1 Introduction

Deformations of quantum field theories
on spacetimes with Killing vector fields

Claudio Dappiaggi, Gandalf Lechner, Eric Morfa-Morales

II. Institut für Theoretische Physik, D-22763 Hamburg, Deutschland

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

Erwin Schrödinger Institute for Mathematical Physics, Vienna, A-1090 Vienna, Austria

claudio.dappiaggi@esi.ac.at, gandalf.lechner@univie.ac.at, emorfamo@esi.ac.at


Abstract. The recent construction and analysis of deformations of quantum field theories by warped convolutions is extended to a class of curved spacetimes. These spacetimes carry a family of wedge-like regions which share the essential causal properties of the Poincaré transforms of the Rindler wedge in Minkowski space. In the setting of deformed quantum field theories, they play the role of typical localization regions of quantum fields and observables. As 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 studied from different points of view in recent years. One motivation for considering such models is a possible noncommutative structure of spacetime at small scales, as suggested by combining classical gravity and the uncertainty principle of quantum physics [DFR95]. Quantum field theories on such noncommutative spaces can then be seen as deformations of usual quantum field theories, and it is hoped that they might capture some aspects of a still elusive theory of quantum gravity (cf. [Sza03] for a review). By now there exist several different types of deformed quantum field theories, see [GW05, BPQV08, Sol08, GL08, BGK08, BDFP10] for some recent papers, and references cited therein.

Certain deformation techniques arising from such considerations can also be used as a device for the construction of new models in the framework of usual quantum field theory on commutative spaces [GL07, BS08, GL08, BLS10, LW10], independent of their connection to the idea of noncommutative spaces. From this point of view, the deformation parameter plays the role of a coupling constant which changes the interaction of the model under consideration, but leaves the classical structure of spacetime unchanged.

Deformations designed for either describing noncommutative spacetimes or for constructing new models on ordinary spacetimes have been studied mostly in the case of a flat manifold, either with a Euclidean or Lorentzian signature. In fact, many approaches rely on a preferred choice of Cartesian coordinates in their very formulation, and do not generalize directly to curved spacetimes. The analysis of the interplay between spacetime curvature and deformations involving noncommutative structures thus presents a challenging problem. As a first step in this direction, we study in the present paper how certain deformed quantum field theories can be formulated in the presence of external gravitational fields (i.e., on curved spacetime manifolds), see also [ABD05, OS09] for other approaches to this question. We will not address here the fundamental question of dynamically coupling the matter fields with a possible noncommutative geometry of spacetime [PV04, Ste07], but rather consider as an intermediate step deformed quantum field theories on a fixed Lorentzian background manifold .

A deformation technique which is well suited for our purposes is that of warped convolutions, see [BS08] and [GL07, GL08] for precursors and related work. Starting from a Hilbert space carrying a representation of , the warped convolution of an operator on is defined as

(1.1)

Here is an antisymmetric -matrix playing the role of deformation parameter, and the integral can be defined in an oscillatory sense if and meet certain regularity requirements. For deformations of a single algebra, the mapping has many features in common with deformation quantization and the Weyl-Moyal product, and in fact was recently shown [BLS10] to be equivalent to specific representations of Rieffel’s deformed -algebras with -action [Rie92]. In application to field theory models, however, one has to deform a whole family of algebras, corresponding to subsystems localized in spacetime, and the parameter has to be replaced by a family of matrices adapted to the geometry of the underlying spacetime.

To apply this scheme to quantum field theories on curved manifolds, we will consider spacetimes with a sufficiently large isometry group containing two commuting Killing fields, which give rise to a representation of as required in (1.1). This setting is wide enough to encompass a number of cosmologically relevant manifolds such as Friedmann-Robertson-Walker spacetimes, or Bianchi models. Making use of the algebraic framework of quantum field theory [Haa96, Ara99], we can then formulate quantum field theories in an operator-algebraic language and study their deformations. Despite the fact that the warped convolution was invented for the deformation of Minkowski space quantum field theories, it turns out that all reference to the particular structure of flat spacetime, such as Poincaré transformations and a Poincaré invariant vacuum state, can be avoided.

