1 Introduction and summary

Dirac Operators on Noncommutative Curved Spacetimes


Dirac Operators on Noncommutative
Curved SpacetimesThis paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html


Alexander SCHENKEL  and Christoph F. UHLEMANN 


A. Schenkel and C.F. Uhlemann


 Fachgruppe Mathematik, Bergische Universität Wuppertal,
 Gaußstraße 20, 42119 Wuppertal, Germany \EmailDschenkel@math.uni-wuppertal.de


 Department of Physics, University of Washington, Seattle, WA 98195-1560, USA \EmailDuhlemann@uw.edu


Received August 09, 2013, in final form December 11, 2013; Published online December 15, 2013


We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples.


Dirac operators; Dirac fields; Drinfeld twists; deformation quantization; noncommutative quantum field theory; quantum field theory on curved spacetimes


81T75; 81T20; 83C65

1 Introduction and summary

Noncommutative geometry has long been of interest from a purely mathematical perspective as a natural generalization of ordinary differential geometry. It is also of crucial interest from a physical perspective, since it generically plays a role when the principles of quantum mechanics are combined with those of general relativity [17, 18]. In both contexts, Dirac operators are of major importance: they are relevant for structural questions in noncommutative geometry [13] and essential for the description of fermionic fields in models for high-energy physics. In this article we focus on the latter point and study Dirac operators as equation of motion operators for Dirac fields on noncommutative spacetimes. The noncommutative geometries which we are going to consider are obtained by formal deformation quantization of smooth manifolds via Abelian Drinfeld twists. Since we are interested in generic curved spacetimes, we will not assume compatibility conditions between the geometry and the twist. In particular, we do not restrict ourselves to twists generated solely by Killing vector fields, as these are just not available on generic spacetimes.

The construction of Dirac operators in our generic setting turns out to be much more involved than in the highly symmetric setup with twists constructed solely from Killing vector fields. As shown in [14], the classical Dirac operator is in this special case also a valid Dirac operator on the noncommutative manifold. In the generic case, however, there are several natural deformations of the classical Dirac operator, which at first seem equally well motivated and are not obviously equivalent. It is thus not clear which operator we should choose, and that state of affairs certainly is not satisfactory. To improve on it, we will propose an abstract characterization of a Dirac operator on noncommutative curved spacetimes, in terms of a minimal set of axioms. Namely, it should be a differential operator of first order in a sense appropriate for noncommutative geometry, it should be constructed from geometric objects like the spin connection and the vielbein alone, and it should have the correct classical limit. We can then study to which extent the various explicit constructions realize these properties, and under which circumstances they turn out to be equivalent.

We find that the minimal set of axioms does in general not uniquely select one of the constructions that we are going to present. That is, we show that two of our operators satisfy the axioms and turn out to be inequivalent. As is to be expected, the classical Dirac operator is not among them. Restricting then to more special classes of deformations, we find that for twists constructed from sufficiently many Killing vector fields (semi-Killing twists, which are important for studying solutions of the noncommutative Einstein equations [2, 26, 33]), our candidates for Dirac operators all agree and fit into our general characterization of noncommutative Dirac operators. The freedom in choosing a Dirac operator is thus reduced drastically in this special class of deformations. Furthermore, as one is thus free to choose the technically most convenient definition, this can simplify explicit calculations considerably. The classical Dirac operator meets our axioms for noncommutative Dirac operators only when restricting to actual Killing twists, and it then coincides with all the deformed constructions. Turning back to the general case of twists which do not necessarily involve Killing vector fields, the question remains which of the (e.g. inequivalent) noncommutative Dirac operators one should prefer. Having in mind the construction of noncommutative quantum field theories, it is natural to also demand formal self-adjointness with respect to a suitable inner product. We will find that this requirement indeed singles out a preferred choice among our candidates, and we will outline the construction of a quantum field theory of noncommutative Dirac fields. A natural next step would be to aim for a complete classification of noncommutative Dirac operators satisfying our axioms, which we leave for future work.

The outline of this paper is as follows: For the coupling of Dirac fields to the noncommutative background geometry we employ techniques of twist-deformed noncommutative geometry and noncommutative vielbein gravity [1], which we review in Section 2. In Section 3 we discuss three well-motivated deformations of the classical Dirac operator, and present our minimal set of axioms for Dirac operators in the noncommutative setting. We then show that two of the proposed operators generally satisfy these axioms. In Section 4 we show that for generic noncommutative curved spacetimes the two noncommutative Dirac operators are inequivalent, and that adding formal self-adjointness as an additional condition selects a unique Dirac operator, at least among the examples we have provided. Furthermore, restricting to a special class of deformations given by semi-Killing twists we show that the ambiguities in defining noncommutative Dirac operators disappear in these highly symmetric models, where the twist contains sufficiently many Killing vector fields. In Section 5 we outline the construction of noncommutative Dirac quantum field theories. To illustrate our constructions we provide in Section 6 explicit examples and formulas for noncommutative Dirac operators on spacetimes of physical interest.

