Exploring the Causal Structures of Almost Commutative Geometries

We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra S(R¹,¹)⊗M₂(C), which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition...

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Дата:	2014
Автори:	Franco, N., Eckstein, M.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2014
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146845
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Цитувати:	Exploring the Causal Structures of Almost Commutative Geometries / N. Franco, M. Eckstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 28 назв. — англ.

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spelling	irk-123456789-1468452019-02-12T01:23:59Z Exploring the Causal Structures of Almost Commutative Geometries Franco, N. Eckstein, M. We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra S(R¹,¹)⊗M₂(C), which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space. 2014 Article Exploring the Causal Structures of Almost Commutative Geometries / N. Franco, M. Eckstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58B34; 53C50; 54F05 DOI:10.3842/SIGMA.2014.010 http://dspace.nbuv.gov.ua/handle/123456789/146845 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra S(R¹,¹)⊗M₂(C), which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space.
format	Article
author	Franco, N. Eckstein, M.
spellingShingle	Franco, N. Eckstein, M. Exploring the Causal Structures of Almost Commutative Geometries Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Franco, N. Eckstein, M.
author_sort	Franco, N.
title	Exploring the Causal Structures of Almost Commutative Geometries
title_short	Exploring the Causal Structures of Almost Commutative Geometries
title_full	Exploring the Causal Structures of Almost Commutative Geometries
title_fullStr	Exploring the Causal Structures of Almost Commutative Geometries
title_full_unstemmed	Exploring the Causal Structures of Almost Commutative Geometries
title_sort	exploring the causal structures of almost commutative geometries
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/146845
citation_txt	Exploring the Causal Structures of Almost Commutative Geometries / N. Franco, M. Eckstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 28 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT francon exploringthecausalstructuresofalmostcommutativegeometries AT ecksteinm exploringthecausalstructuresofalmostcommutativegeometries
first_indexed	2025-07-11T00:45:45Z
last_indexed	2025-07-11T00:45:45Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 010, 23 pages Exploring the Causal Structures of Almost Commutative Geometries? Nicolas FRANCO † and Micha l ECKSTEIN ‡† † Copernicus Center for Interdisciplinary Studies, ul. S lawkowska 17, 31-016 Kraków, Poland E-mail: nicolas.franco@math.unamur.be, nicolas.franco@im.uj.edu.pl ‡ Faculty of Mathematics and Computer Science, Jagellonian University, ul. Lojasiewicza 6, 30-348 Kraków, Poland E-mail: michal.eckstein@uj.edu.pl Received October 31, 2013, in final form January 20, 2014; Published online January 28, 2014 http://dx.doi.org/10.3842/SIGMA.2014.010 Abstract. We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra S(R1,1) ⊗M2(C), which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space. Key words: noncommutative geometry; causal structures; Lorentzian spectral triples 2010 Mathematics Subject Classification: 58B34; 53C50; 54F05 1 Introduction Noncommutative geometry à la Connes [10] offers new, vast horizons for both pure mathema- tics, as well as physical models. In its original formulation it provides a significant generalisa- tion of Riemannian differential geometry. However, for physical applications one should prefer Lorentzian rather than Riemannian structures. This is especially relevant for applications re- lated to gravitation. To this end the theory of Lorentzian spectral triples, a generalisation of the notion of spectral triples based on Krein space approach, has been developed by several authors [13, 14, 15, 23, 24, 25, 27, 28]. We can also mention several applications of noncommu- tative geometry in physics and cosmology in a Lorentzian setting, that use other techniques (for instance Wick rotation) [1, 4, 5, 6, 19, 20, 22]. One of the most important elements of Lorentzian geometry, that is absent in Riemannian theory, is the causal structure. For a mathematician this is a partial order relation defined between the points of a Lorentzian manifold. On the other hand, it is of crucial important for physicists since it determines which of the space-time events can be linked by a physical (not faster than light) signal. The generalisation of the causal structure to noncommutative spaces is not at all straightfor- ward because of the lack of the notion of points (or events). One of the possible counterparts of points in noncommutative geometry are pure states on algebras. By the famous Gelfand– Naimark theorem, pure states on a commutative C∗-algebra A are in one to one correspondence with points of some locally compact Hausdorff space Spec(A). In [15] we proposed a definition of a causal structure for the space of states in the general framework of Lorentzian spectral triples. ?This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html mailto:nicolas.franco@math.unamur.be mailto:nicolas.franco@im.uj.edu.pl mailto:michal.eckstein@uj.edu.pl http://dx.doi.org/10.3842/SIGMA.2014.010 http://www.emis.de/journals/SIGMA/Rieffel.html 2 N. Franco and M. Eckstein We have succeeded in showing that the usual causal relations are recovered whenever the spectral triple is commutative and constructed from a globally hyperbolic Lorentzian manifold. The purpose of this paper is to investigate the properties of such generalised causal struc- tures for a class of noncommutative spaces described by almost commutative geometries. The latter consists in considering spectral triples based on noncommutative algebras of the form C∞(M)⊗AF , with AF being a finite direct sum of matrix algebras. Almost commutative geometries are of crucial importance in physical applications of noncommutative geometry as they provide a firm framework for models of fundamental interactions [9, 10, 11, 26]. The plan of the paper presents itself as follows. Section 2 contains some general results on almost commutative Lorentzian geometries and their causal structure. Later, in Section 3, we present in details the causal structure of a particular spectral triple based on the algebra S(R1,1)⊗M2(C). We unravel the causal relations between the pure states (Section 3.1) as well as for a preferred subclass of mixed states (Section 3.2). We end with an outlook for further exploration of causal structures of almost commutative geometries along with their physical interpretation. 