We are interested in understanding to what extent the familiar structure of quantum field theories on curved spacetimes is preserved under such deformations, and investigate in particular covariance and localization properties. Concerning locality, it is known that in warped models on Minkowski space, point-like localization is weakened to localization in certain infinitely extended, wedge-shaped regions [GL07, BS08, GL08, BLS10]. These regions are defined as Poincaré transforms of the Rindler wedge

(1.2)

Because of their intimate relation to the Poincaré symmetry of Minkowski spacetime, it is not 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 on special manifolds have been studied by many authors in the literature [Kay85, BB99, Reh00, 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 methods in [BLS10], and thus carry two commuting Killing fields, there also exists a family of wedges with causal properties analogous to the Minkowski space wedges. Because of the prominent role wedges play in many areas of Minkowski space quantum field theory [BW75, Bor92, Bor00, BDFS00, BGL02], this geometric and manifestly covariant construction is also of interest independently of its relation to deformations.

In Section 3, we then consider quantum field theories on curved spacetimes, and deform them by warped convolution. We first show that these deformations can be carried through in a model-independent, operator-algebraic framework, and that the emerging models share many structural properties with deformations of field theories on flat spacetime (Section 3.1). In particular, deformed quantum fields are localized in the wedges of the considered spacetime. These and further aspects of deformed quantum field theories are also discussed in the concrete example of a Dirac field in Section 3.2. Section 4 contains our conclusions.

2 Geometric setup

To prepare the ground for our discussion of deformations of quantum field theories on curved backgrounds, we introduce in this section a suitable class of spacetimes and study their geometrical properties. In particular, we show how the concept of wedges, known from Minkowski space, generalizes to these manifolds. Recall in preparation that a wedge in four-dimensional Minkowski space is a region which is bounded by two non-parallel characteristic hyperplanes [TW97], or, equivalently, a region which is a connected component of the causal complement of a two-dimensional spacelike plane. The latter definition has a natural analogue in the curved setting. Making use of this observation, we construct corresponding wedge regions in Section 2.1, and analyse their covariance, causality and inclusion properties. At the end of that section, we compare 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 2.2, the abstract analysis of wedge regions is complemented by a number of concrete examples of spacetimes fulfilling our assumptions.

2.1 Edges and wedges in curved spacetimes

In the following, a spacetime is understood to be a four-dimensional, Hausdorff, (arcwise) connected, smooth manifold endowed with a smooth, Lorentzian metric whose signature is . Notice that it is automatically guaranteed that is also paracompact and second countable [Ger68, Ger70]. The (open) causal complement of a set is defined as

(2.1)

where is the causal future respectively past of in [Wal84, Section 8.1].

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 determined by a second order hyperbolic partial differential equation, we will restrict ourselves to globally hyperbolic spacetimes. So in particular, is orientable and time-orientable, and we fix both orientations. While this setting is standard in quantum field theory on curved backgrounds, we will make additional assumptions regarding the structure of the isometry group of , motivated by our desire to define wedges in which resemble those in Minkowski space.

Our most important assumption on the structure of is that it admits two linearly independent, spacelike, complete, commuting smooth Killing fields , which will later be essential in the context of deformed quantum field theories. We refer here and in the following always to pointwise linear independence, which entails in particular that these vector fields have no zeros. Denoting the flows of by , the orbit of a point is a smooth two-dimensional spacelike embedded submanifold of ,

(2.2)

where are the flow parameters of .

Since is globally hyperbolic, it is isometric to a smooth product manifold , where is a smooth, three-dimensional embedded Cauchy hypersurface. It is known that the metric splits according to with a temporal function and a positive function , while induces a smooth Riemannian metric on [BS05, Thm. 2.1]. We assume that, with as in (2.2), the Cauchy surface is smoothly homeomorphic to a product manifold , where is an open interval or the full real line. Thus , and we require in addition that there exists a smooth embedding . By our assumption on the topology of , it follows that is a globally hyperbolic spacetime without null focal points, a feature that we will need in the subsequent construction of wedge regions.

Definition 2.1

A spacetime is called admissible if it admits two linearly independent, spacelike, complete, commuting, smooth Killing fields and the corresponding split , with defined in (2.2), has the properties described above.

The set of all ordered pairs satisfying these conditions for a given admissible spacetime is denoted . The elements of will be referred to as Killing pairs.

For the remainder of this section, we will work with an arbitrary but fixed admissible spacetime , and usually drop the -dependence of various objects in our notation, e.g., write instead of for the set of Killing pairs, and in place of for the isometry group. Concrete examples of admissible spacetimes, such as Friedmann-Robertson-Walker-, Kasner- and Bianchi-spacetimes, will be discussed in Section 2.2.

