Deformations of quantum field theories on de Sitter spacetime

Eric Morfa-Morales

Mathematical Physics Group, University of Vienna,

Boltzmanngasse 5, A-1090, Vienna, Austria

Abstract: Quantum field theories on de Sitter spacetime with global gauge symmetry are deformed using the joint action of the internal symmetry group and a one-parameter group of boosts. The resulting theory turns out to be wedge-local and non-isomorphic to the initial one for a class of theories, including the free charged Dirac field. The properties of deformed models coming from inclusions of -algebras are studied in detail.

1 Introduction

The construction of interacting quantum field theories by deformations using suitable actions of has recently attracted much attention [GL07], [BS08], [GL08], [BLS10], [DT10], [DLM10]. A first example of this kind was found for the scalar free field on Minkowski space by Grosse and Lechner [GL07], where the deformed operators fulfill a weakened form of locality (wedge-locality) and the two-particle scattering is non-trivial. Later on this deformation method (warped convolution) was generalized to arbitrary quantum field theories on Minkowski space by Buchholz and Summers [BS08]. It is formulated in terms of an action of the translation subgroup of the Poincaré group and the resulting theory has similar properties as in the scalar free field case. Within a Wightman setting this deformation manifests itself as a deformation of the tensor product of the underlying Borchers-Uhlmann algebra [GL08]. Subsequently [BLS10] it was realized that there is in fact a close connection between warped convolutions and a well known deformation method for C-algebras in mathematics, namely, Rieffel deformations [Rie93]. It turns out that the warped algebra forms a representation of the Rieffel deformed algebra for a fixed deformation parameter. Later on this deformation scheme was applied to various situations in quantum field theory. In the chiral conformal case the first examples of massless models which are interacting and asymptotically complete were constructed by these methods [DT10]. Quantum field theories on a class of curved spacetimes relevant to cosmology can also be deformed using the flow of suitable Killing vector fields instead of translations [DLM10].

The de Sitter spacetime does not belong to the class of spacetimes considered in [DLM10]. It is the purpose of this paper to apply the warped convolution deformation procedure to quantum field theories on this spacetime. We use a combination of external and internal symmetries, consisting of a one-parameter group of boosts associated with a wedge and a global gauge symmetry, as an -action to define the deformation. The resulting theory is wedge-local and unitarily inequivalent to the undeformed one for a class of theories, including the free charged Dirac field.

In Section 2 the basic notions concerning the geometry and causal structure of de Sitter space are recalled and we discuss the de Sitter group together with its universal covering. After that, the covariance and inclusion properties of a class of distinguished regions (wedges) in de Sitter space are studied.

In Section 3 we consider quantum field theories with global gauge symmetry within the algebraic setting (field nets) and we show how to reconstruct a wedge-local field net from an inclusion of two C-algebras, which are in a suitable relative position to a wedge. Then the warped convolution deformation is applied to a field net with global gauge symmetry and the properties of the resulting theory are studied.

In Section 3.3 a particular class of field nets is investigated in more detail, namely, nets of -algebras. For these theories the deformed operators can be computed explicitly. The fixed-points of the deformation map are determined and it is shown that the deformed and undeformed field nets are non-isomorphic.

In the conclusions we comment on warped convolutions in terms of purely external and internal symmetries using other Abelian subgroups of the de Sitter and gauge group.

2 de Sitter spacetime

2.1 Geometry and causal structure

The de Sitter spacetime is a vacuum solution of Einstein’s equation with positive cosmological constant. It is maximally symmetric, so it admits 10 Killing vector fields, which is the maximum number for a spacetime of dimension four. It is also globally hyperbolic, so the Cauchy problem for partial differential equations of hyperbolic type, such as the Klein-Gordon and Dirac equation, is well-posed. Furthermore, it is a special case of the Friedmann-Robertson-Walker spacetimes which describe a spatially homogeneous and isotropic universe and it plays a prominent role in many inflationary scenarios for the early universe [Lin09].