2 Lorentzian almost commutative geometries and causal structures The basic objects of Riemannian noncommutative geometry are spectral triples [10, 11]. Their generalisation to a pseudo-Riemannian setting is based on a working set of axioms, which slightly varies among the different approaches [13, 14, 23, 25, 28]. We base our work essentially on the formulation adopted in [13, 14, 15]. However, our results on causal structure can be accommo- dated also in the other approaches as already argued in [15]. Definition 1. A Lorentzian spectral triple is given by the data (A, Ã,H, D,J ) with: • A Hilbert space H. • A non-unital pre-C∗-algebra A, with a faithful representation as bounded operators on H. • A preferred unitisation Ã of A, which is also a pre-C∗-algebra with a faithful representation as bounded operators on H and such that A is an ideal of Ã. • An unbounded operator D, densely defined on H, such that: – ∀ a ∈ Ã, [D, a] extends to a bounded operator on H, – ∀ a ∈ A, a(1 + 〈D〉2)− 1 2 is compact, with 〈D〉2 = 1 2(DD∗ +D∗D). • A bounded operator J on H such that: – J 2 = 1, – J ∗ = J , – [J , a] = 0, ∀ a ∈ Ã, – D∗ = −JDJ , – J = −[D, T ] for some unbounded self-adjoint operator T with domain Dom(T ) ⊂ H, such that ( 1 + T 2 )− 1 2 ∈ Ã. The role of the operator J , called fundamental symmetry, is to turn the Hilbert space H into a Krein space on which the operator iD is self-adjoint [7, 24]. In the commutative case the condition J = −[D, T ] guarantees the correct signature, but restricts the admissible Lorentzian manifolds to the ones equipped with a global time function [14]. It is sufficient for the pur- poses of this paper since the considered almost commutative geometries are based on globally Exploring the Causal Structures of Almost Commutative Geometries 3 hyperbolic space-times. For a more general definition of J suitable in the pseudo-Riemannian context see [23] or [25]. Definition 2. A Lorentzian spectral triple is even if there exists a Z2-grading γ such that γ∗ = γ, γ2 = 1, [γ, a] = 0 ∀ a ∈ Ã, γJ = −J γ and γD = −Dγ. Let us now consider a locally compact complete globally hyperbolic Lorentzian manifold M of dimension n with a spin structure S. By a complete Lorentzian manifold we understand the following: there exists a spacelike reflection – i.e. an automorphism r of the tangent bun- dle respecting r2 = 1, g(r·, r·) = g(·, ·) – such that M equipped with a Riemannian metric gr(·, ·) = g(·, r·) is complete in the usual Lebesgue sense. Given a globally hyperbolic Lorentzian manifoldM one can construct a commutative Lorentzian spectral triple in the following way [15]: • H = L2(M, S) is the Hilbert space of square integrable sections of the spinor bundle over M. • D = −i(ĉ◦∇S) = −ieµaγa∇Sµ is the Dirac operator associated with the spin connection ∇S (the Einstein summation convention is in use and eµa stand for vielbeins). • A ⊂ C∞0 (M) and Ã ⊂ C∞b (M) with pointwise multiplication are some appropriate sub- algebras of the algebra of smooth functions vanishing at infinity and the algebra of smooth bounded functions respectively. Ã must be such that ∀ a ∈ Ã, [D, a] extends to a bounded operator on H. The representation is given by standard multiplication operators on H: (π(a)ψ)(x) = a(x)ψ(x) for all x ∈M. • J = iγ0, where γ0 is the first flat gamma matrix1. If n is even, the Z2-grading is given by the chirality element: γ = (−i) n 2 +1γ0 · · · γn−1. An even Lorentzian spectral triple can be combined with a Riemannian spectral triple in order to obtain a new Lorentzian spectral triple. Only one of the two spectral triples should be Lorentzian, otherwise the obtained spectral triple would be pseudo-Riemannian with a more general signature (as defined in [24]). When the Lorentzian spectral triple is commutative and the Riemanian one is noncommutative and finite, we obtain an almost commutative geometry with Lorentzian signature. Theorem 1. Let us consider an even Lorentzian spectral triple (AM, ÃM,HM, DM,JM) with Z2-grading γM and a (compact) Riemannian spectral triple (AF ,HF , DF ), then the product A = AM ⊗AF , Ã = ÃM ⊗AF , H = HM ⊗HF , D = DM ⊗ 1 + γM ⊗DF , J = JM ⊗ 1 is a Lorentzian spectral triple. Proof. Since the above definition is completely analogous to the standard product between two Riemannian spectral triples [10, 11], we only need to show that the conditions related to the fundamental symmetry J are fulfilled. The conditions J 2 = 1, J ∗ = J and [J , a] = 0 ∀ a ∈ Ã are easily derived from the properties J 2 M = 1, J ∗M = JM and [JM, a] = 0 ∀ a ∈ ÃM. 1Conventions used in the paper are (−,+,+,+, · · · ) for the signature of the metric and {γa, γb} = 2ηab for the flat gamma matrices, with γ0 anti-Hermitian and γa Hermitian for a > 0. 4 N. Franco and M. Eckstein The Hermicity condition of the Dirac operator is met: JDJ = JMDMJM ⊗ 1 + JMγMJM ⊗DF = −D∗M ⊗ 1∗ − γ∗M ⊗D∗F = −D∗, since JMγMJM = −J 2 MγM = −γM = −γ∗M and DF is self-adjoint. We have JM = −[D, TM ] for some unbounded self-adjoint operator TM (which commutes with γM), so by setting T = TM ⊗ 1 we find: −[D, T ] = −DMTM ⊗ 1− γMTM ⊗DF + TMDM ⊗ 1 + TMγM ⊗DF = −[DM, TM]⊗ 1 = JM ⊗ 1 = J . Hence, the resulting spectral triple is indeed of the Lorentzian signature. � Let us remark that, if the Riemannian spectral triple is even with Z2-grading γF , then the product can also be defined with an alternative operator D = DM ⊗ γF + 1⊗DF . In this case the new fundamental symmetry must be set to J = JM⊗ γF . If both spectral triples are even, then the product is even with the Z2-grading γ = γM ⊗ γF . To make the paper self-contained we shall recollect the basic definitions and properties con- cerning the causal structure for Lorentzian spectral triples. As mentioned in the Introduction, the causal relations are defined between the states on the algebra Ã, which can be viewed as a noncommutative generalisation of events (points of space-time). Let us recall that a (mixed) state on a C∗-algebra is a positive linear functional (automatically continuous) of norm one [18]. For our purposes we consider the states on the algebra Ã – the C∗-completion of Ã – restricted to Ã. We shall denote the set of such states as S(Ã). It is a closed convex set for the weak-∗ topology and its extremal points form a set of pure states denoted by P (Ã). The causal relation between the states is defined in the following way [15]: Definition 3. Let us consider the cone C of all Hermitian elements a ∈ Ã respecting ∀φ ∈ H 〈φ,J [D, a]φ〉 ≤ 0, (1) where 〈·, ·〉 is the inner product on H. If the following condition is fulfilled: spanC(C) = Ã, (2) then C is called a causal cone and defines a partial order relation on S(Ã) by ∀ω, η ∈ S(Ã) ω � η iff ∀ a ∈ C ω(a) ≤ η(a). Since the definition of a Lorentzian spectral triple allows for a preferred choice of the uni- tisation Ã, one may actually use the condition (2) to determine it, as (2) may, in general, be false for an arbitrary unitisation. We shall use the following procedure in order to determine the preferred unitisation: 1. Chose Ã0 as the biggest possible unitisation fulfilling the axioms of Definition 1. 2. Define C to be the set of all of the elements in Ã0 fulfilling the condition (1). 