aa r X i v : . [ m a t h - ph ] A ug Temporal Lorentzian Spectral Triples
Nicolas Franco
Copernicus Center for Interdisciplinary Studies ∗ ,Jagiellonian University,ul. S lawkowska 17, PL-31-016 Krak´ow, PolandUniversity of Namur, Department of Mathematics,Rempart de la Vierge 8, B-5000 Namur, [email protected] Abstract
We present the notion of temporal Lorentzian spectral triple which isan extension of the notion of pseudo-Riemannian spectral triple with away to ensure that the signature of the metric is Lorentzian. A temporalLorentzian spectral triple corresponds to a specific 3+1 decomposition ofa possibly noncommutative Lorentzian space. This structure introducesa notion of global time in noncommutative geometry. As an example, weconstruct a temporal Lorentzian spectral triple over a Moyal–Minkowskispacetime. We show that, when time is commutative, the algebra can beextended to unbounded elements. Using such an extension, it is possibleto define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.
The theory of noncommutative geometry is related to the duality existing be-tween algebra and geometry. Thanks to Gel’fand–Naimark theorem, we knowthat unital commutative C ∗ -algebras are equivalent in a category-theoreticalsense to algebras of continuous functions on compact Hausdorff spaces. A sim-ilar result is valid for locally compact Hausdorff spaces being equivalent to notnecessarily unital commutative algebras. The extension of this duality to non-commutative C ∗ -algebras opens the way to the definition of noncommutativespaces.Behind those topological considerations, some metric information can beadded by using Connes’ notion of spectral triples [1, 2, 3]. A spectral triple( A , H , D ) is composed of a Hilbert space H , a unital pre- C ∗ -algebra A witha faithful representation as bounded multiplicative operators on H and a self-adjoint operator D densely defined on H with compact resolvent and such thatall commutators [ D, a ] are bounded for every a ∈ A . ∗ supported by a grant from the John Templeton Foundation
1n the commutative case, the correspondence with Riemannian geometry isgiven by the so-called reconstruction theorem [4, 5]. For every compact Rieman-nian spin manifold, a spectral triple can be constructed by taking the Hilbertspace H = L ( M , S ) of square integrable spinor sections over M , the unital pre- C ∗ -algebra A = C ∞ ( M ) (with pointwise multiplication and supremum norm)and the Dirac operator D = − i ( c ◦ ∇ S ), where c represents the Clifford action.The reconstruction theorem states that, under some additional axioms, commu-tative unital spectral triples correspond to existing compact Riemannian spinmanifolds. Information on the differential structure can be extracted using thedata given by a spectral triple. As an example, a notion of Riemannian distancebetween two states d ( ξ, η ), corresponding to the usual notion of distance, canbe defined as d ( ξ, η ) = sup {| ξ ( a ) − η ( a ) | : a ∈ A , k [ D, a ] k ≤ } .One goal of Connes’ noncommutative geometry is to provide a new mathe-matical background which would be suitable to describe the fundamental forcesof physics in a similar formalism. Such a background is given by the productof two spectral triples, a commutative one representing Euclidean gravitation(Euclidean in the sense of positive signature), and a noncommutative one rep-resenting a (still classical) standard model [2, 6].At this time, the theory of spectral triples is only completely defined forspaces with positive signature for the metric. However, this does not correspondto physical reality. A Lorentzian counterpart would be very useful especiallyfor models dealing with gravitation. The extension to pseudo-Riemannian andLorentzian spaces of the notion of a spectral triple is far from being complete,but some propositions exist [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].In this paper we present a new way to construct Lorentzian spectral triples,starting from the interesting proposition by Strohmaier [12] on building pseudo-Riemannian spectral triples based on Krein spaces. We show that it is possibleto construct a spectral triple representing a (possibly noncommutative) glob-ally hyperbolic Lorentzian manifold with a specific ”3+1” decomposition andwith a specific element of the algebra representing a global time. Such a con-struction is called a temporal Lorentzian spectral triple. The time elementhas the particularity to completely define a fundamental symmetry needed torecover the self-adjointness of the Dirac operator. As an improvement of thepseudo-Riemannian approach [12, 16], we show that a specific construction of afundamental symmetry, defined as the commutator between the Dirac operatorand a time element, ensures that the signature of the metric corresponds to aLorentzian space. This is a crucial property in order to recover the informa-tion coming from the Lorentzian nature of physical spacetimes like causality orLorentzian distance as in [17].Furthermore, the definition of a Lorentzian distance formula as constructedin [17, 18] requires the use of unbounded functions and its noncommutativegeneralization cannot fit into the usual formalism of C ∗ -algebras. Starting withour axioms of temporal Lorentzian spectral triple, we show that, when timeis commutative, such a construction admits an extension of the algebra to un-bounded elements by the way of a filtration. Then we show that a well-definednoncommutative formulation of a Lorentzian distance formula is possible in this2ontext by defining a notion of distance between the unique extension of twogiven pure states.The plan of this paper is the following. In Section 2 we recall the basic no-tions about Krein spaces and pseudo-Riemannian spectral triples. In Section 3we present the axioms of a temporal Lorentzian spectral triple and the methodfor generating a filtered algebra of unbounded elements. In Section 4 we con-struct an example of noncommutative temporal Lorentzian spectral triple asthe Moyal–Minkowski spacetime. In Section 5, we illustrate how the extensionto unbounded elements can be used to define a Lorentzian distance formula fornoncommutative temporal Lorentzian spectral triples with commutative time. We start with a review of the tools introduced by Strohmaier in order to adaptthe construction of spectral triples to pseudo-Riemannian manifolds. We willjust be interested here in the results and we refer the reader to [12] for theproofs and details. Some results about Dirac operators in pseudo-Riemanniangeometry can also be found in [19, 20].In the same way as a Riemannian spectral triple, a pseudo-Riemannian spec-tral triple is a triple ( A , H , D ) which corresponds in the commutative case tothe algebra A = C ∞ ( M ) of smooth functions over a pseudo-Riemannian spinmanifold M of signature ( p, q ) (with q ≥ H consistingof square integrable sections of the spinor bundle over M and on which thereexists a representation of A as multiplicative bounded operators, and to theDirac operator D = − i ( c ◦ ∇ S ) acting on the space H .The Hilbert space H is endowed with the positive definite Hermitian struc-ture ( ψ, φ ) = Z M ψ ∗ φ dµ g where dµ g = p | det g | d n x is the pseudo-Riemannian density on M .However, this structure does not admit any self-adjoint pseudo-RiemannianDirac operator related to this positive definite inner product. Instead, the Diracoperator corresponding to the pseudo-Riemannian metric is an essentially Krein-self-adjoint operator if we transform H into a Krein space [21], which is a spacewith indefinite inner product. Definition 1.