The flow of a Killing pair is written as

(2.3)

where are the parameters of the (complete) flows of . By construction, each is an isometric -action by diffeomorphisms on , i.e., and for all .

On the set , the isometry group and the general linear group act in a natural manner.

Lemma 2.2

Let , , and define, ,

(2.4)
(2.5)

These operations are commuting group actions of   and on , respectively. The -action transforms the flow of according to, ,

(2.6)

If for some , , , then .

Proof: Due to the standard properties of isometries, acts on the Lie algebra of Killing fields by the push-forward isomorphisms [O’N83]. Therefore, for any , also the vector fields are spacelike, complete, commuting, linearly independent, smooth Killing fields. The demanded properties of the splitting directly carry over to the corresponding split with respect to . So maps onto , and since for , we have an action of .

The second map, , amounts to taking linear combinations of the Killing fields . The relation (2.6) holds because commute and are complete, which entails that the respective flows can be constructed via the exponential map. Since , the two components of are still linearly independent, and since (2.2) is invariant under , the splitting is the same for and . Hence , i.e., is a -action on , and since the push-forward is linear, it is clear that the two actions commute.

To prove the last statement, we consider the submanifold (2.2) together with its induced metric. Since the Killing fields are tangent to , their flows are isometries of . Since and is two-dimensional, it follows that acts as an isometry on the tangent space , . But as is spacelike and two-dimensional, we can assume without loss of generality that the metric of is the Euclidean metric, and therefore has the two-dimensional Euclidean group as its isometry group. Thus , i.e., .

The -transformation given by the flip matrix will play a special role later on. We therefore reserve the name inverted Killing pair of for

(2.7)

Note that since we consider ordered tuples , the Killing pairs and are not identical. Clearly, the map is an involution on , i.e., .

After these preparations, we turn to the construction of wedge regions in , and begin by specifying their edges.

Definition 2.3

An edge is a subset of which has the form

(2.8)

for some , . Any spacelike vector which completes the gradient of the chosen temporal function and the Killing vectors to a positively oriented basis of is called an oriented normal of .

It is clear from this definition that each edge is a two-dimensional, spacelike, smooth submanifold of . Our definition of admissible spacetimes explicitly restricts the topology of , but not of the edge (2.2), which can be homeomorphic to a plane, cylinder, or torus.

Note also that the description of the edge in terms of and is somewhat redundant: Replacing the Killing fields by linear combinations , , or replacing by with some , results in the same manifold .

Before we define wedges as connected components of causal complements of edges, we have to prove the following key lemma, from which the relevant properties of wedges follow. For its proof, it might be helpful to visualize the geometrical situation as sketched in Figure 1.


Figure 1: Three-dimensional sketch of the wedge and its edge
Lemma 2.4

The causal complement of an edge is the disjoint union of two connected components, which are causal complements of each other.

Proof: We first show that any point is connected to the base point by a smooth, spacelike curve. Since is globally hyperbolic, there exist Cauchy surfaces passing through and , respectively. We pick two compact subsets , containing , and , containing . If are chosen sufficiently small, their union is an acausal, compact, codimension one submanifold of . It thus fulfils the hypothesis of Thm. 1.1 in [BS06], which guarantees that there exists a spacelike Cauchy surface containing the said union. In particular, there exists a smooth, spacelike curve connecting and . Picking spacelike vectors and , we have the freedom of choosing in such a way that and . If and are chosen linearly independent from and , respectively, these vectors are oriented normals of respectively , and we can select such that it intersects the edge only in .

Let us define the region

(2.9)

and, exchanging with the inverted Killing pair , we correspondingly define the region . It is clear from the above argument that , and that we can prescribe arbitrary normals of , as initial respectively final tangent vectors of the curve connecting to .

The proof of the lemma consists in establishing that and are disjoint, and causal complements of each other. To prove disjointness of , assume there exists a point . Then can be connected with the base point by two spacelike curves, whose tangent vectors satisfy the conditions in (2.1) with respectively . By joining these two curves, we have identified a continuous loop in . As an oriented normal, the tangent vector at is linearly independent of , so that intersects only in .

