relations to quantum field theoryRev Math Phys 8 (1996) 229ndash270

[LR07] P Lauridsen-Ribeiro Structural and Dynamical Aspects of the AdSCFT Correspondencea Rigorous Approach PhD thesis Sao Paulo 2007 [open access]

[LW10] R Longo and E Witten An Algebraic Construction of Boundary Quantum Field The-oryPreprint (April 2010) [open access]

[MFB92] V F Mukhanov H A Feldman and R H Brandenberger Theory of cosmological per-turbations Part 1 Classical perturbations Part 2 Quantum theory of perturbations Part3 Extensions Phys Rept 215 (1992) 203ndash333

[Mo10] E Morfa-Morales work in progress

[OrsquoN83] B OrsquoNeill Semi-Riemannian Geometry Academic Press 1983

[OS09] T Ohl and A Schenkel Algebraic approach to quantum field theory on a class of non-commutative curved spacetimes Preprint (2009) [open access]

[Ped79] GK Pedersen C-Algebras and their Automorphism Groups Academic Press 1979

[PV04] M Paschke and R Verch Local covariant quantum field theory over spectral geometriesClassQuantGrav 21 (2004) 5299ndash5316 [open access]

[Reh00] K-H Rehren Algebraic Holography Annales Henri Poincare 1 (2000) 607ndash623 [open

access]

[Rie92] M A Rieffel Deformation Quantization for Actions of Rd volume 106 of Memoirs ofthe Amerian Mathematical Society American Mathematical Society Providence RhodeIsland 1992

[San08] K Sanders Aspects of locally covariant quantum field theory PhD thesis University ofYork September 2008 [open access]

[San09] K Sanders On the Reeh-Schlieder Property in Curved Spacetime CommunMathPhys288 (2009) 271ndash285 [open access]

[San10] K Sanders The locally covariant Dirac field Rev Math Phys 22 (2010) no 4 381ndash430[open access]

[Sol08] M A Soloviev On the failure of microcausality in noncommutative field theories PhysRev D77 (2008) 125013 [open access]

[Ste07] H Steinacker Emergent Gravity from Noncommutative Gauge Theory JHEP 0712 (2007)049 [open access]

[Str08] R Strich Passive States for Essential Observers J Math Phys 49 (2008) no 022301 [open access]

[Sza03] R J Szabo Quantum field theory on noncommutative spaces Phys Rept 378 (2003)207ndash299 [open access]

[Tay86] M E Taylor Noncommutative Harmonic Analysis American Mathematical Society 1986

[TW97] L J Thomas and E H Wichmann On the causal structure of Minkowski space-time JMath Phys 38 (1997) 5044ndash5086

[Wal84] R M Wald General Relativity University of Chicago Press 1984

1 Introduction

2 Geometric setup






4 Conclusions

challenging problem As a first step in this direction we study in the present paper howcertain deformed quantum field theories can be formulated in the presence of externalgravitational fields (ie on curved spacetime manifolds) see also [ABD+05 OS09] forother approaches to this question We will not address here the fundamental questionof dynamically coupling the matter fields with a possible noncommutative geometry ofspacetime [PV04 Ste07] but rather consider as an intermediate step deformed quantumfield theories on a fixed Lorentzian background manifold M

A deformation technique which is well suited for our purposes is that of warped convo-lutions see [BS08] and [GL07 GL08] for precursors and related work Starting from aHilbert space H carrying a representation U of IRn the warped convolution AQ of anoperator A on H is defined as

AQ = (2π)minusnintdnx dny eminusixy U(Qx)AU(y minusQx) (11)

Here Q is an antisymmetric (ntimes n)-matrix playing the role of deformation parameterand the integral can be defined in an oscillatory sense if A and U meet certain regularityrequirements For deformations of a single algebra the mapping A rarr AQ has manyfeatures in common with deformation quantization and the Weyl-Moyal product and infact was recently shown [BLS10] to be equivalent to specific representations of Rieffelrsquosdeformed Clowast-algebras with IRn-action [Rie92] In application to field theory modelshowever one has to deform a whole family of algebras corresponding to subsystemslocalized in spacetime and the parameter Q has to be replaced by a family of matricesQ adapted to the geometry of the underlying spacetime