An indefinite inner product on a vector space V is a map V × V → C which satisfies:( v, λw + µw ) = λ ( v, w ) + µ ( v, w ) , ( v, w ) = ( w, v ) . An indefinite inner product is nondegenerate if( v, w ) = 0 ∀ v ∈ V ⇒ w = 0 . V can be written as the direct sum of two orthogonalspaces V = V + ⊕ V − such that the inner product is positive definite on V + andnegative definite on V − . Then the two spaces V + and V − are two pre-Hilbertspaces by the induced inner product (with a multiplication by − Definition 2 ([21]) . If the two subspaces V + and V − are complete in the norminduced on them and if the indefinite inner product on V is nondegenerate, thenthe space V = V + ⊕ V − is called a Krein space. The indefinite inner productis called a Krein inner product. Definition 3 ([21]) . For every decomposition V = V + ⊕ V − the operator J =id V + ⊕ − id V − respecting the property J = 1 is called a fundamental sym-metry. Such an operator defines a positive definite inner product on V by h · , · i J = ( · , J · ).Each fundamental symmetry of a Krein space V defines a Hilbert spacestructure, and two norms associated with two different fundamental symmetriesare equivalent. So it is natural to define the space of bounded operators as thespace of bounded operators on the Hilbert space defined for any fundamentalsymmetry. Definition 4 ([21]) . If A is a densely defined linear operator on V , the Krein-adjoint A + of A is the adjoint operator defined for the Krein inner product( · , · ) (with the usual definition and domain). An operator A is called Krein-self-adjoint if A = A + .Of course, for any fundamental symmetry J we can define an adjoint A ∗ for the positive definite inner product h · , · i J . In this case, the Krein-adjointis related to it by A + = J A ∗ J , and an operator A is Krein-self-adjoint if andonly if J A or A J are self-adjoint in the Hilbert space.Now the question is how we could define a Krein space structure from aspin manifold with pseudo-Riemannian metric. This is done by using spacelikereflections. Definition 5 ([12]) . A spacelike reflection r is an automorphism of the vectorbundle T M such that: • g ( r · , r · ) = g ( · , · ), • r = id, • g r ( · , · ) = ( · , r · ) is a positive definite metric on T M .It is clear that, for every pseudo-Riemannian metric of signature ( p, q ), thetangent bundle can be split into an orthogonal direct sum T M = T M p + ⊕ T M q − where the metric is positive definite on the p -dimensional bundle T M p + andnegative on the q -dimensional bundle T M q − , thus a spacelike reflection is auto-matically associated by defining r ( v + | x ⊕ v −| x ) = v + | x ⊕ − v −| x . This splittingis transposed to the cotangent bundle T M ∗ = T M ∗ p + ⊕ T M ∗ q − by isomorphism.4 roposition 6 ([12]) . For each spacelike reflection r , there is an associatedfundamental symmetry J r defined from the Clifford action c on a local orientedorthonormal basis { e , e , . . . , e q } of T M ∗ q − by: J r = i q ( q +1)2 c ( e ) c ( e ) . . . c ( e q ) = i q ( q +1)2 γ γ . . . γ q . This definition is independent of the choice of the local basis. With such afundamental symmetry, the space H of square integrable sections of the spinorbundle becomes a Krein space endowed with the indefinite inner product( ψ, φ ) = Z M ψ ∗ J φ dµ g . Actually, this operation is similar to a Wick rotation, but performed at analgebraic level.In the special case of a 4-dimensional Lorentzian manifold, with signature( − , + , + , +), a canonical fundamental symmetry is just given by J = iγ . We have then the following result concerning the Dirac operator:
Proposition 7 ([12, 19]) . If there exists a spacelike reflection such that theRiemannian metric g r associated is complete and if D is the Dirac operator,then i q D is essentially Krein-self-adjoint. In particular, if M is compact, then i q D is always essentially Krein-self-adjoint. From all these properties, we can introduce the definition of a pseudo-Riemannian spectral triple:
Definition 8 ([12]) . A pseudo-Riemannian Spectral Triple ( A , H , D ) is thedata of: • A Krein space H , • A pre- C ∗ -algebra A with a faithful representation as bounded multiplica-tive operators on H and such that a ∗ = a + , • A Krein-self-adjoint operator D on H such that all commutators [ D, a ] arebounded for every a ∈ A .In addition, we can also assume the existence of a fundamental symmetry J which commutes with all elements in A . In this case, A becomes a subalgebraof B ( H ) and the involution a ∗ corresponds to the adjoint for the Hilbert spacedefined by J . We can notice that in the commutative case, with this definition ofpseudo-Riemannian spectral triple as given in [12], the operator D correspondsto the usual Dirac operator times a factor i q .The compact resolvent condition is not present in this definition. This comesfrom the fact that, on a pseudo-Riemannian manifold, the Dirac operator D is With our choice of signature, the flat Dirac matrix γ is such that (cid:0) γ (cid:1) = − (cid:0) γ (cid:1) ∗ = − γ , so J = iγ respects the conditions of a fundamental symmetry. The other flatDirac matrices respect ( γ a ) = 1 and ( γ a ) ∗ = γ a for a = 1 , , σ D ( ξ ) = c ( ξ ) = g ( ξ, ξ )and so it is not invertible any more. In order to recover a similar condition, wedefine ∆ J = (cid:0) D ] J (cid:1) with [ D ] J = DD ∗ + D ∗ D . This operator is elliptic of order 1 since σ ∆ J ( ξ ) = g r ( ξ, ξ ), and is self-adjoint for the J -product. The compact condition is thenrequired on ∆ − J = (cid:0) D ] J (cid:1) − and this is independent of the choice of thefundamental symmetry J [12]. Pseudo-Riemannian spectral triples are a good tool in order to deal with generalpseudo-Riemannian signatures, but the definition in itself does not allow us tocontrol the signature of the metric. In particular, it is not possible to highlightspectral triples corresponding to Lorentzian spaces, which contain interestingnew properties like causality and are of great importance to physicists. Wepresent here an adaptation of the axioms of pseudo-Riemannian spectral triplescorresponding to noncompact globally hyperbolic Lorentzian manifolds. Thedefinition of temporal Lorentzian spectral triples is presented as a working basis,since the set of axioms is not fixed and could evolve to include more properties.We start by some considerations about the construction of the algebra. Fora noncompact manifold, the dual algebra for Gel’fand theory is the nonunitalalgebra C ( M ) of continuous functions vanishing at infinity. The noncompact-ness of the manifold is mandatory in order to have a causal structure. Indeed,for a Lorentzian manifold without boundary, compactness implies the existenceof points which are in the future of themselves [22, 23], which is not physicallyrealistic.In order to deal with the noncompactness of the manifold, we will follow theconstruction in [24] for noncompact Riemannian spectral triples. We considertwo pre- C ∗ -algebras A ⊂ e A with a similar faithful representation on a Hilbertspace H . The algebra A is non-unital while the algebra e A is a preferred uniti-zation