3. Set Ã = spanC(C) and check that it is a valid unitisation for A. This procedure assures that C is a causal cone which is automatically the maximal one. The latter is important, since in the commutative case the maximal causal cone is needed to recover the classical causal structure on space-time [15]. In the case of a globally hyperbolic Lorentzian manifold, such a suitable unitisation always exists [3, 15]. We shall see that for almost commutative geometries the described procedure is also sound. Motivated by the following theorem, we call the partial order relation � on S(Ã) a causal relation between the states. Exploring the Causal Structures of Almost Commutative Geometries 5 Theorem 2 ([15]). Let (A, Ã,H, D,J ) be a commutative Lorentzian spectral triple constructed from a globally hyperbolic Lorentzian manifold M, and let us define the following subset of pure states: M(Ã) = { ω ∈ P (Ã) : A 6⊂ kerω } ⊂ S(Ã). Then the causal relation � on S(Ã) restricted to M(Ã) ∼= M corresponds to the usual causal relation on M. The proof is based on the one-to-one correspondence between the elements of C and the causal functions, which are the real-valued functions onM non-decreasing along future directed causal curves. This theorem shows the necessity of adding a unitisation in the definition of a Lorentzian spectral triple, since the subset of causal functions within the initial non-unital algebra only contains the null function which is not sufficient to characterise causality. The role of the restriction of the space of states to M(Ã) is to avoid the states localised at infinity. In fact we have M(Ã) ∼= P (A) which, by the Gelfand–Naimark theorem, is isomorphic to the locally compact manifold M itself. The complete proof of this theorem, as well as more details about the definition of causality in the space of states, can be found in [15]. Let us now take a look at the space of states of almost commutative geometries. In full analogy to the Riemannian case (see for instance [26]) we define an almost commutative Lorentzian spectral triple as a product defined in Theorem 1 with (AM, ÃM,HM, DM,JM) constructed from a globally hyperbolic manifold and (AF ,HF , DF ) based on a finite direct sum of matrix algebras. Proposition 1. Let (A, Ã,H, D,J ) be an almost commutative Lorentzian spectral triple and let P (A) denote the space of pure states (extremal points of S(A)) on A. Then P (Ã) ∼= P (ÃM)× P (AF ), P (A) ∼=M× P (AF ). Proof. This facts follows from [18, Theorem 11.3.7] (see also [12]) and the locally compact version of the Gelfand–Naimark theorem. � This motivates us to define the following space of physical states of almost commutative Lorentzian geometries M(Ã) = { χ⊗ ξ ∣∣χ ∈ P (ÃM), AM 6⊂ kerχ, ξ ∈ P (AF ) } ∼= P (A). (3) Let us stress that the fact that all of the pure states on Ã are separable (i.e. have the form of simple tensor product) is a consequence of AM being commutative. Hence, it is a genuine property of all almost commutative geometries. On the other hand, the space of all (mixed) states S(Ã) is far more complicated to determine. We shall not attempt to address this prob- lem here, some results on the causal structure of a subspace of mixed states are presented in Section 3.2. Proposition 1 has a useful consequence, which simplifies the examination of the causal struc- ture and remains valid for all almost commutative Lorentzian geometries. Proposition 2. Let (A, Ã,H, D,J ) be an almost commutative Lorentzian spectral triple and let D̃ = DM ⊗ 1 + γM ⊗ D̃F , with D̃F = UDFU ∗, be a unitary transformation of the finite-part Dirac operator. If we denote by � and �̃ the partial order relations associated with the basic and transformed Dirac operators respectively, then for any ω, η ∈M(Ã) ω � η ⇐⇒ (1⊗ U)ω �̃ (1⊗ U)η. 6 N. Franco and M. Eckstein Proof. On the strength of Proposition 1 any state ω ∈ M(Ã) can be uniquely written as ω = ωp,ξ = ωp ⊗ ωξ with ωp ∈ P (AM) ∼= M and ωξ ∈ P (AF ). Moreover, since AF is a finite direct sum of matrix algebras, every state in P (AF ) is a vector state [17], i.e. ∀ωξ ∈ P (AF ) ∃ ξ ∈ HF 〈ξ, ξ〉 = 1 : ∀ a ∈ AF , ωξ(a) = 〈ξ, aξ〉 . As a digression let us note that if AF is not a single matrix algebra but a direct product, then not every normalised vector in HF defines a pure state on AF . Now, we have ∀ a ∈ C̃, (ωp,Uξ − ωq,Uϕ)(a) ≤ 0 ⇐⇒ ∀ a ∈ C̃, (ωp,ξ − ωq,ϕ) ( (1⊗ U∗)a(1⊗ U) ) ≤ 0, where C̃ denotes the causal cone associated with the Dirac operator D̃. The causality condition reads ∀φ ∈ H, 〈 φ,J [D̃, a]φ 〉 ≤ 0 ⇐⇒ ∀φ ∈ H, 〈φ, (J ⊗ 1)[DM ⊗ 1 + γM ⊗ UDFU ∗, a]φ〉 ≤ 0 ⇐⇒ ∀φ ∈ H, 〈 φ, (J ⊗ 1)[(1⊗ U)D ( 1⊗ U∗ ) , a]φ 〉 ≤ 0 ⇐⇒ ∀φ ∈ H, 〈 φ, (J ⊗ 1)(1⊗ U)[D, ( 1⊗ U∗ ) a(1⊗ U)](1⊗ U∗)φ 〉 ≤ 0 ⇐⇒ ∀ φ̃ = ( 1⊗ U∗ ) φ ∈ H, 〈 φ̃, (J ⊗ 1)[D, ( 1⊗ U∗ ) a(1⊗ U)]φ̃ 〉 ≤ 0 ⇐⇒ ( 1⊗ U∗ ) a(1⊗ U) ∈ C. Thus, we have (1⊗ U)ωp,ξ �̃ (1⊗ U)ωq,ϕ ⇐⇒ ∀ a ∈ C̃, (ωp,ξ − ωq,ϕ) ( (1⊗ U∗)a(1⊗ U) ) ≤ 0 ⇐⇒ ∀ ( 1⊗ U∗ ) a(1⊗ U) ∈ C, (ωp,ξ − ωq,ϕ) ( (1⊗ U∗)a(1⊗ U) ) ≤ 0 ⇐⇒ ωp,ξ � ωq,ϕ. � We conclude this section by investigating the impact of conformal transformations in almost commutative geometries. It is a well known fact that a conformal rescaling of the metric on a Lorentzian manifold does not change its causal structure. It turns out that for this result to hold true in the almost commutative case one needs to transform the Dirac operator DF accordingly. Proposition 3. Let Ω be a positive smooth bounded function on M and let (A, Ã,H, D,J ) be an almost commutative Lorentzian spectral triple. The causal structures of (A, Ã,H, D,J ) and (A, Ã,H, D̃,J ), with D̃ = ΩDΩ, are isomorphic, i.e. ∀ω, η ∈ S(Ã), ω � η ⇐⇒ ω �̃ η, where � and �̃ denote the causal relations on S(Ã) associated with D and D̃ respectively. Proof. Let us first recall that a conformal rescaling of the metric tensor onM, g̃ = Ω4g, results in the change of the Dirac operator of the form D̃M = ΩDMΩ [21]. The condition for an element a ∈ Ã to be in the causal cone C̃ associated with D̃ reads ∀φ ∈ H, 〈 φ,J [D̃, a]φ 〉 ≤ 0 ⇐⇒ ∀φ ∈ H, 〈φ,J [Ω(DM ⊗ 1 + γM ⊗DF )Ω, a]φ〉 ≤ 0 ⇐⇒ ∀φ ∈ H, 〈Ωφ,J [DM ⊗ 1 + γM ⊗DF , a]Ωφ〉 ≤ 0 ⇐⇒ ∀ φ̃ = Ωφ ∈ H, 〈 φ̃,J [DM ⊗ 1 + γM ⊗DF , a]φ̃ 〉 ≤ 0. We have used the fact that Ω commutes with every element of Ã as well as with γM and J . � Exploring the Causal Structures of Almost Commutative Geometries 7 3 Two-dimensional flat almost commutative space-time In this section, we construct a two-dimensional Lorentzian almost commutative geometry as the product of a two-dimensional Minkowski space-time and a noncommutative algebra of complex 2×2 matrices. This is the simplest example of a noncommutative space-time, the causal structure of which exhibits new properties in comparison to a classical Lorentzian manifold. We shall present the explicit calculation of the causal structure for pure states as well as for a class of mixed states. The commutative part is constructed as follows: • HM = L2(R1,1)⊗C2 is the Hilbert space of square integrable sections of the spinor bundle over the two-dimensional Minkowski space-time. • AM = S(R1,1) is the algebra of Schwartz functions (rapidly decreasing at infinity together with all derivatives) with pointwise multiplication. • ÃM = spanC(CM) ⊂ B(R1,1) is a sub-algebra of the algebra of smooth bounded func- tions with all derivatives bounded with pointwise multiplication. CM represents the set of smooth bounded causal functions (real-valued functions non-decreasing along future directed causal curves) on R1,1 with all derivatives bounded. • DM = −iγµ∂µ is the flat Dirac operator. • JM = −[D,x0] = ic(dx0) = iγ0 where x0 is the global time coordinate and c the Clifford action. • γM = γ0γ1. This construction is standard for non-unital spectral triples (see [16] or [14]) except that the unitisation ÃM is slightly restricted in order to meet the condition (2). ÃM is a unital algebra which corresponds to a particular compactification of R1,1 [3, 15], hence a valid unitisation of AM respecting the axioms. On compact ordered spaces, monotonic functions which are order-preserving (isotone functions) can span the entire algebra of functions as a consequence of the Stone-Weierstrass theorem, whereas on noncompact spaces such functions must respect some limit condition in order to remain bounded. In our construction, ÃM represents the sub-algebra of functions in B(R1,1) with existing limits along every causal curve. The finite and noncommutative part is constructed as a compact Riemannian spectral triple: • HF = C2; • AF = M2(C) is the algebra of 2 × 2 complex matrices with natural multiplication and representation on HF ; • DF = diag(d1, d2), with d1, d2 ∈ R. We can restrict our considerations to DF diagonal since, on the strength of Proposition 2, the results are easily translated to any other selfadjoint operator with a unitary transformation U ∈M2(C). The final almost commutative Lorentzian spectral triple is constructed as follows: • H = L2(R1,1)⊗ C2 ⊗ C2 ∼= L2(R1,1)⊗ C4, • A = S(R1,1)⊗M2(C), • Ã ⊂ Ã0 = ÃM ⊗M2(C) ⊂ B(R1,1)⊗M2(C), • D = −iγµ∂µ ⊗ 1 + γ0γ1 ⊗ diag(d1, d2) = diag(−iγµ∂µ + γ0γ1di)i∈{1,2}, • J = iγ0 ⊗ 1 = diag(iγ0). 