To apply this scheme to quantum field theories on curved manifolds we will considerspacetimes with a sufficiently large isometry group containing two commuting Killingfields which give rise to a representation of IR2 as required in (11) This setting is wideenough to encompass a number of cosmologically relevant manifolds such as Friedmann-Robertson-Walker spacetimes or Bianchi models Making use of the algebraic frame-work of quantum field theory [Haa96 Ara99] we can then formulate quantum fieldtheories in an operator-algebraic language and study their deformations Despite thefact that the warped convolution was invented for the deformation of Minkowski spacequantum field theories it turns out that all reference to the particular structure of flatspacetime such as Poincare transformations and a Poincare invariant vacuum state canbe avoided

We are interested in understanding to what extent the familiar structure of quantumfield theories on curved spacetimes is preserved under such deformations and investigatein particular covariance and localization properties Concerning locality it is known thatin warped models on Minkowski space point-like localization is weakened to localizationin certain infinitely extended wedge-shaped regions [GL07 BS08 GL08 BLS10] Theseregions are defined as Poincare transforms of the Rindler wedge

WR = (x0 x1 x2 x3) isin IR4 x1 gt |x0| (12)

Because of their intimate relation to the Poincare symmetry of Minkowski spacetime it isnot obvious what a good replacement for such a collection of regions is in the presence

2 Geometric setup

] (21)

(Nξ)(p) = N(ξ1(p) ξ2(p)) p isinM (25)

0 11 0

ξprime = Πξ = (ξ2 ξ1) (27)

prime follows

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

2 Geometric setup

] (21)

(Nξ)(p) = N(ξ1(p) ξ2(p)) p isinM (25)

0 11 0

ξprime = Πξ = (ξ2 ξ1) (27)

prime follows

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

(Nξ)(p) = N(ξ1(p) ξ2(p)) p isinM (25)

0 11 0

ξprime = Πξ = (ξ2 ξ1) (27)

prime follows

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

(Nξ)(p) = N(ξ1(p) ξ2(p)) p isinM (25)

0 11 0

ξprime = Πξ = (ξ2 ξ1) (27)

prime follows

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

prime follows

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

prime follows

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

W =⊔[ξ]

take the form

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

] (219)

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

Z =1radic2

(1minus iV ) (31)

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

(0 1minus1 0

) (35)

Fξλ =1

4π2limεrarr0

F timesξλ G =1

4π2limεrarr0

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

(f ξλ g)(p) =1

4π2limεrarr0

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

In particular

αh(Fξλ)Ψ =1

4π2limεrarr0

FNξλΨ =1

4π2limεrarr0

0 11 0

)which has

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

⊔[ξ]W[ξ] of W

ξp)λ) (315)

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

F(O)λ =⋂WsupO

F(W )λ (322)

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

y yvA(W )

ρ RΛ = RΛ ρ Λ isin SO0(3 1) (324)

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

(I2 00 minusI2

) γa =

(0 σaminusσa 0

) a = 1 2 3 (326)

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

B(f) =1radic2

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

4π2limεrarr0

F ωξλΩ

functions

ωξλ2 (f1 f2) = ω2(f1 f2) (338)

minus 1

4π2limεrarr0

prime)f4)

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

intds eminusi(s1s

intds1 ds

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

)B(f4)Ω〉

4 (f1 f f f4)

= (w013 λ w

ϕ24)(0)

w013 w

024(0)minus wϕ13 w

ϕ24(0) = 〈f1 P

ζ1 f〉〈f P

ζ2 f〉〈f P

ζ1 f4〉

1 f4〉

4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

References

access]

1 Introduction

2 Geometric setup






4 Conclusions

access]

1 Introduction

2 Geometric setup






4 Conclusions

access]

1 Introduction

2 Geometric setup






4 Conclusions

1 Introduction

2 Geometric setup






4 Conclusions

Documents

Deformations of quantum eld theories on spacetimes with ... - Dappiaggi, Lechner, Morfa-Mor… · The recent construction and analysis of deformations of quantum eld theories by warped