of A . Moreover, we require that A is an ideal of e A (note that a weakeraxiom can be used by requiring that A is an ideal of e A ). Such a preferredunitization can be seen as a sub-algebra of the multiplier algebra of A .In the commutative case, those algebras can be constructed in the followingway. Let us consider a noncompact Lorentzian spin manifold M . We considerthe Hilbert space H = L ( M , S ) of square integrable spinor sections over M and the Dirac operator D = − i ( c ◦ ∇ S ) = − iγ µ ∇ Sµ . Then A ⊂ C ∞ ( M )and e A ⊂ C ∞ b ( M ) are some appropriate sub-algebras of the algebra of smoothfunctions vanishing at infinity and the algebra of smooth bounded functions inorder to keep the property that ∀ a ∈ e A , [ D, a ] is bounded. As an example, onMinkowski spacetime, one can take A to be the space of Schwartz functions and e A the space of smooth bounded functions with bounded derivatives [24].6he main difficulty in defining a pseudo-Riemannian spectral triple corre-sponding to a Lorentzian manifold is to give a constraint on the signature suchthat only one timelike direction remains possible. In the construction we sug-gest here, such a constraint will take the form of an element representing aglobal time and defining by itself the needed fundamental symmetry in order torecover the self-adjointness of the Dirac operator. In the commutative case, wewill suppose that the corresponding manifold respects the condition of globalhyperbolicity.From now, we suppose that M is a noncompact globally hyperbolic Lorentzianmanifold admitting a spin structure. Since M is globally hyperbolic, M ad-mits at least a Cauchy temporal function T , which is a smooth time function(function strictly increasing along each future-directed causal curve) with past-directed timelike gradient everywhere and such that level sets are Cauchy sur-faces [25, 26, 27]. The Lorentzian metric admits a globally defined orthogonalsplitting g = − u d T + g T , where g T is a Riemannian metric on each level set of T and u is a smoothpositive function on M (we use here a signature of the type ( − , + , + , + , . . . )).The function T is in general unbounded and thus does not belong to e A but itssmooth inverse (cid:0) T (cid:1) − is a smooth bounded function.At first, we consider a specific conformal transformation of the metric ˜ g = u g and we suppose that the spin structure corresponds to the metric ˜ g . Such aconformal transformation induces an orthonormal splitting of the metric˜ g = − d T + ˜ g T . This splitting on the metric induces a splitting on the tangent (and the cotan-gent) bundle T M = T M − ⊕ T M + , where the subbundle T M − has a one di-mensional fiber generated by the gradient ∇T . So the temporal function T defines a spacelike reflection, with the associated Riemannian metric being˜ g r = d T + ˜ g T . Since ∇T is a generating element of T M − of norm one, d T is a generating element of T M ∗− of norm one. From the Proposition 6, weknow that each spacelike reflection generates a fundamental symmetry J de-fined from the Clifford action c on a local oriented orthonormal basis { e } of T M ∗− by J = ic ( e ). Since { d T } is an orthonormal basis of T M ∗− , we have that J = ic ( d T ) = iγ is a fundamental symmetry, and if the manifold M is complete under the metric˜ g r = d T + ˜ g T , by the Proposition 7 iD is Krein-self-adjoint for the Krein spacedefined by J , which is equivalent to the fact that J D is a skew-self-adjointoperator in H .Then, we can recall that the Dirac operator respects the following knownproperty [3]: ∀ f ∈ C ∞ ( M ) , [ D, f ] = − i c ( df ) . J = − [ D, T ]is a fundamental symmetry of the Krein space. The condition on the Krein-self-adjointness of iD becomes a condition on the skew-self-adjointness of [ D, T ] D on the Hilbert space H and can be written as D ∗ = −J D J = − [ D, T ] D [ D, T ].Such a construction is satisfactory if we want information about the causalstructure of the Lorentzian manifold, since the causality is unchanged by con-formal transformation. However, in order to keep the metric aspect, one wouldlike to extend the construction to more general conformal transformations. Thiscan be done by relaxing the condition J = 1. Indeed, for a general metric g = − u d T + g T and the corresponding spin structure, we can define a moregeneral fundamental operator J u = − [ D, T ] = iγ where γ is now the curvedDirac matrix such that γ γ = g − u
1. Since the fundamental symmetryis only modified by a Hermitian element
J → J u = u J with u >
0, it still de-fines a Krein space where the operator iD is Krein-self-adjoint, but it has lost itssymmetric property and respects instead the more general condition J = u C ∗ -algebra formalism, we will restrict the conformal factors u to elements in e A , such that J can be seen as a positive (Hermitian invertible)element of e A .We now propose the following set of axioms for a Lorentzian spectral triplecorresponding to a (possibly noncommutative) globally hyperbolic Lorentzianmanifold. Definition 9.
A Temporal Lorentzian Spectral Triple ( A , e A , H , D, T ) is thedata of: • A Hilbert space H , • A nonunital pre- C ∗ -algebra A with a faithful representation as boundedmultiplicative operators on H , • A preferred unitization e A of A with a similar faithful representation asbounded multiplicative operators on H and such that A is an ideal of e A , • An unbounded self-adjoint operator T on H such that (cid:0) T (cid:1) − ∈ e A , • An unbounded operator D densely defined on H such that: – all commutators [ D, a ] are bounded for every a ∈ e A , – [ D, T ] is a self-adjoint operator on H which commutes with everyelement in e A and such that [ D, T ] = u ∈ e A with u > – [ D, T ] D is a skew-self-adjoint operator on H , – ∀ a ∈ A , a (1 + h D i ) − is compact, with h D i = − ( D [ D, T ] D [ D, T ] + [ D, T ] D [ D, T ] D ).8 efinition 10. A temporal Lorentzian spectral triple ( A , e A , H , D, T ) is evenif there exists a Z -grading γ such that γ ∗ = γ , γ = 1, [ γ, a ] = 0 ∀ a ∈ e A , γ T = T γ and γD = − Dγ .A temporal Lorentzian spectral triple is a pseudo-Riemannian spectral triplecorresponding to a manifold with Lorentzian signature and admitting a globaltime, where the usual fundamental symmetry (usual in the sense of J = 1)is completely determined by the Dirac operator D and the global time T by J = − u − [ D, T ] with u = [ D, T ] . We can notice that, when the algebra isnoncommutative, the commutativity condition of [ D, T ] restricts the possibilityof conformal transformations by requiring that u belongs to the center of thealgebra Z ( e A ). When the center is trivial, this reduces to the usual fundamentalsymmetry with [ D, T ] = 1 up to a constant scale. An example of constructionof a temporal Lorentzian spectral triple in the noncommutative case will begiven in Section 4.While the construction of a temporal Lorentzian spectral triple comes fromthe properties of Lorentzian manifolds with global hyperbolicity, the followingproposition shows us that the Lorentzian characteristics are recovered by thischoice of axioms. Proposition 11.