Most conveniently it can be represented as the embedded submanifold

of five-dimensional Minkowski space , where is identified with , . The signature of is and the de Sitter radius is fixed to one. The metric on is the induced metric from the ambient space, i.e., , where is the embedding map. We use the ambient space notation to parametrize the de Sitter hyperboloid, so we write , for points in , subject to the relation , where is a Cartesian coordinate system of .

Since the metric on is the induced metric from the ambient Minkowski space, the causal structure is also inherited. Hence points in are called timelike, spacelike or null related, if they are so as points in , respectively. We fix a time orientation in once and for all. The interior of the causal complement of a spacetime region is denoted by .

For the generators of the Clifford algebra which is associated with the quadratic form on the vector space we use the representation [Gaz07]

where and are the Pauli matrices. We have and is a basis of the quaternions . Note that this representation is in fact not faithful, since . Similar to the case of four-dimensional Minkowski space, where points are parametrized by hermitian matrices, we parametrize points on the de Sitter hyperboloid by quaternionic matrices. This parametrization is useful for the discussion of the covering group of the de Sitter group later on. Define

where is the quaternionic conjugate of . Conversely, every matrix of the above form determines a point in de Sitter space via

The map defines an isomorphism between and , where are the hermitian matrices over . Furthermore, there holds , where is the transpose of the quaternionic conjugate of .

2.2 The de Sitter group and its covering

The isometry group of is

and its action on is given by the action of the Lorentz group in the ambient Minkowski space. This group is a ten-dimensional, non-compact, non-connected and real Lie group which has four connected components. The connected component which contains the identity is denoted by . This group is called de Sitter group (proper orthochronous Lorentz group) and its elements preserve the orientation and time orientation of .

Since we also want to treat quantum fields with half-integer spin we consider the two-fold (and universal) covering of , which is the spin group . Hence there exists a short exact sequence of group homomorphisms

There holds and is simply connected. Note that the Lie group is isomorphic to the pseudo-symplectic group [Tak63]

Equivalently, if and only if . In this representation the covering homomorphism is given by

and acts on by conjugation .

2.3 de Sitter wedges

Now we discuss the typical localization regions of the deformed quantum fields from Section 3. In [BB99] a de Sitter wedge is defined as the causal completion of the worldline of a uniformly accelerated observer (timelike geodesic) in de Sitter space. Equivalently, they can be characterized as intersections of wedges in the ambient Minkowski space [TW97] and the de Sitter hyperboloid. Hence we specify a reference (or right) wedge by

and define the family of wedges as the set of all de Sitter transforms of :

By definition, acts transitively on . Each wedge has an attached edge which is a two-sphere. We have and for . The wedge coincides with a connected component of the causal complement of the edge [BB99]. For the stabilizer of the wedge we write .

From the properties of wedges in follows that the causal complement of a wedge is again a wedge and that every is causally complete, i.e., . Furthermore, the family is causally separating, so given spacelike separated double cones , there exists a such that (see [TW97]).

Remark 2.1.

Wedges are frequently used as localization regions in quantum field theory [BW75], [Bor00], [BD00]. Although a net of observables over wedges is to a certain degree non-local, it is possible to construct a local theory over double cones from it. This is achieved by taking suitable intersections of algebras associated with wedges. Since is causally separating the resulting theory satisfies, in particular, local commutativity (see [BS08, BLS10]). To decide whether these intersections are in fact non-trivial remains a challenging task in four dimensions.

For every there exists a one-parameter group , such that each maps onto and for all . Moreover


Associated with is a future-directed Killing vector field in the wedge and the worldline from which the wedge is constructed is an integral curve of (a portion of) this vector field. Furthermore, for every there exists a reflection which maps onto and satisfies


Since acts transitively on we only need to specify these maps for . We choose


and note that is the associated Killing vector field.

Remark 2.2.