8 N. Franco and M. Eckstein Following the procedure described in Seciton 2 we first chose the maximal possible unitisation Ã0 = spanC(CM)⊗M2(C). Let us now characterise the set C = { a ∈ Ã0 : a = a∗ and ∀φ ∈ H, 〈φ,J [D,a]φ〉 ≤ 0 } . For further convenience we shall write an element a ∈ C as ( a −c −c∗ b ) . Proposition 4. Let a = ( a −c −c∗ b ) ∈ Ã0 be a Hermitian element, then the following conditions are equivalent: (a) a ∈ C, i.e. ∀φ ∈ H, 〈φ,J [D,a]φ〉 ≤ 0. (b) At every point of R1,1, the matrix a,0 + a,1 0 −c,0 − c,1 −(d1 − d2)c 0 a,0 − a,1 (d1 − d2)c −c,0 + c,1 −c∗,0 − c∗,1 (d1 − d2)c∗ b,0 + b,1 0 −(d1 − d2)c∗ −c∗,0 + c∗,1 0 b,0 − b,1  is positive semi-definite. (c) At every point of R1,1, ∀α1, α2, α3, α4 ∈ C, \|α1\|2 (a,0 + a,1) + \|α2\|2 (a,0 − a,1) + \|α3\|2 (b,0 + b,1) + \|α4\|2 (b,0 − b,1) ≥ 2<{α∗1α3(c,0 + c,1) + α∗2α4(c,0 − c,1) + ( α∗1α4 − α∗2α3 ) \|d1 − d2\| c } . We use here the notation f,µ = ∂µf = ∂f ∂xµ . Proof. We work with the following basis of flat gamma matrices: γ0 = ( 0 i i 0 ) , γ1 = ( 0 −i i 0 ) , which implies γ0γ1 = γM = ( −1 0 0 1 ) · Using the isomorphism C2 ⊗ C2 ∼= C4, we have the following matrices: [D,a] = ( −iγµ∂µa iγµ∂µc− γM(d1 − d2)c iγµ∂µc ∗ + γM(d1 − d2)c∗ −iγµ∂µb ) , J = ( iγ0 0 0 iγ0 ) , −J [D,a] = ( −γ0γµ∂µa γ0γµ∂µc− iγ1(d1 − d2)c γ0γµ∂µc ∗ + iγ1(d1 − d2)c∗ −γ0γµ∂µb ) · With our choice of gamma matrices, the above matrix gives precisely the one displayed in (b). Now, let us check that the condition ∀φ ∈ H, 〈φ,J [D,a]φ〉 ≤ 0 is equivalent to having J [D,a] ≤ 0 at every point of R1,1, i.e. −J [D,a] is a positive semi-definite matrix at every point of R1,1. Indeed, the second condition implies the first one, and if the second one is false at some particular point p ∈ R1,1, then by continuity it is false on some open set Up ⊂ R1,1 and the first condition is false for some non-null spinor φ ∈ H the support of which is included in Up. Hence, (a) is equivalent to (b). The condition (c) is just a reformulation of the condition (b) with an arbitrary spinor φ = (α1, α2, α3, α4) taken at a fixed point of R1,1. � Let us now enlighten the properties of the set C by considering some of its subsets. Exploring the Causal Structures of Almost Commutative Geometries 9 Lemma 1. The following elements in Ã0 belong to C: (a) a = ( a 0 0 b ) , where a and b are two causal functions on R1,1 (a, b ∈ CM). (b) b = ( a −c −c∗ a ) , with a,0 − \|a,1\| ≥ \|c,0\|+ \|c,1\|+ \|d1 − d2\| \|c\|. Proof. (a) Since a causal function onM is non-decreasing along future directed causal curves, a, b ∈ CM is equivalent to a,0 ± a,1 ≥ 0 and b,0 ± b,1 ≥ 0, so Proposition 4(c) is respected for c = 0. We notice that, by choosing αi = 1 and αj = 0 (j 6= i) for i = 1, . . . , 4, this condition is necessary and sufficient whenever c = 0. (b) By Proposition 4(c), we must check for all α1, α2, α3, α4 ∈ C the following condition:( \|α1\|2 + \|α3\|2 ) (a,0 + a,1) + ( \|α2\|2 + \|α4\|2 ) (a,0 − a,1) ≥ 2<{α∗1α3(c,0 + c,1) + α∗2α4(c,0 − c,1) + ( α∗1α4 − α∗2α3 ) \|d1 − d2\| c } . Using AM-GM inequality, we have( \|α1\|2 + \|α3\|2 ) (a,0 + a,1) + ( \|α2\|2 + \|α4\|2 ) (a,0 − a,1) ≥ ( \|α1\|2 + \|α3\|2 + \|α2\|2 + \|α4\|2 ) (\|c,0\|+ \|c,1\|+ \|d1 − d2\| \|c\|) ≥ ( \|α1\|2 + \|α3\|2 ) \|c,0 + c,1\|+ ( \|α2\|2 + \|α4\|2 ) \|c,0 − c,1\| + (( \|α1\|2 + \|α4\|2 ) + ( \|α2\|2 + \|α3\|2 )) \|d1 − d2\| \|c\| ≥ 2 \|α1\| \|α3\| \|c,0 + c,1\|+ 2 \|α2\| \|α4\| \|c,0 − c,1\|+ 2(\|α1\| \|α4\|+ \|α2\| \|α3\|) \|d1 − d2\| \|c\|) ≥ 2<{α∗1α3(c,0 + c,1)}+ 2<{α∗2α4(c,0 − c,1)}+ 2<{(α∗1α4 − α∗2α3) \|d1 − d2\| c} . � Proposition 5. If the unitisation Ã ⊂ Ã0 is restricted to matrices with off-diagonal entries in AM = S(R1,1) ⊂ spanC(CM), then C is a causal cone for the Lorentzian spectral triple (A, Ã,H, D,J ). Proof. Ã is a unital algebra which contains A as an ideal. Let us designate the sub-algebra of Hermitian elements in Ã by ÃH . Any element in ÃH can be written as a = ( a −c −c∗ b ) with a, b ∈ spanR(CM) and c ∈ S(R1,1). Then a can be decomposed as: a = ( a −c −c∗ b ) = ( a 0 0 b ) + ( ã −c −c∗ ã ) − ( ã 0 0 ã ) , where ã is a suitable causal function in CM such that ã,0 ≥ \|c,0\| + \|c,1\| + \|d1 − d2\| \|c\| and ã,1 = 0. Such function always exists since \|c,0\| + \|c,1\| + \|d1 − d2\| \|c\| is bounded and rapidly decreasing at infinity. From Lemma 1, the last two matrices belong to C and the first one belongs to spanR(C), so a ∈ spanR(C) which implies ÃH = spanR(C). We conclude by noting that Ã = ÃH + iÃH = spanC(C). � Hence, we define the ultimate unitisation Ã as follows: Ã ∼= ( ÃM AM AM ÃM ) = ( spanC(CM) S ( R1,1 ) S ( R1,1 ) spanC(CM) ) . Thus we have a well-defined causal structure on the two-dimensional flat almost commu- tative space-time (A, Ã,H, D,J ) determined by the set of causal elements C characterised in Proposition 4. The subsets of causal elements defined in Lemma 1 give us some information about the general behaviour of causal elements. The diagonal entries must be causal functions on R1,1 (i.e. real and non-decreasing along causal curves), while the off-diagonal entry is a com- plex function which respects a bound depending on the growth rate of the diagonal entries along causal curves. The Fig. 1 illustrates this behaviour. 10 N. Franco and M. Eckstein Figure 1. Typical behaviour of the functions constituting a causal element a ∈ C along a causal curve. 3.1 Causal structure for pure states In this section, we make a complete characterisation of the causal structure of the two-dimen- sional flat almost commutative space-time (A, Ã,H, D,J ) at the level of pure states. We shall focus on the space of physical pure statesM(Ã), defined by (3), that do not localise at infinity. On the strength of Proposition 1 a pure state ωp,ξ ∈M(Ã) is determined by two entries: • a point p ∈ R1,1, • a normalised complex vector ξ ∈ C2. The value of a pure state on some element a ∈ Ã is: ωp,ξ(a) = ξ∗a(p)ξ, where a(p) is an application of the evaluation map at p on each entry in a. It is clear that two normalised vectors determine the same state if and only if they are equal up to a phase, so the set of normalised vectors corresponds to CP 1 ∼= S2. Hence, on the set- theoretic level, the space of pure states M(Ã) is isomorphic to the product R1,1 × S2. We will refer to R1,1 as the continuous space and to S2 as the internal space. Hoverer, one should be aware that the geometry of the internal space, resulting from the finite noncommutative algebra, is quite different from the usual Riemannian geometry of S2 (see [17]). Proposition 6. Let us take two pure states ωp,ξ, ωq,ϕ ∈M(Ã). Then ωp,ξ � ωq,ϕ if and only if ∀a = ( a −c −c∗ b ) ∈ C, \|ϕ1\|2 a(q)− \|ξ1\|2 