Let us assume that a temporal Lorentzian spectral triple whosealgebra is commutative ( A , e A , H , D, T ) corresponds to a pseudo-Riemannianspin manifold ( M , g ) in the usual way. Then the geometry is Lorentzian andthe metric admits a global splitting.Proof. We work with a spin structure on ( M , g ) such that the Dirac operatorreads D = − iγ µ ∇ Sµ where the gamma matrices γ µ = c ( dx µ ) are chosen to beeither Hermitian or anti-Hermitian depending on the signs of the metric andwhere the Clifford relations are γ µ γ ν + γ ν γ µ = 2 g µν .For arbitrary coordinates, we have [ D, T ] = − iγ µ ∂ µ T , so the condition[ D, T ] = u > T vanishes nowhere.Hence, T can be chosen as a first global coordinate x = T and we get [ D, T ] = − iγ . Then [ D, T ] = − g > ⇒ g <
0. We can notice that γ is forcedto be anti-Hermitian due to the self-adjointness of [ D, T ].From the skew-self-adjoint condition ([ D, T ] D ) ∗ + [ D, T ] D = 0 and discardinga divergence term (in a similar way to [3, Proposition 9.13]), we get − i ( γ µ ) ∗ γ ( − i ∇ Sµ ) ∗ − iγ γ µ ( − i ∇ Sµ ) = − (cid:0) ( γ µ ) ∗ γ + γ γ µ (cid:1) ∇ Sµ = 0 . Here we can discuss the Hermicity of the matrices γ µ for µ >
0. If some matrix γ µ has the same Hermicity that γ (so is anti-Hermitian), we get γ µ γ = γ γ µ which implies γ µ γ = g µ γ γ µ cannot be a multiple of the identity matrix for µ = 0. Hence all matrices γ µ are Hermitian for µ > γ µ γ + γ γ µ = 0 = g µ g µ J = − u − [ D, T ] restricts the possibilities tothose manifolds with Lorentzian signature. Also, the chosen class of fundamen-tal symmetries does not cover all possible Lorentzian spin manifolds but onlythose which admit a global splitting.In the rest of this section, we will show that the above definition of a temporalLorentzian spectral triple is suitable for the definition of an extension of thealgebra to unbounded elements. Indeed, one key element about Lorentziannoncommutative geometry is the set of causal functions, which are real-valuedfunctions that do not decrease along every causal future-directed curve. Inparticular, the sets of time or temporal functions are subsets of the set of causalfunctions. Those causal functions are used in order to give information aboutcausality and metric aspects (see [17, 18, 28, 29, 30, 31]). In particular, theconstruction of a Lorentzian distance function requires some functions witha minimum growth rate along causal future-directed curves g ( ∇ f, ∇ f ) ≤ − e A . So we would like to construct an algebra which could contain unboundedcausal functions in order to keep information about time, Lorentzian distanceand causality. However, in this case, the supremum norm (and a fortiori L p norms) cannot be used to define a Banach algebra. The needed unboundedelements are clearly in the ∗ -algebra C ∞ ( M ). Since smooth functions are locallyintegrable, we can use a structure defined for the space of locally integrablefunctions L ( M ). Such a structure is provided by the theory of partial innerproduct spaces (PIP-spaces). We give here a few elements of this theory. Acomplete introduction, including the topological aspects, can be found in [32]. Definition 12 ([32]) . A linear compatibility relation on a vector space V is asymmetric binary relation f g which preserves linearity. For S ⊂ V , we candefine the vector subspace S = { g ∈ V : g f, ∀ f ∈ S } ⊂ V which respectsthe inclusion property S ⊂ (cid:0) S (cid:1) . Vector subspaces such that S = (cid:0) S (cid:1) arecalled assaying subspaces.The family of assaying subspaces forms a lattice with the inclusion order,and the meet and join operations given by S ∧ S = S ∩ S and S ∨ S =( S + S ) . Usually, it is enough to consider only an indexed sublattice I = { S r : r ∈ I} which covers V , with an involution defined on the index I by ( S r ) = S r . Definition 13 ([32]) . A partial inner product on ( V, h · , · i defined exactly on compatible pairs of vectors f g . A partial inner prod-uct space (PIP-space) is a vector space V equipped with a linear compatibilityrelation and a partial inner product. An indexed PIP-space is a PIP-space witha generating involutive indexed lattice of assaying subspaces.10he space L ( M ) is a PIP-space with the compatibility relation given by f g ⇐⇒ Z M | f g | dµ g < ∞ and the partial inner product h f, g i = Z M f ∗ g dµ g . We can define an involutive indexed lattice of assaying subspaces by us-ing weight functions. If, for r, r − ∈ L ( M ) with r a.e. positive Hermitian,we define the space L ( r ) of measurable functions f such that f r − is squareintegrable, i.e. L ( r ) = (cid:26) f ∈ L ( M ) : Z M | f | r − dµ g < ∞ (cid:27) , then the family I = (cid:8) L ( r ) (cid:9) r respects L = S r L ( r ) and is a generatinglattice of assaying subspaces, with an involution defined by r = r − since L ( r ) = L ( r − ).Actually, we have a realization of this indexed PIP-space as a lattice ofHilbert spaces {H r } r where each H r = L ( r ) is endowed with the positive defi-nite Hermitian inner product h f, g i r = Z M f ∗ g r − dµ g . In the following, we will denote such an indexed PIP-space by S r H r , with theso called center space H being the space of square integrable functions.The space C ∞ b ( M ) is a unital pre- C ∗ -algebra which acts as bounded multi-plicative operators on H , with the operator norm corresponding to the supre-mum norm k f k ∞ = sup x ∈M | f ( x ) | . We can make a similar modification to thesupremum norm by introducing a smooth weight function r . We denote the al-gebra of bounded smooth functions by e A = C ∞ b ( M ). We define the followinglattice of spaces: e A r = (cid:26) f ∈ C ∞ ( M ) : sup x ∈M (cid:12)(cid:12) f ( x ) r − ( x ) (cid:12)(cid:12) < ∞ (cid:27) · Except for the center algebra e A , each e A r is a vector space which doesnot have the structure of an algebra. Instead, those spaces have a structureof partial *-algebra, which means that the product f g ∈ e A r is well definedonly for a bilinear subset of e A r × e A r . Actually, we have the grading property f g ∈ e A r if f ∈ e A s and g ∈ e A t , with r = st . Each e A r is endowed with a norm k · k r = (cid:13)(cid:13) · r − (cid:13)(cid:13) , where k · k is the operator norm on