Within the context of applications of Tomita-Takesaki modular theory in quantum field theory the standard choice for the reflection is , which is an element of the extended symmetry group . In this paper we have no intention to use these techniques and the choice (2.3) appears to be more natural since we restrict our considerations to . However, all of our results can be generalized to the group in a straightforward manner.

The following lemma collects the basic properties of these maps.

Lemma 2.3.

Let and , be as above. Then

  1. ,

  2. ,


a): For holds for all by (2.1). b) follows from and (2.1), (2.2). ∎

The stabilizer of has the form , where are rotations in . Hence coincides with the center of . From b) follows that the Killing vector fields associated with and differ only by temporal orientation.

The following lemma shows that the possible causal configurations of wedges are very much constrained in de Sitter space.

Lemma 2.4.

Let and . Then .


The wedges can be written as , , where is a wedge in the ambient Minkowski space. Since the causal closure of in coincides with , there follows from . As the edges both contain the origin, there follows and also since . The assertion follows from the assumption together with the fact that is a connected component of the causal complement of . ∎

Remark 2.5.

All the previous statements carry over to the covering in a straightforward manner. Define an action of on with the covering homomorphism


which is transitive, since acts transitively. The one-parameter group lifts to a unique one-parameter group and for its elements we write . Again, since acts transitively on , we only need to specify these maps for . We have [Tak63, p.368]

Clearly, and for all , with respect to the action (2.4). For the lift of the reflection we choose

Again, and for all , with respect to (2.4). Hence analogous statements as in Lemma 2.3 hold for with replaced by , i.e.,


for all and .

3 Deformations of quantum field theories on de Sitter spacetime

3.1 Field nets

We work in the operator-algebraic approach to quantum field theory on curved spacetimes [Dim80] adapted to the concrete case of de Sitter space [BB99]. To this end, we consider a C-algebra (field algebra) whose elements are physically interpreted as (bounded functions of) quantum fields on . We equip with a local structure and focus on localization in wedges, since this turns out to be stable under the deformation. Hence we associate to each a C-subalgebra . Due to the trivial inclusion properties of wedges in de Sitter space (see Lemma 2.4) the usual isotony condition reduces to well-definedness of .

We assume that there exists a strongly continuous representation of by automorphisms on , such that

  • (De Sitter Covariance): for all , holds

Furthermore, we assume that there is a Lie group (global gauge group) and a strongly continuous representation of by automorphisms on , such that

  • (Gauge Invariance): for all , , holds


We assume that there exists a distinguished element such that satisfies


This (grading) automorphism can be used to separate an operator into its Bose() and Fermi() part via .

Remark 3.1.

For convenience, we assume that the datum is faithfully and covariantly represented on a Hilbert space . So to each corresponds a norm-closed -subalgebra of and the automorphisms are implemented by the adjoint action of unitary operators on , respectively. Note that this is no loss of generality since we can either use the covariant representation which exists for C-dynamical systems (see [BLS10, DLM10] and references therein) or we work in the GNS-representation of a de Sitter- and gauge-invariant state. In the former case we assume that is separable, as it is the case in a variety of concrete models.

We assume that the grading satisfies . With the operator a unitary twisting map is defined to treat the (anti)commutation relations between the Bose/Fermi parts of a field on the same footing [DHR69]. Let and

The map is an isomorphism of and we have [Foi83]

where the commutant is understood as the relative commutant in . Locality is now formulated in the following way

  • (Twisted Locality): .

Twisted locality is equivalent to the ordinary (anti)commutation relations between the Bose/Fermi parts of fields, i.e., for , , (see [DHR69]).

For later reference we define the joint action of the external and internal symmetry group on by


The unitary which implements this action is .

Remark 3.2.

A datum which satisfies conditions 1) 3) is referred to as a wedge-local field net. We simply write to denote it, if no confusion can arise. Examples are nets of -algebras with gauge symmetry, such as the free charged Dirac field.

Remark 3.3.