a(p) + \|ϕ2\|2 b(q)− \|ξ2\|2 b(p) ≥ 2< { ϕ∗1ϕ2c(q)− ξ∗1ξ2c(p) } , where ξ = ( ξ1 ξ2 ) and ϕ = ( ϕ1 ϕ2 ) . Proof. From Definition 3, ωp,ξ � ωq,ϕ if and only if ∀a ∈ C, ωp,ξ(a) ≤ ωq,ϕ(a) ⇐⇒ ξ∗a(p)ξ ≤ ϕ∗a(q)ϕ ⇐⇒ ( ξ∗1 ξ∗2 )( a(p) −c(p) −c(p)∗ b(p) )( ξ1 ξ2 ) ≤ ( ϕ∗1 ϕ∗2 )( a(q) −c(q) −c(q)∗ b(q) )( ϕ1 ϕ2 ) ⇐⇒ \|ξ1\|2 a(p)− 2< { ξ∗1ξ2c(p) } + \|ξ2\|2 b(p) ≤ \|ϕ1\|2 a(q)− 2< { ϕ∗1ϕ2c(q) } + \|ϕ2\|2 b(q). � Proposition 7. Two pure states ωp,ξ, ωq,ϕ ∈M(Ã) are causally related with ωp,ξ � ωq,ϕ only if p � q in R1,1. Proof. From Lemma 1, a = ( a 0 0 a ) ∈ C where a is a causal function on R1,1. Using Proposition 6 with a we find ( \|ϕ1\|2 + \|ϕ2\|2 ) a(q)− ( \|ξ1\|2 + \|ξ2\|2 ) a(p) ≥ 0 =⇒ a(p) ≤ a(q). Since the above inequality is valid for all a ∈ CM, we have p � q from Theorem 2. � Exploring the Causal Structures of Almost Commutative Geometries 11 This first necessary condition tells us that the coupling of R1,1 with the internal space S2 does not induce any violation on the usual causal structure on R1,1 and the result does not depend on the particular form of DF . Let us now show that if the finite-part Dirac operator is degenerate (i.e. d1 = d2), then no changes in the internal space are allowed by the causal structure. Proposition 8. If the finite-part Dirac operator is degenerate (d1 = d2), then two pure states ωp,ξ, ωq,ϕ ∈M(Ã) are causally related with ωp,ξ � ωq,ϕ only if p � q in R1,1 and ξ = ϕ. Proof. If d1 = d2, then the element a = ( a −c −c∗ b ) ∈ Ã, with a, b and c being constant functions, is in C on the strength of Proposition 4. For such element a the causality condition form Proposition 6 reads( \|ϕ1\|2 − \|ξ1\|2 ) a+ ( \|ϕ2\|2 − \|ξ2\|2 ) b ≥ 2< { (ϕ∗1ϕ2 − ξ∗1ξ2)c } . Since a, b ∈ R and c ∈ C are arbitrary the above inequality is always fulfilled only if ξ = ϕ. � From now on we will assume that the finite-part Dirac operator is not degenerate (i.e. d1 6= d2). Under this assumption, the movements in the internal space are allowed, but they turn out to be further constrained by the causal structure. In order to express the second necessary condition it is convenient to write down explicitly the isomorphism between CP 1 and S2. A nor- malised vector ξ = (ξ1, ξ2) ∈ CP 1 can be associated with the point (xξ, yξ, zξ) of S2 embedded in R3 by the relations: xξ = 2<{ξ∗1ξ2}, yξ = 2={ξ∗1ξ2}, zξ = \|ξ1\|2 − \|ξ2\|2 . We will refer to zξ as the latitude (or the altitude) of the vector ξ on S2. Proposition 9. Two pure states ωp,ξ, ωq,ϕ ∈ M(Ã) are causally related with ωp,ξ � ωq,ϕ only if ξ and ϕ have the same latitude, i.e. zξ = zϕ. Proof. From Lemma 1, a = ( a 0 0 0 ) ∈ C where a is a constant function on R1,1. Using Proposi- tion 6 with a we find: \|ϕ1\|2 a(q)− \|ξ1\|2 a(p) = ( \|ϕ1\|2 − \|ξ1\|2 ) a ≥ 0. Since this inequality must be valid for all a ∈ R, we must have \|ϕ1\|2 = \|ξ1\|2. Since the vectors are normalised we must have \|ϕ2\|2 = \|ξ2\|2, hence zξ = zϕ. � This condition is completely coherent with the results obtained in [17] using Connes’ Rie- mannian distance formula on the noncommutative space M2(C). The parallels of latitude within the sphere S2 were found to be separated by an infinite distance. This can be viewed as a geo- metrical characteristic of our internal space, which forbids any movement between disconnected parallels. From Proposition 2, it follows that a unitary transformation of the finite-part Dirac operator DF is equivalent to a rotation in the internal space of states. Hence, the notion of the latitude is completely determined by the diagonalisation of the operator DF . We should remark that the infinite distance separation between the parallels is a property of the representation of M2(C) on C2, which is the usual type of representation for the non- commutative standard model [9, 26]. It has been shown in [8] that one can obtain a finite distance between the parallels by using instead a representation on M2(C) and by defining an operator D as a truncation of the usual Dirac operator on the Moyal plane. It is also not known if the correspondence between causally disconnected states and infinite Riemannian distance is just a coincidence of this particular model or a more general property. 12 N. Franco and M. Eckstein We now come to our main result which is the complete characterisation of the motions in the internal space that do not violate the causality conditions. It turns out that the possible movements within a particular parallel of fixed latitude in the internal space are intrinsically connected to the motion in the space-time. In order to simplify the expressions, we can assume without loss of generality that ξ1 and ϕ1 are always real and non-negative since ξ and ϕ are defined up to phase. Since ξ and ϕ have the same latitude and are normalised vectors, we find that ξ1 = ϕ1 and \|ξ2\| = \|ϕ2\|, so we can set ξ2 = \|ϕ2\| eiθξ and ϕ2 = \|ϕ2\| eiθϕ with θξ and θϕ representing the position of the vectors on the parallel of latitude. Theorem 3. Two pure states ωp,ξ, ωq,ϕ ∈ M(Ã) are causally related with ωp,ξ � ωq,ϕ if and only if the following conditions are respected: • p � q in R1,1; • ξ and ϕ have the same latitude; • l(γ) ≥ \|θϕ−θξ\|\|d1−d2\| , where l(γ) represents the length of a causal curve γ going from p to q on R1,1. The proof of this theorem will be completed in the remaining part of this section. The necessary parts of the two first conditions have already been treated in Propositions 7 and 9. The sufficient conditions are proven in Proposition 10 while the necessary condition on l(γ) is proven in Proposition 11. Figure 2. Representation of a path in the space of pure states. Theorem 3 gives a full description of the causal structure of the two-dimensional flat almost commutative space-time. A graphical presentation of the space of pure states with a visualisation of a causal path is given in Fig. 2. Whenever the curve γ is not null, the inequality between the distances in the internal and continuous spaces can be written as: \|θϕ − θξ\| l(γ) ≤ \|d1 − d2\| . (4) \|θϕ − θξ\| represents a measure of the distance within the parallel of latitude between two vectors in the internal space. l(γ) = ∫ √ −gγ(γ̇(t), γ̇(t))dt physically represents the proper time of γ, i.e. the time measured by a clock moving along γ in the Minkowski space-time. \|d1 − d2\| is a constant defined by the eigenvalues of the operator DF of the internal space. The condition (4) can be seen as a constant upper bound to the ratio between the distance in the internal space and the proper time in the continuous space. Note that if one moves along a light-like geodesic Exploring the Causal Structures of Almost Commutative Geometries 13 in the Minkowski space-time (i.e. l(γ) = 0), then no motion in the internal space is allowed by the causal structure. We now come to the technical proof of Theorem 3. At first, we consider a timelike curve γ linking the space-time events p and q. The following Proposition will give us the sufficient part of Theorem 3. Proposition 10. Suppose that p � q with γ a future directed timelike curve going from p to q and that ξ, ϕ have the same latitude, with ξ1 = ϕ1 ∈ R∗ and ξ2 = \|ϕ2\| eiθξ , ϕ2 = \|ϕ2\| eiθϕ. If l(γ) ≥ \|θϕ−θξ\|\|d1−d2\| , then ωp,ξ � ωq,ϕ. Proof. Since ϕ1 = ξ1 ∈ R and \|ϕ2\| = \|ξ2\|, from Proposition 6 we need to prove the following inequality ∀a = ( a −c −c∗ b ) ∈ C, \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) ≥ 2< { ϕ1ϕ2c(q)− ϕ1ξ2c(p) } . (5) Let