Recall that according to Definition 2.1, splits as the product , with an open interval which is smoothly embedded in . Hence we can consider the projection of the loop onto , which is a closed interval because the simple connectedness of rules out the possibility that forms a loop, and on account of the linear independence of , the projection cannot be just a single point. Yet, as is a loop, there exists such that . We also know that is contained in and, since and are causally separated, the only possibility left is that they both lie on the same edge. Yet, per construction, we know that the loop intersects the edge only once at and, thus, and must coincide, which is the sought contradiction.

To verify the claim about causal complements, assume there exist points , and a causal curve connecting them, , . By definition of the causal complement, it is clear that does not intersect . In view of our restriction on the topology of , it follows that intersects either or . These two cases are completely analogous, and we consider the latter one, where there exists a point . In this situation, we have a causal curve connecting with , and since , it follows that must be past directed. As the time orientation of is the same for the whole curve, it follows that also the part of connecting and is past directed. Hence , which is a contradiction to . Thus .

To show that coincides with , let . Yet is not possible since and is open. So , i.e., we have shown , and the claimed identity follows.


in a Lorentz cylinder

Lemma 2.4 does not hold if the topological requirements on are dropped. As an example, consider a cylinder universe , the product of the Lorentz cylinder [O’N83] and the Euclidean plane . The translations in the last factor define spacelike, complete, commuting, linearly independent Killing fields . Yet the causal complement of the edge has only a single connected component, which has empty causal complement. In this situation, wedges lose many of the useful properties which we establish below for admissible spacetimes.

In view of Lemma 2.4, wedges in can be defined as follows.

Definition 2.5

(Wedges)
A wedge is a subset of which is a connected component of the causal complement of an edge in . Given , , we denote by the component of which intersects the curves , , for any oriented normal of . The family of all wedges is denoted

(2.10)

As explained in the proof of Lemma 2.4, the condition that the curve intersects a connected component of is independent of the chosen normal , and each such curve intersects precisely one of the two components of .

Some properties of wedges which immediately follow from the construction carried out in the proof of Lemma 2.4 are listed in the following proposition.

Proposition 2.6

(Properties of wedges)
Let be a wedge. Then

  1. is causally complete, i.e., , and hence globally hyperbolic.

  2. The causal complement of a wedge is given by inverting its Killing pair,

    (2.11)
  3. A wedge is invariant under the Killing flow generating its edge,

    (2.12)

Proof: a) By Lemma 2.4, is the causal complement of another wedge , and therefore causally complete: . Since is globally hyperbolic, this implies that is globally hyperbolic, too [Key96, Prop. 12.5].

b) This statement has already been checked in the proof of Lemma 2.4.

c) By definition of the edge (2.8), we have for any , and since the are isometries, it follows that . Continuity of the flow implies that also the two connected components of this set are invariant.

Corollary 2.7

(Properties of the family of wedge regions)
The family of wedge regions is invariant under the isometry group and under taking causal complements. For , it holds

(2.13)

Proof: Since isometries preserve the causal structure of a spacetime, we only need to look at the action of isometries on edges. We find

(2.14)

by using the well-known fact that conjugation of flows by isometries amounts to the push-forward by the isometry of the associated vector field. Since for any , (Lemma 2.2), the family is invariant under the action of the isometry group. Closedness of under causal complementation is clear from Prop. 2.6 b).

In contrast to the situation in flat spacetime, the isometry group does not act transitively on for generic admissible , and there is no isometry mapping a given wedge onto its causal complement. This can be seen explicitly in the examples discussed in Section 2.2. To keep track of this structure of , we decompose into orbits under the - and -actions.

Definition 2.8

Two Killing pairs are equivalent, written , if there exist and such that .

As and are commuting group actions, is an equivalence relation. According to Lemma 2.2 and Prop. 2.6 b), c), acting with on either leaves invariant (if ) or exchanges this wedge with its causal complement, (if ). Therefore the “coherent”111See [BS07] for a related notion on Minkowski space. subfamilies arising in the decomposition of the family of all wedges along the equivalence classes ,

(2.15)

take the form

(2.16)

In particular, each subfamily is invariant under the action of the isometry group and causal complementation.

In our later applications to quantum field theory, it will be important to have control over causal configurations and inclusions of wedges . Since is closed under taking causal complements, it is sufficient to consider inclusions. Note that the following proposition states in particular that inclusions can only occur between wedges from the same coherent subfamily .

Proposition 2.9

(Inclusions of wedges).
Let , . The wedges and form an inclusion, , if and only if and there exists with , such that .