2 Preliminaries

In the following we review techniques from deformation quantization of smooth manifolds by Abelian Drinfeld twists and the framework of noncommutative vielbein gravity, as far as they will be relevant for the main part.

2.1 Twist-deformed noncommutative geometry

Let be a -dimensional manifold. The noncommutative geometries that we shall consider are those which arise as deformations of by an Abelian Drinfeld twist


where is an antisymmetric, real and constant matrix (not necessarily of rank ) and are mutually commuting real vector fields on , i.e.  for all , .111The reason for restricting to Abelian Drinfeld twists is the validity of the graded cyclicity property (2.3), which does not hold true for generic Drinfeld twists. The deformation parameter is assumed to be infinitesimally small, i.e. we work in formal deformation quantization. In this setup a formal power series extension of the complex numbers, as well as of all vector spaces, algebras, etc., has to be performed, but for notational simplicity we will suppress the square brackets denoting these extensions. We can assume, without loss of generality, that is of the canonical (Darboux) form


It is convenient to introduce the following notation.

Definition 2.1.

A twisted manifold is a pair , where is a -dimensional manifold and  is an Abelian Drinfeld twist, cf. (2.1).

On any twisted manifold we can develop a canonical noncommutative differential geometry. As a first step, consider , the space of all complex-valued smooth functions on . By using the twist , we can equip this space with an associative and noncommutative product (the -product)

where is the usual point-wise product and the action of the vector fields on the functions  is via the (Lie) derivative.

Moreover, the de Rham calculus on can be deformed by  into a differential calculus on the -product algebra . Explicitly, we define the -product

where the action of the vector fields on the differential forms , is via the Lie derivative. The undeformed differential satisfies the graded Leibniz rule with respect to the -product, i.e.  with denoting the degree of , and hence is a differential calculus over . We extend the involution on , which is given by point-wise complex conjugation, to a graded involution on by applying the rules and . The undeformed integral satisfies the graded cyclicity property. This means that, for all with compact overlapping support and such that ,


Finally, we deform the contraction operator (interior product) between vector fields and one-forms on by the twist . The resulting -contraction operator is given by (cf. [4])


where again the vector fields act via the Lie derivative on vector fields  and one-forms .

This completes our snapshot review of twist-deformed noncommutative geometry and we refer the reader to [29] for a more detailed discussion.

2.2 Noncommutative vielbein gravity

The explicit constructions and examples of the main part will involve spacetimes of dimension  and , and we therefore give the formalism of noncommutative vielbein gravity for both cases in the following. Throughout, will denote a twisted manifold of appropriate dimension. Following [1], we describe the noncommutative gravitational field by a noncommutative vielbein field and a noncommutative spin connection . Both are Clifford-algebra valued one-forms. Our gamma-matrix conventions are collected for easy reference in Appendix A. We say that the noncommutative spin connection is -torsion free if , where is the -anticommutator. The -torsion constraint is part of the equations of motion of noncommutative vielbein gravity [1]. For reasons of generality, we do not assume the -torsion constraint for our general constructions and we shall clearly indicate at which later step it is used.

. We can expand and in terms of the gamma-matrix basis of the -dimensional Clifford algebra as


We further demand the reality conditions and . Notice that noncommutative vielbein gravity contains more fields than its commutative counterpart, where . The reason is that (Lorentz) -gauge transformations do not close and have to be extended to -gauge transformations. The -gauge transformations act on and by


where is a Clifford algebra valued function and is the -commutator. We impose the reality condition .

As in [1] we require that , such that the commutative limit yields a usual commutative vierbein and spin connection. We shall use the notation .

Definition 2.2.

Let be a -dimensional twisted manifold. A noncommutative Cartan geometry on is a pair of Clifford algebra valued one-forms , satisfying the expansion (2.5), the reality conditions , and the limit .

Let us now consider Dirac fields, i.e. functions valued in the fundamental representation of the Clifford algebra. We denote the Dirac adjoint by . The -gauge transformations act on and by and , respectively. Notice that the matrix transforms in the adjoint representation, . For all Dirac fields  with compact overlapping support we define the inner product


which is -gauge invariant due to (2.3), (2.6) and the cyclicity of the matrix trace .