us first take a timelike curve γ : [0, T ]→ R1,1, with γ(0) = p, γ(T ) = q and l(γ) = \|θϕ−θξ\| d , where d = \|d1 − d2\| and T > 0. We define the following functions for t ∈ [0, T ]: • v(t) = γ̇1(t) γ̇0(t) ∈]−1, 1[ with γ̇(t) = (γ̇0(t), γ̇1(t)), which is well defined since γ is future directed timelike, so γ̇0(t) > 0 and γ̇0(t) > \|γ̇1(t)\|. • λ1(t) = 1+v(t) 2 , λ2(t) = 1−v(t) 2 ∈]0, 1[. • χ(t) = ξ2e i sgn(θϕ−θξ)dlγ(t) where lγ(t) = ∫ t 0 √ −gγ(s)(γ̇(s), γ̇(s))ds represents the length of γ restricted to the interval [0, t], with t ≤ T . We have obviously that \|χ(t)\| = \|ξ2\| = \|ϕ2\| ∀ t ∈ [0, T ], χ(0) = ξ2 and χ(T ) = ϕ2. Using the second fundamental theorem of calculus, we have the following equality: a(q)− a(p) = a(γ(T ))− a(γ(0)) = ∫ T 0 daγ(t)(γ̇(t))dt = ∫ T 0 γ̇0(t) (a,0(γ(t)) + v(t)a,1(γ(t))) dt = ∫ T 0 γ̇0(t) [λ1(t) (a,0(γ(t)) + a,1(γ(t))) + λ2(t) (a,0(γ(t))− a,1(γ(t)))] dt. In the following, we shall omit the arguments t and γ(t) in order to have more readable expres- sions. An analogous equality can be obtained for the function b: b(q)− b(p) = ∫ T 0 γ̇0 [λ1(b,0 + b,1) + λ2(b,0 − b,1)] dt. Now, the l.h.s. of (5) can be written as: \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) = ∫ T 0 γ̇0 [ \|ϕ1\|2 (λ1(a,0 + a,1) + λ2(a,0 − a,1)) + \|ϕ2\|2 (λ1(b,0 + b,1) + λ2(b,0 − b,1)) ] dt = ∫ T 0 γ̇0 [ \|ϕ1\|2 λ1(a,0 + a,1) + \|ϕ1\|2 λ2(a,0 − a,1) + \|χ\|2 λ1(b,0 + b,1) + \|χ\|2 λ2(b,0 − b,1) ] dt. (6) If we apply Proposition 4(c) to the integrand of (6) with α1 = √ λ1ϕ1e i∆1 , α2 = √ λ2ϕ1e i∆2 , α3 = √ λ1χe i∆1 , α4 = √ λ2χe i∆2 14 N. Franco and M. Eckstein we get \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) ≥ ∫ T 0 γ̇0 [ 2< { ϕ1χλ1(c,0 + c,1) + ϕ1χλ2(c,0 − c,1) } + 2< {√ λ1λ2 ( ei(∆2−∆1) − e−i(∆2−∆1) ) ϕ1χdc }] dt = ∫ T 0 [ 2< { ϕ1χ dc dt } + 2< { ϕ1cχi sin(∆2 −∆1)d2γ̇0 √ λ1λ2 }] dt. (7) Now, let us note that: 2γ̇0(t) √ λ1(t)λ2(t) = γ̇0(t) √ 1− v2(t) = √ γ̇2 0(t)− γ̇2 1(t) = √ −gγ(t)(γ̇(t), γ̇(t)) and dχ(t) dt = χ(t)i sgn(θϕ − θξ)d dlγ(t) dt = χ(t)i sgn(θϕ − θξ)d √ −gγ(t)(γ̇(t), γ̇(t)) = χ(t)i sgn(θϕ − θξ)d2γ̇0(t) √ λ1(t)λ2(t). (8) So if we choose ∆1, ∆2 to have sin(∆2 −∆1) = sgn(θϕ − θξ), we can substitute (8) into (7) to get: \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) ≥ ∫ T 0 [ 2< { ϕ1χ dc dt } + 2< { ϕ1c dχ dt }] dt = ∫ T 0 2< { ϕ1 d dt (χc) } dt = 2< { ϕ1χ(T )c(γ(T ))− ϕ1χ(0)c(γ(0)) } = 2< { ϕ1ϕ2c(q)− ϕ1ξ2c(p) } (9) and we have proven (5). In order to complete the proof, we must show that this result is still valid when l(γ) ≥ \|θϕ−θξ\|d . Since the causal order is a well defined partial order, it is sufficient to prove that, if ξ = ϕ, then the causal relation is valid for every causal curve γ with length l(γ) ≥ 0. This can be done by using (7) with χ constant and ∆2 = ∆1: \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) ≥ ∫ T 0 [ 2< { ϕ1χ dc dt } + 0 ] dt = 2< { ϕ1ϕ2c(q)− ϕ1ξ2c(p) } . � Proposition 11. Let us suppose that p � q and that ξ, ϕ have the same latitude, with ξ1 = ϕ1 ∈ R∗ and ξ2 = \|ϕ2\| eiθξ , ϕ2 = \|ϕ2\| eiθϕ, with 0 < \|θϕ − θξ\| ≤ π. If ωp,ξ � ωq,ϕ, then there exists a future directed timelike curve γ going from p to q such that l(γ) ≥ \|θϕ−θξ\|\|d1−d2\| . Proof. At first, we shall assume that \|θϕ − θξ\| < π. Let us note, that we can exclude the cases where ϕ1 = 0 or ϕ2 = 0 since they correspond to the poles of the sphere S2 where no movement is possible within the internal space, so every future directed timelike curve would be convenient. We will prove this Proposition by contradiction. Let us suppose that for every future directed timelike curve γ such that γ(0) = p and γ(T ) = q we have l(γ) < \|θϕ−θξ\| d with d = \|d1 − d2\|. In the same way as in the proof of Proposition 10, we define λ1(t) = 1+v(t) 2 and λ2(t) = 1−v(t) 2 with v(t) = γ̇1(t) γ̇0(t) for t ∈ [0, T ]. Exploring the Causal Structures of Almost Commutative Geometries 15 By Proposition 6, in order to demonstrate that ωp,ξ � ωq,ϕ is false we only need to find one causal element a = ( a −c −c∗ b ) ∈ C such that: \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) < 2< { ϕ1ϕ2c(q)− ϕ1ξ2c(p) } . (10) We shall define such an element along the curve γ and consider its implicit extension to the whole space R1,1. We first define the off-diagonal terms as: c(γ(t)) = − csc(dlγ(t) + ε)ei(sgn(θϕ−θξ)ε−θξ), where lγ(t) = ∫ t 0 √ −gγ(s)(γ̇(s), γ̇(s))ds and ε > 0 is chosen such that \|θϕ − θξ\| + ε < π. Since dlγ(T ) < \|θϕ − θξ\|, this implies that c(γ(t)) is well defined ∀ t ∈ [0, T ]. The diagonal terms a and b are defined up to a constant along the curve γ as follows: a,0(γ(t)) = d 2 √ λ1(t)λ2(t) \|ϕ2\| \|ϕ1\| csc2(dlγ(t) + ε), a,1(γ(t)) = −v(t)d 2 √ λ1(t)λ2(t) \|ϕ2\| \|ϕ1\| csc2(dlγ(t) + ε), b,0(γ(t)) = d 2 √ λ1(t)λ2(t) \|ϕ1\| \|ϕ2\| csc2(dlγ(t) + ε), b,1(γ(t)) = −v(t)d 2 √ λ1(t)λ2(t) \|ϕ1\| \|ϕ2\| csc2(dlγ(t) + ε). (11) The proof of a being positive semi-definite along the curve γ is a straightforward algebraic computation and we shift to the Appendix. Next, we extend a, b and c to the whole R1,1 in such a way that the matrix a remains positive semi-definite at each point of the Minkowski space- time. Such an extension is always possible since a, b and c were only defined on a compact set, so c(γ) can be extended to c ∈ S(R1,1) and the extension of a and b can be constructed using e.g. the inequalities in Lemma 1. Since such an extension has no impact on the following arguments, we do not need to specify it explicitly. By making use of the proof of Proposition 10 we obtain the following equality: \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) = ∫ T 0 γ̇0 [ \|ϕ1\|2 λ1(a,0 + a,1) + \|ϕ1\|2 λ2(a,0 − a,1) + \|ϕ2\|2 λ1(b,0 + b,1) + \|ϕ2\|2 λ2(b,0 − b,1) ] dt = ∫ T 0 γ̇0 [ 4 √ λ1(t)λ2(t)d \|ϕ1ϕ2\| csc2(dlγ(t) + ε) ] dt = 2 \|ϕ1ϕ2\| ∫ T 0 d2γ̇0 √ λ1(t)λ2(t) csc2(dlγ(t) + ε)dt. Since 2γ̇0 √ λ1(t)λ2(t) = √ γ̇2 0(t)− γ̇2 1(t) = dlγ(t) dt , we can compute the integral explicitly: \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) = 2 \|ϕ1ϕ2\| (− cot(dlγ(T ) + ε) + cot(ε)) . (12) Now we can also make explicit the r.h.s. of (10): 2< { ϕ1ϕ2c(q)− ϕ1ξ2c(p) } = 2 \|ϕ1ϕ2\| < { eiθϕc(γ(T ))− eiθξc(γ(0)) } = 2 \|ϕ1ϕ2\| ( −< { ei(θϕ−θξ+sgn(θϕ−θξ)ε) csc(dlγ(T ) + ε) } + < { ei sgn(θϕ−θξ)ε csc(ε) }) 16 N. Franco and M. Eckstein = 2 \|ϕ1ϕ2\| ( − cos(θϕ − θξ + sgn(θϕ − θξ)ε) sin(dlγ(T ) + ε) + cos(sgn(θϕ − θξ)ε) sin(ε) ) = 2 \|ϕ1ϕ2\| ( − cos(\|θϕ − θξ\|+ ε) sin(dlγ(T ) + ε) + cot(ε) ) . (13) Since the cosine function is strictly decreasing on ]0, π[ and since from our assumption we have 0 < dlγ(T ) + ε < \|θϕ − θξ\|+ ε < π, we get the following strict inequality: cos(dlγ(T ) + ε) > cos(\|θϕ − θξ\|+ ε) =⇒ cot(dlγ(T ) + ε) > cos(\|θϕ − θξ\|+ ε) sin(dlγ(T ) + ε) =⇒ 2 \|ϕ1ϕ2\| (− cot(dlγ(T ) + ε) + cot(ε)) < 2 \|ϕ1ϕ2\| ( − cos(\|θϕ − θξ\|+ ε) sin(dlγ(T ) + ε) + cot(ε) ) =⇒ \|ϕ1\|2 (a(q)− a(p)) + \|ϕ2\|2 (b(q)− b(p)) < 2< { ϕ1ϕ2c(q)− ϕ1ξ2c(p) } by using the equalities (12) and (13). Hence (10) is true and the proof is complete for \|θϕ−θξ\|<π. The case θϕ = θξ + π can be dealt with by using the transitivity of the causal order. Once more we suppose by contradiction that dlγ(T ) = π − δ < π. We can construct an extension of the curve γ to [0, T ′] with T ′ > T and q � q′ = γ(T ′) such that dlγ(T ′) = π − 2 3δ. From Proposition 10, the causal relation ωq,ϕ � ωq′,ϕ′ for θϕ′ = θϕ− 1 3 δ is allowed since the length of γ between q and q′ is 1 3δ. Since ∣∣θϕ′ − θξ∣∣ = π − 1 3δ < π, we can apply the previous