Proof: () Let us assume that holds for some with , and . In this case, the Killing fields in are linear combinations of those in , and consequently, the edges and intersect if and only if they coincide, i.e. if . If the edges coincide, we clearly have . If they do not coincide, it follows from that and are either spacelike separated or they can be connected by a null geodesic.

Consider now the case that and are spacelike separated, i.e. . Pick a point and recall that can be characterized by equation (2.1). Since and , there exist curves and , which connect the pairs of points and , respectively, and comply with the conditions in (2.1). By joining and we obtain a curve which connects and . The tangent vectors and are oriented normals of and we choose and in such a way that these tangent vectors coincide. Due to the properties of and , the joint curve also complies with the conditions in (2.1), from which we conclude , and thus .

Consider now the case that and are connected by null geodesics, i.e. . Let be the point in which can be connected by a null geodesic with and pick a point . The intersection yields another null curve, say , and the intersection is non-empty since and are spacelike separated and . The null curve is chosen future directed and parametrized in such a way that and . By taking we find and which entails .

() Let us assume that we have an inclusion of wedges . Then clearly . Since is four-dimensional and are all spacelike, they cannot be linearly independent. Let us first assume that three of them are linearly independent, and without loss of generality, let and with three linearly independent spacelike Killing fields . Picking points , these can be written as and in the global coordinate system of flow parameters constructed from and the gradient of the temporal function.

For suitable flow parameters , we have and . Clearly, the points and are connected by a timelike curve, e.g. the curve whose tangent vector field is given by the gradient of the temporal function. But a timelike curve connecting the edges of is a contradiction to these wedges forming an inclusion. So no three of the vector fields can be linearly independent.

Hence with an invertible matrix . It remains to establish the correct sign of , and to this end, we assume . Then we have , by (Prop. 2.6 b)) and the () statement in this proof, since and are related by a positive determinant transformation and . This yields that both, and its causal complement, must be contained in , a contradiction. Hence , and the proof is finished.

Having derived the structural properties of the set of wedges needed later, we now compare our wedge regions to the Minkowski wedges and to other definitions proposed in the literature.

The flat Minkowski spacetime clearly belongs to the class of admissible spacetimes, with translations along spacelike directions and rotations in the standard time zero Cauchy surface as its complete spacelike Killing fields. However, as Killing pairs consist of non-vanishing vector fields, and each rotation leaves its rotation axis invariant, the set consists precisely of all pairs such that the flows , are translations along two linearly independent spacelike directions. Hence the set of all edges in Minkowski space coincides with the set of all two-dimensional spacelike planes. Consequently, each wedge is bounded by two non-parallel characteristic three-dimensional planes. This is precisely the family of wedges usually considered in Minkowski space222Note that we would get a “too large” family of wedges in Minkowski space if we would drop the requirement that the vector fields generating edges are Killing. However, the assumption that edges are generated by commuting Killing fields is motivated by the application to deformations of quantum field theories, and one could generalize our framework to spacetimes with edges generated by complete, linearly independent smooth Killing fields. (see, for example, [TW97]).

Besides the features we established above in the general admissible setting, the family of Minkowski wedges has the following well-known properties:

  1. Each wedge is the causal completion of the world line of a uniformly accelerated observer.

  2. Each wedge is the union of a family of double cones whose tips lie on two fixed lightrays.

  3. The isometry group (the Poincaré group) acts transitively on .

  4. is causally separating in the sense that given any two spacelike separated double cones , then there exists a wedge such that [TW97]. is a subbase for the topology of .

All these properties a)–d) do not hold for the class of wedges on a general admissible spacetime, but some hold for certain subclasses, as can be seen from the explicit examples in the subsequent section.

There exist a number of different constructions for wedges in curved spacetimes in the literature, mostly for special manifolds. On de Sitter respectively anti de Sitter space Borchers and Buchholz [BB99] respectively Buchholz and Summers [BS04] construct wedges by taking property a) as their defining feature, see also the generalization by Strich [Str08]. In the de Sitter case, this definition is equivalent to our definition of a wedge as a connected component of the causal complement of an edge [BMS01]. But as two-dimensional spheres, the de Sitter edges do not admit two linearly independent commuting Killing fields. Apart from this difference due to our restriction to commuting, linearly independent, Killing fields, the de Sitter wedges can be constructed in the same way as presented here. Thanks to the maximal symmetry of the de Sitter and anti de Sitter spacetimes, the respective isometry groups act transitively on the corresponding wedge families (c), and causally separate in the sense of d).