Lemma 2.3.

Let be a noncommutative Cartan geometry on a -dimensional twisted manifold . Then the inner product (2.7) is hermitian, it reduces to the canonical commutative one for and it is non-degenerate, i.e.:

  • .

  • , where .

  • If for all , then .


We show by the following short calculation

In the second equality we have used (graded) cyclicity, the reality condition , , and . In the third equality we have used , and .

To show let us set in (2.7) and use that (i.e. that vanishes at order ). Using further that the antisymmetrized product of 4 gamma-matrices is and that we obtain the desired result.

is a consequence of and the fact that the classical inner product is non-degenerate. ∎

. The noncommutative twobein and spin connection have the following expansion in terms of the gamma-matrix basis of the -dimensional Clifford algebra


We define for the -gauge transformations and . As in the case of , we had to introduce the extra fields and such that the -gauge transformations close. Note, however, that we do not need additional terms in the twobein field and thus the interpretation of as a soldering form remains valid in . This will facilitate the study of noncommutative Dirac operators in , as compared to . We again impose the reality conditions , and .

Definition 2.4.

Let be a -dimensional twisted manifold. A noncommutative Cartan geometry on is a pair of Clifford algebra valued one-forms , satisfying the expansion (2.8), the reality conditions , and the limit .

Let us now consider Dirac fields, which in the case of are functions with values in the fundamental representation of the Clifford algebra. The Dirac adjoint is and -gauge transformations act on and via and . We define in analogy to (2.7) a -gauge invariant and hermitian inner product


For we obtain the usual inner product , since .

3 Noncommutative Dirac operators

As a first step, we will give three explicit candidate definitions for noncommutative Dirac operators in and . These are obtained by using techniques of noncommutative differential geometry and twist deformation quantization. Depending on taste and point of view, either of them may be seen as a valid extension of the classical Dirac operator to the noncommutative setting. This shows that a more abstract characterization of noncommutative Dirac operators is needed, and we develop in the second step what we believe is a minimal set of axioms for such operators. This will already rule out the classical Dirac operator along with one of our candidates, and we show that the remaining two indeed meet our criteria for noncommutative Dirac operators.

3.1 Explicit candidates

The following set of candidates for noncommutative Dirac operators should show, how focusing on different aspects of noncommutative differential geometry and twist deformation quantization leads to different constructions. It is not meant to be exhaustive.

The Aschieri–Castellani Dirac operator. The first operator we consider is motivated by the noncommutative Dirac field action proposed in [1], which reads


where is the -covariant differential acting on Dirac fields. Since the inner product (2.7) is non-degenerate, we can define a differential operator by requiring that, for all of compact support,


This yields exactly the equation of motion operator which is obtained by varying the action (3.1) with respect to . The construction in is fully analogous. The action then reads

and since the inner product (2.9) is also non-degenerate, we can define a differential operator by requiring that, for all of compact support,


The contraction Dirac operator. For our next operator we shall follow closely the usual construction of a Dirac operator on commutative spacetimes, which goes as follows: Let be a classical vielbein, a classical spin connection and let us denote by the inverse vielbein. The classical Dirac operator is , where we have expressed in the vielbein basis . Notice that this operator can be written in an index-free form , where is the classical contraction operator (interior product).

Using the deformed contraction operator between vector fields and one-forms as defined in (2.4), we generalize the above construction to the noncommutative setting. For this we define the -inverse vielbein by the -contraction condition . We collect all in the Clifford algebra valued vector field . Following the same strategy as in the classical case, we define a differential operator by


The construction outlined above is valid as it stands in . For the following remark is in order: As seen in (2.5), the noncommutative vierbein field has an extra field . This implies that is locally a -matrix and hence there is no unique -inverse . Since invertibility of the vielbein is essential in classical vielbein gravity, this may be seen as a shortcoming of the noncommutative vielbein gravity formulated in [1]. There have been attempts to overcome this issue by using Seiberg–Witten maps [3], which, however, obscure the noncommutative differential geometry and in practice require an expansion in the deformation parameter to some fixed order, so they are not convenient for our purpose. We shall instead take the following approach: We restrict the class of allowed noncommutative Cartan geometries to those satisfying . We are aware that this restriction is not invariant under -gauge transformations (in ) and hence it is not convenient for dynamical noncommutative vielbein gravity. However, for a given fixed noncommutative Cartan geometry it certainly makes sense to assume a  of this special form.