result to get the necessary condition dlγ(T ′) ≥ π − 1 3δ which contradicts dlγ(T ′) = π − 2 3δ. Hence dlγ(T ) ≥ π. � The Propositions 10 and 11 complete the proof of Theorem 3, except for the cases where the curve γ is null (or partially null). Those particular cases can be treated by the same kind of arguments as above using transitivity of the causal order and continuity. Let us stress the importance of using the maximal cone of causal elements. Indeed, following the results in [15], one can define a partial order structure on the space of states using only a subcone of C, as long as the condition spanC(C) = Ã is still respected for some suitable unitisation Ã. For example, in the model considered in this section, if we use the subcone defined by the causal elements described in Lemma 1, the global causal structure would be similar, but with a different constraint on l(γ), allowing for more causal relations. Indeed, one can check that the causal function a used in the proof of Proposition 11 does not belong to such a subcone. Since in the commutative case the usual causal structure is recovered by using the maximal cone of causal functions [15], the maximal cone should also be favoured in the noncommutative case. 3.2 Causal structure for a class of mixed states As mentioned in Section 2, the space of general states on A may contain states that mix the space-time with the internal degrees of freedom. The physical interpretation of such states is unclear and we shall not discuss it here. On the other hand, there exists a subclass of S(A) of particular interest: P (AM) × S(AF ) = M(ÃM) × S(AF ), which we shall denote by N (Ã). In [17] it has been shown that the Riemannian distance in the space of states on P (M2(C)) ∼= S2 is equal to the Euclidean distance between the corresponding points of S2, whenever the two states in consideration have the same latitude. This means that the shortest path between two pure states leads necessarily through mixed states. Surprisingly enough, this is no longer the case in our model. Note that the path of states [0, T ] 3 t 7→ χ(t) used in the proof of Proposition 10 leads entirely through the space of pure states. The purpose of this section is to show that indeed there is no “short-cut” through the space of internal mixed states. Exploring the Causal Structures of Almost Commutative Geometries 17 Let us recall that a mixed state can always be written as a convex combination of two pure states. Since P (AF ) ∼= S2, we have S(AF ) ∼= B2, i.e. a solid 2-dimensional ball of radius one. Again, let us stress that the isomorphism S(AF ) ∼= B2 is only set-theoretic, the metric aspect of B2 is ruled by the noncommutative geometry of M2(C). It is convenient to express the mixed states in terms of density matrices rather than convex combination. Any mixed state in the subclass N (Ã) can be written as ωp,ρ with ωp,ρ(a) = Tr ρa(p). The density matrices ρ need to satisfy Tr ρ = 1 and are conveniently parametrised [2] by ρ = 1 2 (1 + ~r · ~σ) , with ~r ∈ B2 ⊂ R3 and ~σ = (σx, σy, σz), where σi are Pauli matrices. We shall first write down an analogue of the Proposition 6. Proposition 12. Let us take two mixed states ωp,ρ, ωq,σ ∈ N (Ã), where ρ and σ are determined by the vectors ~r,~s ∈ B2 respectively. Then ωp,ρ � ωq,σ if and only if ∀a = ( a −c −c∗ b ) ∈ C, (1 + sz)a(q)− (1 + rz)a(p) + (1− sz)b(q)− (1− rz)b(p) ≥ 2< { sxc(q)− rxc(p) } − 2= { syc(q)− ryc(p) } . Proof. The Definition 3 implies that ωp,ρ � ωq,σ if and only if ∀a ∈ C, ωp,ρ(a) ≤ ωq,σ(a) ⇐⇒ Tr (ρa(p)) ≤ Tr (σa(q)) ⇐⇒ 1 2 Tr ( 1 + rz rx − iry rx + iry 1− rz )( a(p) −c(p) −c(p)∗ b(p) ) − 1 2 Tr ( 1 + sz sx − isy sx + isy 1− sz )( a(q) −c(q) −c(q)∗ b(q) ) ≤ 0. � The Propositions 8, 7 and 9 are easily generalised using the same elements of C as in the proofs for pure states: Proposition 13. If the finite-part Dirac operator is degenerate (i.e. d1 = d2), then two mixed states ωp,ρ, ωq,σ ∈ N (Ã) are causally related with ωp,ρ � ωq,σ only if p � q in R1,1 and ρ = σ. Proposition 14. Two states ωp,ρ, ωq,σ ∈ N (Ã), with ρ and σ associated with the vectors ~r,~s ∈ B2 respectively, are causally related with ωp,ρ � ωq,σ only if p � q in R1,1 and rz = sz. The above Proposition encourages us to restrict the further investigation to the mixed states which have the same latitude and change the parametrisation of the vectors ~r and ~s as follows: rz = sz = z, rx ± iry = re±iθr , sx ± isy = se±iθs with r, s ∈ [ 0, √ 1− z2 ] . In this case, the condition for ωp,ρ � ωq,σ simplifies to (1 + z) (a(q)− a(p)) + (1− z) (b(q)− b(p)) ≥ 2< { c(q)seiθs − c(p)reiθr } . (14) The main theorem of this section is the following generalisation of the Theorem 3. Theorem 4. Two states ωp,ρ, ωq,σ ∈ N (Ã), with ρ and σ associated with the vectors ~r,~s ∈ B2 respectively, are causally related with ωp,ρ � ωq,σ if and only if the following conditions are respected: • p � q in R1,1; • rz = sz = z; 18 N. Franco and M. Eckstein • l(γ) ≥ supθ∈R ∣∣∣∣arccos ( s√ 1−z2 cos(θs+θ) ) −arccos ( r√ 1−z2 cos(θr+θ) )∣∣∣∣ \|d1−d2\| , where l(γ) represents the length of a causal curve γ going from p to q on R1,1. Proof. To prove the necessary condition for the causal relation to hold we make use of Propo- sition 14. Hence, it is sufficient to show that p � q, rz = sz and ωp,ρ � ωq,σ imply the existence of a causal curve linking p and q with l(γ) ≥ ∣∣θ̃s − θ̃r∣∣ d , where θ̃s\|r = arccos ( s\|r√ 1− z2 cos(θs\|r + θ) ) , d = \|d1 − d2\| and θ ∈ [0, 2π[ is chosen such that ∣∣θ̃s − θ̃r∣∣ is maximal. The argument by contradiction used in the proof of Proposition 11 can be applied, so we repeat the same reasoning and stress only the differences. Since the argument of arccos in the definition of θ̃s\|r is in the interval [−1, 1], we have θ̃s, θ̃r ∈ [0, π]. We shall consider θ̃s, θ̃r ∈]0, π[ and the limiting cases can be dealt with by the argument of transitivity of the causal order as in Proposition 11. We assume that l(γ) < \|θ̃s−θ̃r\| d for every future directed timelike curve γ with γ(0) = p and γ(T ) = q and we show that there exists a causal element a = ( a −c −c∗ b ) ∈ C such that: (1 + z) (a(q)− a(p)) + (1− z) (b(q)− b(p)) < 2< { c(q)seiθs − c(p)reiθr } . The off-diagonal terms are defined as: c(γ(t)) = − csc(dlγ(t) + ε)eiθc , where { ε = θ̃r, θc = θ if θ̃s − θ̃r > 0, ε = −θ̃r + π, θc = θ + π if θ̃s − θ̃r < 0. The diagonal terms are defined as in the formulas (11) using \|ϕ1\| = √ 1 + z and \|ϕ2\| = √ 1− z. Then the equality corresponding to (12) is (1 + z) (a(q)− a(p)) + (1− z) (b(q)− b(p)) = 2 √ 1− z2 (− cot(dlγ(T ) + ε) + cot(ε)) (15) and since (s\|r) cos(θs\|r + θc) = sgn(θ̃s − θ̃r) √ 1− z2 cos ( ± θ̃s\|r ) we have 2< { c(q)seiθs − c(p)reiθr } = 2 ( − s cos(θs + θc) sin(dlγ(T ) + ε) + r cos(θr + θc) sin(ε) ) = 2 √ 1− z2 ( −sgn(θ̃s − θ̃r) cos(±θ̃s) sin(dlγ(T ) + ε) + cot(ε) ) . (16) By comparing (15) and (16) it remains to prove that cos(dlγ(T ) + ε) > sgn(θ̃s − θ̃r) cos(±θ̃s). Since cosine is decreasing on ]0, π[ we have: • If θ̃s − θ̃r > 0, we get cos(dlγ(T ) + θ̃r) > cos(θ̃s) which is true since 0 < dlγ(T ) + θ̃r <∣∣θ̃s − θ̃r∣∣+ θ̃r = θ̃s < π. • If θ̃s − θ̃r < 0, we get cos(dlγ(T ) − θ̃r + π) > cos(−θ̃s + π) which is true since 0 < dlγ(T )− θ̃r + π < ∣∣θ̃s − θ̃r∣∣− θ̃r + π = −θ̃s + π < π. So the necessary condition is completed. In order to prove the sufficient condition, we use the proof of Proposition 10 with the following changes: Exploring the Causal Structures of Almost