e A . In the same wayas the lattice of Hilbert spaces, we define the ∗ -algebra S r e A r which obviously11orresponds to the algebra of smooth functions C ∞ ( M ) but endowed with agradation.It is trivial that S r e A r acts on the PIP-space S r H r as multiplicative op-erators. Moreover, we have that each a ∈ A Ms acts as a bounded operator a : H r → H rs if we consider the weighted norm.So we see that it is possible to consider C ∞ ( M ) as a graded algebra ofoperators on some PIP-space, with a set of norms defined on the gradation.However, we do not need the whole space C ∞ ( M ). Indeed, the needed causalfunctions giving metric information are only unbounded ”in the direction oftime”, which means that we do not really need functions which are growingindefinitely on spacelike surfaces. So we propose the following idea: the set ofweight functions can be restricted in such a way that the filtered algebra S r e A r contains some functions which are growing indefinitely only along causal curves.A typical way is to restrict C ∞ ( M ) to the functions which satisfy a particulargrowth rate along causal curves. The final algebra can be constructed by theuse of the global time function T : • e A = C ∞ b ( M ) is the unital pre- C ∗ -algebra of smooth bounded functionson M . • e A n = n f ∈ C ∞ ( M ) : sup x ∈M (cid:12)(cid:12)(cid:12) f ( x ) (cid:0) T ( x ) (cid:1) n (cid:12)(cid:12)(cid:12) < ∞ o , n ∈ N . • S n ∈ Z e A n is the unital filtered algebra of smooth functions on M of poly-nomial growth along timelike curves which are bounded on each Cauchysurface T − ( t ), t ∈ R .This algebra has a large number of interesting properties, easily derived fromthe definition: • Each set e A n is simply generated from e A by f ∈ e A n if and only if f (cid:0) T ( x ) (cid:1) n ∈ e A . • Each set e A n is endowed with a norm k · k n = (cid:13)(cid:13)(cid:13) · (cid:0) T ( x ) (cid:1) n (cid:13)(cid:13)(cid:13) where k · k is the norm on e A . • We have the (descending) filtration properties e A n ⊂ e A n − ∀ n ∈ Z and ab ∈ e A n ⇐⇒ ∃ m, l ∈ Z : n = m + l , a ∈ e A m , b ∈ e A l . • T ∈ S n ∈ Z e A n , and more precisely T ∈ e A − . • All smooth causal functions with polynomial growth are included in thealgebra S n ∈ Z e A n . • All continuous a.e. differentiable causal functions with polynomial growthare in the closure algebra S n ∈ Z e A n , where the closure is taken in eachsubset with respect to each norm k · k n .12he polynomial growth is just a particular choice. For example, one candefine a similar filtered algebra with causal functions of exponential growth bytaking a weight function like e α |T | , α ∈ R . The choice of the center unitizationis also arbitrary since e A can be replaced by another suitable unitization.Now, we can remember that the pre- C ∗ -algebra A = C ∞ ( M ), as well asits unitization e A = e A = C ∞ b ( M ), act as multiplicative bounded operators onthe Hilbert space H = L ( M , S ) of square integrable spinor sections over M .From that, we can construct an indexed PIP-space S n ∈ Z H n of spinor sectionsover M which are square integrable under the weighted norm, and on which S n ∈ Z e A n acts as a family of bounded operators.Such a space corresponds to a scale of Hilbert spaces generated by the self-adjoint operator (cid:0) T (cid:1) ≥
1. Indeed, T : Dom ( T ) → H is an unboundedself-adjoint operator acting on H = H = L ( M , S ) with domain Dom ( T ) ⊂ H ,and (cid:0) T (cid:1) is an unbounded positive self-adjoint operator with similar do-main Dom ( T ) = Dom (cid:16)(cid:0) T (cid:1) (cid:17) .We can define the following Hilbert spaces: H n = n \ k =0 Dom (cid:16)(cid:0) T (cid:1) k (cid:17) = n \ k =0 Dom( T k ) ∀ n ∈ N . The conjugate spaces H n = H − n , n ∈ N are just the topological duals of thespaces H n (properly speaking, there are the identifications of the topologicalduals under the Hermitian inner product on H ).Then we have the discrete scale of Hilbert spaces: \ n ∈ Z H n ⊂ · · · ⊂ H ⊂ H ⊂ H ⊂ H − ⊂ H − ⊂ · · · ⊂ [ n ∈ Z H n . Each H n is endowed with the positive definite Hermitian inner product h ψ, φ i n = D ψ, (cid:0) T (cid:1) n φ E = Z M ψ ∗ φ (cid:0) T (cid:1) n dµ g . Then S n ∈ Z e A n acts as a family of multiplicative operators on S n ∈ Z H n with a : H n → H n + m bounded if a ∈ e A m and for each n ∈ Z .This construction can be generalized to noncommutative spaces using tem-poral Lorentzian spectral triples. However, in order to conserve the properties ofthis construction, we need to impose some additional commutative conditions. Definition 14.
A temporal Lorentzian spectral triple ( A , e A , H , D, T ) has com-mutative time if the operators (cid:0) T (cid:1) and [ D, (cid:0) T (cid:1) ] commute with allelements in e A . 13 roposition 15. If a temporal Lorentzian spectral triple ( A , e A , H , D, T ) hascommutative time, then: • T generates a filtered algebra S n ∈ Z e A n by e A = e A , a ∈ e A n +1 if and only if (cid:0) T (cid:1) a ∈ e A n , with a norm k · k n = (cid:13)(cid:13) (1 + T ) n · (cid:13)(cid:13) defined on each e A n . • S n ∈ Z e A n has a faithful representation as a family of multiplicative opera-tors on the indexed PIP-space S n ∈ Z H n defined by H = H , H n = n \ k =0 Dom ( T k ) ∀ n > and H − n = H n the conjugate dual of H n with a : H n → H n + m bounded if a ∈ e A m and with a positive definite innerproduct h · , · i n = (cid:10) · , (cid:0) T (cid:1) n · (cid:11) defined on each H n .Moreover, the norms k a k m = (cid:13)(cid:13) (1 + T ) m a (cid:13)(cid:13) and k [ D, a ] k m = (cid:13)(cid:13) (1 + T ) m [ D, a ] (cid:13)(cid:13) for a ∈ e A m correspond to the operator norm on S n ∈ Z H n .Proof. We know that (cid:0) T (cid:1) commutes with all elements in e A . This im-plies that (cid:0) T (cid:1) − ∈ e A commutes with all elements in e A since ∀ a ∈ e A ,[ (cid:0) T (cid:1) − , a ] (cid:0) T (cid:1) = 0 and the codomain of (cid:0) T (cid:1) is H . The fil-tration property a ∈ e A m , b ∈ e A n = ⇒ ab ∈ e A m + n and ba ∈ e A m + n comes from the fact that S n ∈ Z e A n is generated by multiple applications of (cid:0) T (cid:1) and (cid:0) T (cid:1) − on e A = e A , and the whole construction is simi-lar to the commutative