The deformed Dirac operator. The last noncommutative Dirac operator is motivated by the framework of Connes for noncommutative spin geometry [13], where the Dirac operator enters as a fundamental degree of freedom of the theory. It is obtained by deforming the classical Dirac operator via the techniques developed in [6]. More precisely, denoting the inverse twist by , we define the deformed Dirac operator by applying the deformation map constructed in [6]

where is the adjoint action.

3.2 Abstract consideration

The variety of different generalizations of the classical Dirac operator to the noncommutative setting provided above clearly calls for a more precise definition of what we actually mean by a noncommutative Dirac operator. As a reasonable starting point, we are looking for linear differential operators acting on Dirac fields ( is the dimension of the fundamental representation of the Clifford algebra), subject to certain conditions generalizing the properties of the classical Dirac operator. Thus, we naturally start by generalizing some relevant notions to the noncommutative setting, beginning with the notion of a first-order differential operator.

Definition 3.1.

Let be a twisted manifold. A differential operator is called a first-order noncommutative differential operator, if for all and ,


where is a spinor-valued vector field on and the -contraction is defined in (2.4).

For we obtain from (3.5) the usual Leibniz rule property of a first-order differential operator. Written in local coordinates it reads . Hence, (3.5) promotes this property to the realm of twisted manifolds. Note that a first-order noncommutative differential operator is not necessarily a first-order differential operator in the usual sense (cf. the examples in the sections below). It can and in general must contain higher order derivatives, but these are restricted by the form of the twist .

The second notion we want to generalize aims to capture more of the essence of the classical Dirac operator. Namely, that it is constructed from purely geometric data in a natural way. We will formalize the requirement that a noncommutative Dirac operator should be constructed only from the data of the noncommutative Cartan geometry and the twisted manifold in a geometric (natural) way as follows222This can be made more precise in a category theoretical framework for noncommutative Cartan geometries on twisted manifolds, where a natural differential operator could be defined in terms of a natural transformation between the section functors of the Dirac bundles.:

Definition 3.2.

Let be a noncommutative Cartan geometry on a twisted manifold . A differential operator is called a geometric noncommutative differential operator if it is constructed from the noncommutative vielbein and the -covariant differential in terms of the operations of twisted noncommutative geometry.

We can now state our general definition for noncommutative Dirac operators, which combines the properties introduced above with the natural demand that the standard Dirac operator should be recovered in the commutative limit:

Definition 3.3.

A noncommutative Dirac operator on a noncommutative Cartan geometry over a twisted manifold is a differential operator , such that

  • is a first-order noncommutative differential operator,

  • is a geometric noncommutative differential operator,

  • reproduces the classical Dirac operator corresponding to for .

Having stated this definition, the first two questions one could ask are, firstly, whether there are noncommutative Dirac operators in this sense at all, and, secondly, whether the requirements are possibly trivial altogether. In the remaining part of this subsection we will answer these two questions, focusing on and .

The first question is easily answered by providing an explicit construction which satisfies the demands of Definition 3.3:

Proposition 3.4.

Let be any - or -dimensional twisted manifold and any noncommutative Cartan geometry. Then the operator defined in (3.3) and (3.2) is a noncommutative Dirac operator according to Definition 3.3.


We give the proof for , and note that it follows analogously for . We have to check the three conditions in Definition 3.3. For property 1) let us evaluate the following inner product

The proof would follow if we could show that the condition


defines a unique spinor-valued vector field on . This is indeed the case by the following argument: Notice that is equal to the classical volume form at order . Since this form is non-degenerate, the condition (3.6) determines a unique spinor-valued function , for any . The operation is right linear under the -multiplication by , for all and ,

Since the space of vector fields is exactly the dual module of the module of one-forms, is a spinor-valued vector field.

Property 2) is clear: We have used only -products, -products, integrals , vielbeins and -covariant differentials in order to define .

To prove property 3) we consider (3.3) at . We expand in the twobein basis , i.e. , where is the inverse of , which is a vector field. We obtain

Bearing in mind the remark on the case below Equation (3.4), also the contraction Dirac operator is valid in the sense of Definition 3.3:

Proposition 3.5.

Let be a twisted manifold of dimension or , and a noncommutative Cartan geometry, such that . Then the operator defined in (3.4) is a noncommutative Dirac operator according to Definition 3.3.


We have to check the three conditions in Definition 3.3. Property 1) follows from a short calculation

where .