A definition related to the previous examples has been given by Lauridsen-Ribeiro for wedges in asymptotically anti de Sitter spacetimes (see Def. 1.5 in [LR07]). Note that these spacetimes are not admissible in our sense since anti de Sitter space is not globally hyperbolic.

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

Wedges in a class of Friedmann-Robertson-Walker spacetimes with spherical spatial sections have been constructed with the help of conformal embeddings into de Sitter space [BMS01]. This construction also yields wedges defined as connected components of causal complements of edges. Here a) does not, but c) and d) do hold, see also our discussion of Friedmann-Robertson-Walker spacetimes with flat spatial sections in the next section.

The idea of constructing wedges as connected components of causal complements of specific two-dimensional submanifolds has also been used in the context of globally hyperbolic spacetimes with a bifurcate Killing horizon [GLRV01], building on earlier work in [Kay85]. Here the edge is given as the fixed point manifold of the Killing flow associated with the horizon.

2.2 Concrete examples

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

The easiest way to construct an admissble spacetime is to take the warped product [O’N83, Chap. 7] of an edge with another manifold. Let be a two-dimensional Riemannian manifold endowed with two complete, commuting, linearly independent, smooth Killing fields, and let be a two-dimensional, globally hyperbolic spacetime diffeomorphic to , with an open interval or the full real line. Then, given a positive smooth function on , consider the warped product , i.e., the product manifold endowed with the metric tensor field

where and are the projections on and . It readily follows that is admissible in the sense of Definition 2.1.

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

Proposition 2.10

Let be a spacetime diffeomorphic to , where is open and simply connected, endowed with a global coordinate system according to which the metric reads

(2.17)

Here runs over the whole , for and do not depend on and . Then is an admissible spacetime in the sense of Definition 2.1.

Proof: Per direct inspection of (2.17), is isometric to with with , where is smooth and positive, and is a metric which depends smoothly on . Furthermore, on the hypersurfaces at constant , and is block-diagonal. If we consider the sub-matrix with , this has a positive determinant and a positive trace. Hence we can conclude that the induced metric on is Riemannian, or, in other words, is a spacelike, smooth, three-dimensional Riemannian hypersurface. Therefore we can apply Theorem 1.1 in [BS05] to conclude that is globally hyperbolic.

Since the metric coefficients are independent from and , the vector fields and are smooth Killing fields which commute and, as they lie tangent to the Riemannian hypersurfaces at constant time, they are also spacelike. Furthermore, since per definition of spacetime, and thus also is connected, we can invoke the Hopf-Rinow-Theorem [O’N83, 5, Thm. 21] to conclude that is complete and, thus, all its Killing fields are complete. As is simply connected by assumption, it follows that is admissible.

Under an additional assumption, also a partial converse of Proposition 2.10 is true. Namely, let be a globally hyperbolic spacetime with two complete, spacelike, commuting, smooth Killing fields, and pick a local coordinate system , where and are the flow parameters of the Killing fields. Then, if the reflection map , , is an isometry, the metric is locally of the form (2.17), as was proven in [Cha83, CF84]. The reflection is used to guarantee the vanishing of the unwanted off-diagonal metric coefficients, namely those associated to “” and “”. Notice that the cited papers allow only to establish a result on the local structure of and no a priori condition is imposed on the topology of , in distinction to Proposition 2.10.

Some of the metrics (2.17) are used in cosmology. For the description of a spatially homogeneous but in general anisotropic universe where (see §5 in [Wal84] and [FPH74]), one puts in (2.17) and takes to depend only on . This yields the metric of Kasner spacetimes respectively Bianchi I models333 The Bianchi models I–IX [Ell06] arise from the classification of three-dimensional real Lie algebras, thought of as Lie subalgebras of the Lie algebra of Killing fields. Only the cases Bianchi I–VII, in which the three-dimensional Lie algebra contains as a subalgebra, are of direct interest here, since only in these cases Killing pairs exist.

(2.18)

Clearly here the isometry group contains three smooth Killing fields, locally given by , which are everywhere linearly independent, complete and commuting. In particular, , and are Killing pairs.