Given a field net, the net of observables is defined as

so observables form the gauge-invariant part of the field net.

Due to the transitive action of on , it is possible to define a wedge-local field net in terms of an inclusion of just two C-algebras which are in a suitable relative position to . This point of view will be advantageous for the warped convolution later on, since the deformation of a wedge-local field net amounts to deforming the relative position of one algebra in the other. Following [BLS10] we make the following definition.

Definition 3.4.

A causal Borchers system relative to consists of

  • an inclusion of concrete C-algebras,

  • commuting representations and which are unitarily implemented,

  • an automorphism on which commutes with and and satisfies ,

such that

  1. ,

  2. ,

  3. , .

Proposition 3.5.

Let be a causal Borchers system relative to . Then


defines a wedge-local field net together with .


We begin by proving well-definedness. From follows and by assumption a). Hence and the assertion follows.

Covariance holds by definition.

Twisted locality is proved in a similar way. Let . Since there holds

where we used condition b), together with the assumption that each , is a homomorphism which commutes with .

The gauge invariance of the local algebras follows immediately:

since the representations commute and by assumption c). ∎

Note that the converse of this proposition is trivially true. Given a wedge-local field net, then satisfies property a) by covariance and b) by twisted locality. Property c) holds by definition.

Remark 3.6.

A causal Borchers system is closely connected to the notion of a causal Borchers triple [BLS10] on Minkowski spacetime (see also [Lec10] for the related notion of a wedge triple). In this setting, is a von Neumann algebra and is the adjoint action of a unitary representation of the Poincaré group. In addition one assumes that the joint spectrum of the generators of the translations is contained in the closed forward lightcone (spectrum condition) and that admits a cyclic and separating vector (existence of a vacuum state). Gauge transformations are absent in this setting since nets of observables are considered.

Remark 3.7.

For the sake of brevity we will write to denote a causal Borchers system relative to .

3.2 Deformations of field nets with gauge symmetry

Now we apply the warped convolution deformation method to our present setting. Let be a causal Borchers system relative to . The basic idea is to define a deformation of the small algebra using a suitable -action (see below) in such a way that is again a causal Borchers system. Then the inclusion gives rise to another wedge-local field net by Proposition 3.5.

For the warped convolution we make the further assumption that the gauge group is . The representation of yields a -periodic -action by automorphisms on . The warped convolution is now defined with the -action coming from the one-parameter group of boosts and the internal symmetry group:

Note that implicitly depends on the Killing field which is associated with (see Section 2.3). We will use the notation

Since the warped convolution is defined is terms of oscillatory integrals of operator-valued functions, we first need to specify suitable smooth elements of the C-algebra for which these integrals are well-defined. The joint action (3.3) is a strongly continuous action of the Lie group which acts automorphically, and therefore isometrically, on . The algebra is, in general, only invariant under the action of the subgroup . Adapted to the present setting, and following [DLM10], we consider the following notion of smoothness with respect to the subgroup .

Definition 3.8.

An operator is called -smooth, if is smooth in the norm topology of . The set of all -smooth operators in is denoted by .

Note that the set is a norm-dense -subalgebra of (see [Tay86]). Another ingredient for the definition of the warped convolution is the antisymmetric (real) matrix

and an arbitrary but fixed real number which plays the role of a deformation parameter.

Definition 3.9.

The warped convolution of an operator is defined as


Here denotes the standard Euclidean inner product of and , is a cutoff function which is necessary to define this operator-valued integral in an oscillatory sense.

From the results in [BLS10] follows that the above limit exists in the strong operator topology of on the dense domain

and is independent of the chosen cutoff function within the specified class. The densely defined operator extends to a bounded and smooth operator, which is denoted by the same symbol.

Definition 3.10.

The space of all vectors which are smooth with respect to the representation is denoted by .