Property 2) is clear: We have only used , , the -inverse (defined via ) and to construct . Furthermore, property 3) is a consequence of the fact that all operations entering (3.4) reduce for to the corresponding classical operations. ∎

To answer the second question and show that the requirements are indeed not vacuous, we show that, quite expectedly, the classical Dirac operator fails to meet our criteria. It will nevertheless be instructive to see which conditions are violated. Furthermore, we will see that the deformed Dirac operator is ruled out as well. Let us first note that property 3), controlling the classical limit, is satisfied by both of these operators. In order to understand if the classical Dirac operator satisfies property 1), we expand up to first order in the deformation parameter , which yields

For to be a first-order noncommutative differential operator the have to commute with  and , i.e. the twist has to be generated completely by Killing vector fields. This shows that for generic noncommutative Cartan geometries on twisted manifolds the classical Dirac operator is not a first-order noncommutative differential operator and in particular not a noncommutative Dirac operator.

The deformed Dirac operator, on the other hand, is a first-order noncommutative differential operator, since

However, like the classical Dirac operator it fails to satisfy property 2), as in the construction of  and  undeformed covariant differentials and contraction operators appear. Summing up, our axioms above are satisfied by and , but the deformed and classical Dirac operator fails in the general to be a noncommutative Dirac operator. We will turn to the question for the remaining freedom to choose a Dirac operator in the next sections.

4 Comparing the noncommutative Dirac operators

4.1 Non-uniqueness in the general case

Having established two examples of noncommutative Dirac operators on noncommutative Cartan geometries over twisted manifolds , we shall now show that they do not coincide in general. Our strategy is to calculate explicitly the two noncommutative Dirac operators and for a simple example of and , from which the desired result can be directly read off.

We will start with the -dimensional case and consider the noncommutative spacetime known as ‘quantum plane’. Let and consider the twist in (2.1), constructed from and , where and are global coordinates. Notice that this twisted manifold leads to the commutation relations of the quantum plane, i.e. . We equip this twisted manifold with the following noncommutative Cartan geometry: and is the unique -torsion free connection. The -inverse of is defined by and it is given by , . We further find for the -covariant differential . This leads to the following contraction Dirac operator (3.4)


In order to compare our two noncommutative Dirac operators, we also evaluate the Aschieri–Castellani Dirac operator (3.3) for this model. Using that , with denoting the volume form, we obtain for the inner product (2.9)


Furthermore, evaluating (3.3) we obtain

which yields the Aschieri–Castellani Dirac operator on the quantum plane


Comparing (4.1) and (4.3) we observe that the noncommutative Dirac operators and do not coincide. Notice that the difference is not just in the overall factor, but the two terms have also acquired different phases. With the -dimensional analog of the quantum plane and a similar calculation, we obtain the following result.

Proposition 4.1.

The two noncommutative Dirac operators and do not coincide for generic noncommutative Cartan geometries over twisted manifolds of dimension  or .

4.2 The formal self-adjointness condition

In regard of the non-uniqueness result above we try to include stronger conditions on noncommutative Dirac operators in order to single out a preferred choice. We shall focus in this subsection only on one strongly motivated extra condition, which is formal self-adjointness. This condition is essential for associating to a noncommutative Dirac operator a quantum field theory of noncommutative Dirac fields, see Section 5. Notice that also in a Riemannian setting, a formal self-adjointness condition is an important ingredient for understanding the spectral theory of Dirac operators.

Definition 4.2.

Let be a twisted manifold and a noncommutative Cartan geometry on . A noncommutative Dirac operator is called formally self-adjoint, if it satisfies , for all Dirac fields , with compactly overlapping support. Here  is the inner product defined for in (2.9) and for in (2.7).

Notice that the classical Dirac operator is formally self-adjoint with respect to the classical inner product if the spin connection is torsion free. For the Aschieri–Castellani Dirac operator we obtain the analogous property in the twisted setting.

Proposition 4.3.

Let be any twisted manifold of dimension or , and let be a noncommutative Cartan geometry that is -torsion free, i.e. . Then the noncommutative Dirac operator defined in (3.3) and (3.2) is formally self-adjoint.


We show this statement by the following calculation for the -dimensional case

In the first equality we have used hermiticity of the inner product. In the second equality we have used that is a graded involution on the deformed differential forms as well as , which follows from the hermiticity condition . Then for the third step we used , and , and in the fourth one graded cyclicity (2.3) and twice . In the fifth equality we made use of the graded Leibniz rule of and the -torsion constraint . The last step is simply Stokes’ theorem. The proof for the -dimensional case is once again fully analogous. ∎

We will now show that our second example of a noncommutative Dirac operator, that is the contraction Dirac operator, does not satisfy the formal self-adjointness condition on generic -torsion free noncommutative Cartan geometries . To this end we again consider the quantum plane as a simple example. The contraction Dirac operator is given in and using the explicit form of the inner product (4.2) we obtain for the formal adjoint of the contraction Dirac operator