A case of great physical relevance arises when specializing the metric further by taking all the functions in (2.18) to coincide. In this case, the metric assumes the so-called Friedmann-Robertson-Walker form

(2.19)

Here the scale factor is defined on some interval , and in the second equality, we have introduced the conformal time , which is implicitely defined by . Notice that, as in the Bianchi I model, the manifold is , i.e., the variable does not need to range over the whole real axis. (This does not affect the property of global hyperbolicity.)

By inspection of (2.19), it is clear that the isometry group of this spacetime contains the three-dimensional Euclidean group . Disregarding the Minkowski case, where and is constant, the isometry group in fact coincides with . Edges in such a Friedmann-Robertson-Walker universe are of the form , where is a two-dimensional plane in and . Here consists of a single coherent family, and the -orbits in are labelled by the time parameter for the corresponding edges. Also note that the family of Friedmann-Robertson-Walker wedges is causally separating in the sense discussed on page 4.

The second form of the metric in (2.19) is manifestly a conformal rescaling of the flat Minkowski metric. Interpreting the coordinates as coordinates of a point in therefore gives rise to a conformal embedding .


Two wedges in FRW spacetime

In this situation, it is interesting to note that the set of all images of edges in the Friedmann-Robertson-Walker spacetime coincides with the set of all Minkowski space edges which lie completely in , provided that does not coincide with . These are just the edges parallel to the standard Cauchy surfaces of constant in . So Friedmann-Robertson-Walker edges can also be characterized in terms of Minkowski space edges and the conformal embedding , analogous to the construction of wedges in Friedmann-Robertson-Walker spacetimes with spherical spatial sections in [BMS01].

3 Quantum field theories on admissible spacetimes

Having discussed the relevant geometric structures, we now fix an admissible spacetime and discuss warped convolution deformations of quantum field theories on it. For models on flat Minkowski space, it is known that this deformation procedure weakens point-like localization to localization in wedges [BLS10], and we will show here that the same holds true for admissible curved spacetimes. For a convenient description of this weakened form of localization, and for a straightforward application of the warped convolution 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 we consider algebras of quantum fields supported in wedges . The main idea underlying the deformation is to apply the formalism developed in [BS08, BLS10], but with the global translation symmetries of Minkowski space replaced by the Killing flow corresponding to the wedge under consideration. In the case of Minkowski spacetime, these deformations reduce to the familiar structure of a noncommutative Minkowski space with commuting time.

The details of the model under consideration will not be important in Section 3.1, since our construction relies only on a few structural properties satisfied in any well-behaved quantum field theory. In Section 3.2, the deformed Dirac quantum field is presented as a particular example.

3.1 Deformations of nets with Killing symmetries

Proceeding to the standard mathematical formalism [Haa96, Ara99], we consider a -algebra , whose elements are interpreted as (bounded functions of) quantum fields on the spacetime . The field algebra has a local structure, and in the present context, we focus on localization in wedges , since this form of localization turns out to be stable under the deformation. Therefore, corresponding to each wedge , we consider the -subalgebra of fields supported in . Furthermore, we assume a strongly continuous action of the isometry group of on , and a Bose/Fermi automorphism whose square is the identity automorphism, and which commutes with . This automorphism will be used to separate the Bose/Fermi parts of fields ; in the model theory of a free Dirac field discussed later, it can be chosen as a rotation by in the Dirac bundle.

To allow for a straightforward application of the results of [BLS10], we will also assume in the following that the field algebra is concretely realized on a separable Hilbert space , which carries a unitary representation of implementing the action , i.e.,

We emphasize that despite working on a Hilbert space, we do not select a state, since we do not make any assumptions regarding -invariant vectors in or the spectrum of subgroups of the representation .444Note that every -dynamical system , where is a concrete -algebra on a separable Hilbert space and is a strongly continuous representation of the locally compact group , has a covariant representation [Ped79, Prop. 7.4.7, Lemma 7.4.9], build out of the left-regular representation on the Hilbert space . The subsequent analysis will be carried out in a -setting, without using the weak closures of the field algebras in .

For convenience, we also require the Bose/Fermi automorphism to be unitarily implemented on , i.e., there exists a unitary such that . We will also use the associated unitary twist operator

(3.1)

Clearly, the unitarily implemented and can be continued to all of . By a slight abuse of notation, these extensions will be denoted by the same symbols.

In terms of the data , the structural properties of a quantum field theory on can be summarized as follows [Haa96, Ara99].

  1. Isotony: whenever .

  2. Covariance under :

    (3.2)
  3. Twisted Locality: With the unitary (3.1), there holds