Furthermore, it is shown in [BLS10] that the warped convolution (3.5) is closely related to Rieffel deformations of C-algebras [Rie93]. In this context one defines, instead of a deformation of the algebra elements, a new product on by

This limit exists in the norm-topology of for all and is again in . The completion of in a suitable norm yields another C-algebra [Rie93].

The following lemma collects the basic properties of the map and shows that the warped operators form a representation of the Rieffel deformed C-algebra for a fixed deformation parameter.

Lemma 3.11 ([Bls10, Dlm10]).

Let and . Then

  1. .

  2. .

  3. If for all , then .

  4. If for all , then .

  5. Let be a unitary which commutes with for all . Then and is -smooth.


Statements a), b), c), e) were shown in [BLS10]. d) is a consequence of Lemma 3.2 in [DLM10] and the fact that commutes with and . ∎

The next lemma lists the transformation properties of warped operators under the de Sitter and gauge group.

Lemma 3.12.

Let , , and . Then

  1. is -smooth and


    where is the push-forward of with respect to .

  2. is -smooth and


Statement a) follows from Lemma 3.3 a) in [DLM10] and the fact that and commute. Statement b) follows from Lemma 3.11 e). ∎

Now we apply the warped convolution deformation method to a causal Borchers system . Define

The following theorem shows that the inclusion gives rise to a wedge-local field net in the sense of Proposition 3.5.

Theorem 3.13.

Let be as above. Then

  1. ,

  2. ,

  3. .


a): Let and . From (2.5) follows that commutes with each . Hence

by Lemma 3.12 a) and by property a) of the undeformed causal Borchers system. Therefore and by taking the norm-closure of the statement follows.

b): From Lemma 3.12 a) and (2.5) follows


together with an elementary substitution in (3.5). We have by property b) of the undeformed causal Borchers system, i.e., for all . Pick some and consider its warped convolution . We have for all since is invariant under . Hence

by (3.8) and Lemma 3.11 e). By taking the norm-closure of the statement follows.

Assertion c) is a consequence of Lemma 3.12 b) and the invariance under gauge transformations. ∎

Remark 3.14.

Note that the minus sign which appears in (3.8) is the main reason why the locality proof works. That this argument is also valid for the extended symmetry group can be seen in the following way. The reflection commutes with boosts in the -direction. Again, a deformed operator transforms under the lift of according to since is represented by an antiunitary operator.

3.3 Example: Deformations of -nets

Now we investigate a particular class of wedge-local field nets in more detail, namely, nets of -algebras. The free charged Dirac field is an example thereof. After it is shown that these models fit into the framework of Section 3.2, the properties of the deformed field operators and observables are studied in detail and it is proved that the deformed and undeformed nets are non-isomorphic.

3.3.1 The selfdual -algebra

We use Araki’s selfdual approach to the -algebra [Ara71]. Let be a separable infinite-dimensional complex Hilbert space with inner product and let be an antiunitary involution on , i.e., and for all . On the -algebra which is algebraically generated by elements and a unit , satisfying

  • is complex linear,

  • ,

  • ,

there exists a unique C-norm satisfying (see [EK98])

Hence each is bounded and is norm-continuous. The C-completion of is denoted by . This C-algebra is simple [Ara71], so all its representations are faithful or trivial.

If is a unitary on which commutes , then defines a -automorphism on . We refer to as Bogolyubov transformation and to as Bogolyubov automorphism.

3.3.2 Quasifree representations

A state on is called quasifree, if

holds for all , where the sum runs over all permutations of satisfying

Let be a bounded linear operator on satisfying

In [Ara71, Lemma 3.3] it is shown that for every such there exists a unique quasifree state satisfying

Conversely, every quasifree state on gives rise to such an operator [Ara71, Lemma 3.2]. Hence quasifree states can be parametrized by this class of operators.

Let be a quasifree state. For the GNS-triple associated with we write . If a Bogolyubov transformation commutes with , then the associated Bogolyubov automorphism can be unitarily implemented, i.e., there exists a unitary operator on , such that