case.The norm k · k m = (cid:13)(cid:13) (1 + T ) m · (cid:13)(cid:13) on e A m is the operator norm. Indeed, ifwe consider a ∈ e A m as an operator a : H n → H n + m for some n ∈ Z , then: k a k op = sup φ ∈H n , φ =0 h a φ, a φ i n + m h φ, φ i n = sup φ ∈H n , φ =0 (cid:10) a φ , (cid:0) T (cid:1) n + m a φ (cid:11) h φ , (1 + T ) n φ i = sup φ ∈H n , φ =0 (cid:10)(cid:0) T (cid:1) m a (cid:0) T (cid:1) n φ , (cid:0) T (cid:1) m a (cid:0) T (cid:1) n φ (cid:11)D (1 + T ) n φ , (1 + T ) n φ E = sup ψ ∈H , ψ =0 (cid:10)(cid:0) T (cid:1) m a ψ , (cid:0) T (cid:1) m a ψ (cid:11) h ψ , ψ i = (cid:13)(cid:13)(cid:0) T (cid:1) m a (cid:13)(cid:13) = k a k m where we use the facts that (cid:0) T (cid:1) and a commute with each other and that (cid:0) T (cid:1) is self-adjoint. The result is clearly independent of the chosen H n .14he condition that [ D, (cid:0) T (cid:1) ] commutes with all elements in e A is equiv-alent to the fact that, for all a ∈ e A , [ D, a ] commutes with (cid:0) T (cid:1) . Indeed,by using the fact that (cid:0) T (cid:1) commutes with a :[ D, (cid:0) T (cid:1) ] a − a [ D, (cid:0) T (cid:1) ]= D (cid:0) T (cid:1) a − (cid:0) T (cid:1) Da − aD (cid:0) T (cid:1) + a (cid:0) T (cid:1) D = Da (cid:0) T (cid:1) − (cid:0) T (cid:1) Da − aD (cid:0) T (cid:1) + (cid:0) T (cid:1) aD = [ D, a ] (cid:0) T (cid:1) − (cid:0) T (cid:1) [ D, a ] . Then, by a reasoning similar to above, the operator norm of [
D, a ] is independentof the chosen H n . Commutative temporal Lorentzian spectral triples can easily be constructedby using the construction presented in the previous section from any suitableglobally hyperbolic Lorentzian manifold. One can wonder however if noncom-mutative examples can explicitly be constructed. In this section we answerthe problem by showing that the Minkowski spacetime endowed with a Moyalproduct is a noncomutative temporal Lorentzian spectral triple.The construction of spectral triples based on Euclidean Moyal planes hasalready been studied extensively in [24]. The algebra A is chosen as the space S ( R n ) of Schwartz functions, which are smooth functions f ∈ C ∞ ( R n ) rapidlyvanishing at infinity together with all derivatives. The product on S ( R n ) isdefined by the Moyal product:( f ⋆ h )( x ) = 1(2 π ) n Z Z f ( x −
12 Θ u ) h ( x + v ) e − iu · v d n u d n v where Θ is a real skewsymmetric n × n constant matrix. This product can beextended to the space L ( R n ) [24, 33] and there is a representation of A onthe space of square integrable spinors L ( R n ) ⊗ C ⌊ n/ ⌋ as a multiple of the leftregular action π ( f ) ψ = ( L ( f ) ⊗ ψ = f ⋆ ψ. A is a nonunital involutive Fr´echet algebra, and if equipped with the operatornorm k a k op = sup b ∈ L ( R n ) ,b =0 k a ⋆ b k k b k , A becomes a nonunital pre- C ∗ -algebra. The pre- C ∗ -algebra A = {S ( R n ) , ⋆ } ,the Hilbert space H = L ( R n ) ⊗ C ⌊ n/ ⌋ and the Dirac operator D = − i∂ µ ⊗ γ µ define a Euclidean noncompact spectral triple of spectral dimension n .15e construct the Moyal–Minkowski spacetime in a similar way. We workwith the space R ,n − , and the Moyal ⋆ product is defined on the space ofSchwartz functions S = S ( R ,n − ) and extended to L ( R ,n − ). We definethe nonunital pre- C ∗ -algebra A = ( S , ⋆ ) acting on the Hilbert space H = L ( R ,n − ) ⊗ C ⌊ n/ ⌋ . We must choose a suitable unitization e A for the tem-poral Lorentzian spectral triple. A first choice could be the (left and right)Moyal multiplier algebra: M ( A ) = { T ∈ S ′ : T ⋆ h ∈ S and h ⋆ T ∈ S for all h ∈ S} where S ′ = S ′ ( R ,n − ) is the dual space of tempered distributions. Coordi-nates functions x , x , . . . x n − belong to this space, and respect the followingcommutation relations with the star product:[ x µ , x ν ] = x µ ⋆ x ν − x ν ⋆ x µ = i Θ µν . However, this unitization cannot be chosen since some elements in M ( A ) areunbounded. Instead we choose the sub-algebra e A = ( B , ⋆ ) ⊂ M ( A ) of smoothfunctions which are bounded together with all derivatives. e A = ( B , ⋆ ) is aunital Fr´echet pre- C ∗ -algebra that contains A as an ideal [24]. To complete thetemporal Lorentzian spectral triple, we choose the temporal element T = x asthe first coordinate function and the usual flat Dirac operator D = − i∂ µ ⊗ γ µ . Proposition 16.
The triple ( A , e A , H , D, T ) = (cid:16)(cid:0) S ( R ,n − ) , ⋆ (cid:1) , (cid:0) B ( R ,n − ) , ⋆ (cid:1) , L ( R ,n − ) ⊗ C ⌊ n/ ⌋ , − i∂ µ ⊗ γ µ , x (cid:17) respects the chosen axioms for a temporal Lorentzian spectral triple.Proof. • H = L ( R ,n − ) ⊗ C ⌊ n/ ⌋ is a Hilbert space with positive definiteinner product defined by the usual inner product on spinors h ψ, φ i = R ψ ∗ φ dµ g . • A = (cid:0) S ( R ,n − ) , ⋆ (cid:1) is a nonunital pre- C ∗ -algebra with a representation π ( f ) = L ( f ) ⊗ H with the norm k · k = k · k op . • e A = (cid:0) B ( R ,n − ) , ⋆ (cid:1) is a preferred unitization of A with the same repre-sentation π on H and with the same operator norm k · k . • T = x acts as an unbounded self-adjoint operator on H . (cid:0) T (cid:1) − = √ x ) is bounded on all R ,n − with bounded derivatives, so it belongsto e A . • D = − i∂ µ ⊗ γ µ is clearly an unbounded operator densely defined on H .The property that [ D, a ] acts as a multiplicative operator D ( a ) = − ic ( da )is still valid in the nonunital case [24], so [ D, a ] is bounded for every a ∈ e A .16 From [
D, x ] = − ic ( dx ) = − iγ , the Krein space conditions can easily bechecked: – (Hermicity) [ D, x ] ∗ = (cid:0) − iγ (cid:1) ∗ = i (cid:0) γ (cid:1) ∗ = − iγ = [ D, x ], – (Commutativity) [ D, x ] = − iγ commutes with every element in e A since − i ∈ Z ( e A ), – (symmetry) [ D, x ] = − g = 1, – (skew-self-adjointness) (cid:0) [ D, x ] D (cid:1) ∗ = − [ D, x ] D ⇐⇒ − ( γ µ ) ∗ γ = γ γ µ for µ ≥ ⇐⇒ γ γ = γ γ and γ µ γ = − γ γ µ for µ > . If the dimension n is even, then this temporal Lorentzian spectral triple iseven by taking the chirality element γ = ( − i ) n +1 γ . . . γ n − . Proposition 17.
The temporal Lorentzian spectral triple on the Moyal–Minkowskispacetime has commutative time if and only if the matrix defining the Moyalproduct is degenerate with Θ µ = Θ µ = 0 for all µ = 0 , . . . , n − . Proof.
The degeneration of the matrix is equivalent to the fact that T = x isin the center of M ( A ), which implies by expansion that p x ) commuteswith all elements in e A ⊂ M ( A ). Moreover, the property that [ D, p x ) ]commutes with all elements a ∈ e A is equivalent to the fact that [ D, a ] com-mutes with p x ) which is true since [ D, a ] acts as a simple multiplicativeoperator.We can notice that, when the Moyal–Minkowski spacetime has commutativetime, this allows us to construct temporal Lorentzian spectral triples correspond-ing to a modification of the metric by a scale factor u ( t ) with [ D, x ] = u ( t ),since u ( t ) belongs to the center of the algebra. We have proposed a definition of temporal Lorentzian spectral triples based ona time element T and shown that such a time element, under some commutativecondition, allows us to extend the algebra to unbounded elements. In this lastsection, we show that such an extension can be used to define a Lorentziandistance formula for noncommutative spacetimes.17e first start by reviewing the results on the Lorentzian distance formulain the commutative case. We recall that a Lorentzian distance on a Lorentzianmanifold M is a function d : M × M → [0 , + ∞ ) ∪ { + ∞} defined by d ( p, q ) = sup l ( γ ) : γ future directed causalpiecewise smooth curvewith γ (0) = p, γ (1) = q if p (cid:22) q p (cid:14) q where l ( γ ) = R p − g γ ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) dt is the length of the curve and (cid:22) denotesthe causal relation between points.If M is globally hyperbolic, this function is finite and continuous, and re-spects the following properties [34]:1. d ( p, p ) = 0,2. d ( p, q ) ≥ p, q ∈ M ,3. If d ( p, q ) >
0, then d ( q, p ) = 0,4. If d ( p, q ) > d ( q, r ) >
0, then d ( p, r ) ≥ d ( p, q ) + d ( q, r ).In the same way as Connes’ distance formula gives an algebraic formula-tion of the Riemannian distance, there exists an algebraic formulation for theLorentzian distance following the results in [17, 18]: Definition 18. [[17]] Let us consider a globally hyperbolic Lorentzian spinmanifold M with even dimension. For every two points p, q ∈ M , we define: e d ( p, q ) := inf f ∈ C ∞ ( M , R ) { max { , f ( q ) − f ( p ) } : ∀ φ ∈ H , h φ, J ([ D, f ] + iγ ) φ i ≤ } where D is the Dirac operator, H is the Hilbert space of square integrable spinorsections over M , γ is the chirality element and J = iγ (curved gamma matrix). Proposition 19 ([17]) . The function e d ( p, q ) respects all the properties of aLorentzian distance and, for every p, q ∈ M , we have d ( p, q ) ≤ e d ( p, q ) . It is still unknown if there is an equality between this algebraic formula andthe usual Lorentzian distance formula for every globally hyperbolic Lorentzianspin manifold. However, as it is shown in [17], the equality can be obtained formanifolds whose Lorentzian distance function can be approximate by suitablesmooth functions. In particular, on the Minkowski spacetime, this formula cor-responds exactly to the usual Lorentzian distance. The proof of the equality formore general manifolds is currently a work in progress. We can conjecture thatthis is valid for a large class of globally hyperbolic Lorentzian spin manifolds.18e want to show that it is possible to adapt this algebraic formula to com-mutative temporal Lorentzian spectral triples with a possible generalization tononcommutative spaces. The difficulty comes from the constraint ∀ φ ∈ H , h φ, J ([ D, f ] + iγ ) φ i ≤ g ( ∇ f, ∇ f ) ≤ − ∇ f is past-directedwhich implies that the needed functions are unbounded along timelike curves,which is not possible in e A . However, some functions of the filtered algebra S n ∈ Z e A n fit our goal, and we can propose the following definition: Definition 20.
Let us consider a commutative even temporal Lorentzian spec-tral triple ( A , e A , H , D, T ) with Z -grading γ constructed from a globally hyper-bolic Lorentzian spin manifold M . For every two points p, q ∈ M , we define: e d ( p, q ) := inf f ∈ S e A n { max { , f ( q ) − f ( p ) } : ∀ φ ∈ H , h φ, [ D, T ]([ D, f ] + iγ ) φ i ≥ } where S e A n is the filtered algebra generated by the time element T .This definition is equivalent to the definition 18. Indeed, since the Lorentziandistance between two points is a local quantity, there is no influence on imposingconditions on the behaviour of the test functions at infinity (as long as their re-spect the condition sup g ( ∇ f, ∇ f ) ≤ − A are positive linear functionals (automatically continuous)of norm one. The space of states is denoted by S ( A ). It is a closed convex set(for the weak- ∗ topology), and extremal points are called pure states, with theset of pure states denoted by P ( A ). In the commutative case, those pure statesare the characters of the algebra (non-zero *-homomorphisms). From Gel’fand–Naimark theory (see e.g. [3]), the pure states on A = C ∞ ( M ) correspond tothe points of the manifold M by the relation χ ∼ p if and only if, ∀ f ∈ A , χ ( f ) = f ( p ) , for χ ∈ P ( A ) and p ∈ M . Those states can be extended uniquelyto S e A n ⊂ C ∞ ( M ) by requiring that ∀ f ∈ S e A n , χ ( f ) = f ( p ), and we have thefollowing definition: Definition 21.
Let us consider a commutative even temporal Lorentzian spec-tral triple ( A , e A , H , D, T ) with Z -grading γ . For every two pure states χ, ξ ∈ P ( A ), we define: e d ( χ, ξ ) := inf f ∈ S e A n { max { , ξ ( f ) − χ ( f ) } : ∀ φ ∈ H , h φ, [ D, T ]([ D, f ] + iγ ) φ i ≥ } where S e A n is the filtered algebra generated by the time element T and where χ and ξ are considered as their unique extension to S e A n .19his definition cannot be generalized to noncommutative spaces in a straightway. We need to study the way to extend the notion of states in the noncom-mutative case without using the notion of points, since such a notion becomesmeaningless. First, we need to notice that, when the algebra is noncommuta-tive, the space P ( A ) and P ( e A ) can be strongly different since the states on A may not necessarily be extended in a unique way to e A . This will impose on usto only define a Lorentzian distance on P ( e A ). In order to extend the definitionof the states to S e A n , we need the following lemma: Lemma 22.
Let e A be a unital noncommutative pre- C ∗ -algebra and let χ ∈ P ( e A ) be a pure state. If a ∈ Z ( e A ) , then ∀ b ∈ e A we have χ ( ab ) = χ ( a ) χ ( b ) .Proof. At first, let us suppose that b ∈ e A is normal, i.e. bb ∗ = b ∗ b . We can definethe unital pre- C ∗ -algebra A ⊂ e A generated by the elements { a, a ∗ , b, b ∗ , } .Since a and a ∗ commute with everything and b is normal, A is a commutativepre- C ∗ -algebra. We have that χ ∈ S ( A ), since χ is a positive linear functionalon A such that χ (1) = 1. We can see that χ is a pure state on A . Indeed, if wesuppose that χ = λχ + (1 − λ ) χ for two states χ , χ ∈ S ( A ) with 0 < λ < χ can be extended to e A as a state with | λχ | ≤ | χ | by the Hahn–Banachtheorem [35] as well as χ by setting χ = − λ ( χ − λχ ), which implies that χ = χ since χ is a pure states on e A . Since A is commutative and χ is apure state, it is a character and respects the *-homomorphism property on A ,so χ ( ab ) = χ ( a ) χ ( b ).Let us suppose now that b is not normal. Then it can be written as b = b + b where b is Hermitian and b anti-Hermitian, so b and b are normal. Then wehave χ ( ab ) = χ ( ab ) + χ ( ab ) = χ ( a ) χ ( b ) + χ ( a ) χ ( b ) = χ ( a ) χ ( b ) . Proposition 23.
Let us consider a temporal Lorentzian spectral triple withcommutative time ( A , e A , H , D, T ) . Then for every pure state χ ∈ P ( e A ) suchthat χ ( (cid:0) T (cid:1) − ) = 0 , χ admits a unique extension to the filtered algebra S n ∈ Z e A n defined by: ∀ a ∈ [ n ∈ Z e A n , χ ( a ) = χ ( (cid:0) T (cid:1) − ) − n χ ( a ) if a = (cid:0) T (cid:1) n a with a ∈ e A . Proof.
Let us recall that, for every a ∈ S n ∈ Z e A n , there exist a ∈ e A and n suchthat a = (cid:0) T (cid:1) n a . Moreover, since (cid:0) T (cid:1) − ∈ e A , for every a, b ∈ S n ∈ Z ,there exist a , b ∈ e A and a common n such that a = (cid:0) T (cid:1) n a and b = (cid:0) T (cid:1) n b . It follows that the extension of χ to S n ∈ Z e A n is linear. We havetrivially that χ is positive with χ (1) = 1.20n order to show that this extension if well defined, we must prove that thedefinition of χ ( a ) is independent of the chosen a . Let us suppose that a = (cid:0) T (cid:1) n a = (cid:0) T (cid:1) m a ′ with a , a ′ ∈ e A . We have a = (cid:0) T (cid:1) m − n a ′ .Since the temporal Lorentzian spectral triple has commutative time, (cid:0) T (cid:1) − is in the center of the algebra, and by the Lemma 22 we have χ ( a ) = χ ( (cid:0) T (cid:1) m − n ) χ ( a ′ ) = χ ( (cid:0) T (cid:1) − ) n − m χ ( a ′ ) . Therefore, χ ( a ) = χ ( (cid:0) T (cid:1) − ) − n χ ( a )= χ ( (cid:0) T (cid:1) − ) − n χ ( (cid:0) T (cid:1) − ) n − m χ ( a ′ )= χ ( (cid:0) T (cid:1) − ) − m χ ( a ′ )and the definition of the extension of χ is independent of the chosen a ∈ e A .We can notice that, in the commutative case, pure states χ ∈ P ( e A ) suchthat χ ( (cid:0) T (cid:1) − ) = 0 do not correspond to pure states in P ( A ) since theydo not correspond to an evaluation map at any point. They are states addedby the unitization process (corresponding to a compactification of the mani-fold) and should be ignored. So it looks natural to ignore them even in thenoncommutative case.We can also see that, even if the requirement to have a commutative timecould seem sometime restrictive, it is a necessary condition in order to guaranteethat the extension of the states to unbounded elements is unique, and so to havea well defined noncommutative generalization of a Lorentzian distance formula.At the end, we have the following definition for a Lorentzian distance formulavalid on both commutative and noncommutative spaces: Definition 24.
Let us consider an even temporal Lorentzian spectral triple( A , e A , H , D, T ) with commutative time and with Z -grading γ . For every twopure states χ, ξ ∈ P ( e A ) such that χ ( (cid:0) T (cid:1) − ) = 0 and ξ ( (cid:0) T (cid:1) − ) = 0,we define: e d ( χ, ξ ) := inf a ∈ S e A n { max { , ξ ( a ) − χ ( a ) } : ∀ φ ∈ H , h φ, [ D, T ]([ D, a ] + iγ ) φ i ≥ } where S e A n is the filtered algebra generated by the time element T and where χ and ξ are considered as their unique extension to S e A n defined by: χ ( a ) = χ ( (cid:0) T (cid:1) − ) − n χ ( a ) and ξ ( a ) = ξ ( (cid:0) T (cid:1) − ) − n ξ ( a )if a = (cid:0) T (cid:1) n a with a ∈ e A . Conclusions
In this paper, we have studied the problem of defining spectral triples corre-sponding to manifolds with Lorentzian signature. We have suggested some ax-ioms to construct temporal Lorentzian spectral triples, which are spectral triplescorresponding, in the commutative case, to globally hyperbolic Lorentzian man-ifolds. Those axioms are based on the existence of an element representing aglobal time and which is sufficient to define a fundamental symmetry neededto recover the self-adjointness of the Dirac operator. This can be shown as akind of ”3+1” decomposition of a possibly noncommutative Lorentzian mani-fold. An explicit construction of a noncommutative example is given in Section4 with a Moyal–Minkowski spacetime. In the commutative case, the Proposi-tion 11 guarantees that, even with an abstract definition of temporal Lorentzianspectral triples, the corresponding metric has a Lorentzian signature and a splitnature.We have shown that, under the requirement that time is commutative, thoseaxioms allow us to extend the usual C ∗ -algebra formalism to unbounded ele-ments by using a technique of filtration and partial inner product spaces (PIP-spaces). This extension is a needed tool in order to extend the definition of aLorentzian distance formula [17] to noncommutative spaces. The noncommu-tative Lorentzian distance (Definition 24) is defined at the level of extensionsof pure states to unbounded elements which are well-defined thanks to the fil-tration properties. At this time, the correspondence between such a formulaand the usual Lorentzian distance in the commutative case is only completelyproven for Minkowski spacetime, the correspondence resulting in an inequalityin the general case. The possibility to define such distance on the space of purestates without requiring a commutative time is still an open question, as well asthe complete proof of the correspondence with the usual Lorentzian distance.We must remark that, for more general manifolds and possibly noncommutativetime, a definition of causality on the space of states can always be defined as itis shown in [17].The axioms of temporal Lorentzian spectral triples are subject to furtherdevelopment. In particular, the question of the smoothness of the Dirac operator(or more precisely of its elliptic modification) could be taken into account. Itcould also be interesting to wonder which temporal elements T generate a similarLorentzian manifold, and in this case to define a set of axioms for Lorentzianspectral triples which should be independent of the choice of a temporal element.The theory of Lorentzian noncommutative geometry is clearly still in its earlystages, and a lot of work should be done in order to have a complete set of axiomsfor Lorentzian spectral triples which could give rise to a possible reconstructiontheorem. Acknowledgement
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