Commutative Geometries 19 • l(γ) = \|θ̃s−θ̃r\| d ; • ξ1 = ϕ1 = √ 1 + z; • χ(t) = √ 1− zei(θ̃r+sgn(θ̃s−θ̃r)dlγ(t)), which implies \|χ(t)\| = √ 1− z ∀ t ∈ [0, T ], χ(0) =√ 1− zeiθ̃r and χ(T ) = √ 1− zeiθ̃s . • For an arbitrary a = ( a −c −c∗ b ) ∈ C we consider instead an associate causal element ã =( a −c̃ −c̃∗ b ) ∈ C with c̃ = ce−iθ. One can check using Proposition 4(c) with α̃3 = α3e −iθ and α̃4 = α4e −iθ that a ∈ C ⇔ ã ∈ C. By repeating the proof of Proposition 10 until (9), we get: (1 + z) (a(q)− a(p)) + (1− z) (b(q)− b(p)) ≥ 2 √ 1 + z< { χ(T )c̃(q)− χ(0)c̃(p) } . From the formula (14), it remains to show that the following inequality is true for at least one θ ∈ [0, 2π): 2 √ 1 + z< { χ(T )c̃(q)− χ(0)c̃(p) } ≥ 2< { c(q)seiθs − c(p)reiθr } . (17) Let us set c(γ(t)) = \|c(γ(t))\| eiθc(t). At first let us assume that θc(0) = θc(T ). Then we can choose θ = θc(0) = θc(T ) which implies c̃(p) = \|c(p)\| ∈ R and c̃(q) = \|c(q)\| ∈ R. Hence, 2 √ 1 + z< { χ(T )c̃(q)− χ(0)c̃(p) } = 2 \|c(q)\| √ 1− z2 cos(θ̃s)− 2 \|c(p)\| √ 1− z2 cos(θ̃r) = 2 \|c(q)\| s cos(θs + θ)− 2 \|c(p)\| r cos(θr + θ) = 2< { \|c(q)\| eiθseiθs − \|c(p)\| eiθreiθr } = 2< { c(q)seiθs − c(p)reiθr } . Let us now assume that θc(0) 6= θc(T ). The l.h.s. of (17) can be expanded as: 2 √ 1 + z< { χ(T )c̃(q)− χ(0)c̃(p) } = 2 √ 1− z2 \|c(q)\| cos(θc(T )− θ + θ̃s)− 2 √ 1− z2 \|c(p)\| cos(θc(0)− θ + θ̃r) = 2 \|c(q)\| cos(θc(T )− θ)s cos(θs + θ)− 2 √ 1− z2 \|c(q)\| sin(θc(T )− θ) sin(θ̃s) − 2 \|c(p)\| cos(θc(0)− θ)r cos(θr + θ) + 2 √ 1− z2 \|c(p)\| sin(θc(0)− θ) sin(θ̃r) (18) and the r.h.s. as: 2< { c(q)seiθs − c(p)reiθr } = 2< { c(q)e−iθsei(θs+θ) − c(p)e−iθrei(θr+θ) } = 2 \|c(q)\| cos(θc(T )− θ)s cos(θs + θ)− 2 \|c(q)\| sin(θc(T )− θ)s sin(θs + θ) − 2 \|c(p)\| cos(θc(0)− θ)r cos(θr + θ) + 2 \|c(p)\| sin(θc(0)− θ)r sin(θr + θ). (19) If we compare the two expressions (18) and (19), in order to respect the inequality (17), we need to prove that there exists at least one θ ∈ R such that: −2 √ 1− z2 \|c(q)\| sin(θc(T )− θ) sin(θ̃s) + 2 √ 1− z2 \|c(p)\| sin(θc(0)− θ) sin(θ̃r) ≥ −2 \|c(q)\| sin(θc(T )− θ)s sin(θs + θ) + 2 \|c(p)\| sin(θc(0)− θ)r sin(θr + θ) ⇐⇒ \|c(p)\| sin(θc(0)− θ) [√ 1− z2 sin(θ̃r)− r sin(θr + θ) ] ≥ \|c(q)\| sin(θc(T )− θ) [√ 1− z2 sin(θ̃s)− s sin(θs + θ) ] . (20) By using the formula sin(arccos(x)) = √ 1− x2 and s, r ≤ √ 1− z2, we have that:√ 1− z2 sin(θ̃s\|r) = √ (1− z2)− (s\|r)2 cos(θs\|r + θ) ≥ √ (s\|r)2 − (s\|r)2 cos(θs\|r + θ) = (s\|r) ∣∣sin(θs\|r + θ) ∣∣ . 20 N. Franco and M. Eckstein So the two brackets inside (20) are always non-negative, and the inequality is manifestly re- spected when sin(θc(0)− θ) > 0 and sin(θc(T )− θ) < 0. Since θc(0) 6= θc(T ), there always exists one θ ∈ R making the first expression positive and the second negative. Therefore, the proof is complete. � The constraint for mixed states can be given a geometrical interpretation. The difference between the arccosine values in ∣∣∣θ̃s − θ̃r∣∣∣ is the length of a projection of the segment ~s−~r on the parallel of radius √ 1− z2. The supremum over the angle θ means that the projection should always be the maximal one. This constraint is thus compatible with the formula (4) for the pure states derived in Section 3.1. It also leads to the conclusion that paths going trough the space of mixed states N (Ã) – in particular the straight line path corresponding to the realisation of the Riemannian distance in the internal space [17] – do not shorten the amount of proper time needed for the path to be causal. Hence, a unitary evolution (i.e. the one involving only pure states) in the internal space is, in a sense, favoured by the system. 4 Conclusions In this paper, we have provided a definition of an almost commutative Lorentzian spectral triple which describes a product of a globally hyperbolic Lorentzian manifold with a finite noncom- mutative space. Following the results in [15], we equipped the space of states of this spectral triple with a partial order relation that generalises the causal structure of Lorentzian manifolds. After some general results we explored in details the causal structure of a simple almost com- mutative space-time based on the algebra S(R1,1) ⊗M2(C). We have shown that the space of pure states is isomorphic to the set R1,1×S2, with the geometry of the S2 component governed by a noncommutative matrix algebra. Our main result – Theorem 3 – shows that the causal structure of this simple almost commutative space-time is highly non-trivial. Its main features are the following: 1. No violation of the classical causality relations in the 2-dimensional Minkowski space-time component occurs. 2. The Riemannian noncommutative geometry of M2(C) represented on C2 is respected, i.e. the parallels of S2 are causally disconnected. 3. The admissible motions in the Minkowski space-time and the internal space S2 are intrin- sically related by an explicit formula (4). 4. The path realising the shortest Riemannian distance between two pure states, which leads through the space of mixed states, is equivalent, from the causal point of view, to a path enclosed entirely in the space of pure states. Naively, one could think that in the setting of almost commutative geometries the motion in the internal space should be completely independent of that in the base space-time. On the contrary, the simple model we have chosen to investigate has revealed a lot of interesting and unexpected features of the causal relations in the space of states. This shows that the causal structure of general almost commutative Lorentzian spectral triples differs significantly from the one of Lorentzian manifolds and its exploration may lead to surprising conclusions. The illustrative example presented in Section 3.1 can be generalised to curved manifolds. One can use Proposition 3 to extend the results of Theorem 3 to a general globally hyperbolic Lorentzian metric on R2, since such a metric is always locally conformally flat [21]. On the other hand, this feature is no longer true when the dimension of space-time is larger then 2. Exploring space-times of larger dimensions is also technically more difficult and does not necessarily lead to a linear characterisation of the causal elements as given in Proposition 4. Exploring the Causal Structures of Almost Commutative Geometries 21 In addition to the possible generalisations of the space-time component it would be desirable to investigate the causal structure of almost commutative geometries with other matrix algebras and direct sums of the latter. Here the main problem, apart from the increasing level of com- plexity of the formulas for higher dimensional matrices, is that the Riemannian noncommutative geometry of the associated space of states has not been studied in details. Finally, let us comment on possible physical applications of the results contained in this paper. As stressed in the Introduction, almost commutative geometries provide a firm mathematical framework for the study of fundamental interactions. Although the physical interpretation of the space of states of such geometries (especially the mixed states) is not clear, our investigations show that one can consider the motion in internal space in a meaningful way. What is more, the possibility of the internal movement turns out to be dependent on the speed assumed in the space-time component. It suggests that the “speed of light constraint” in this model should be understood in terms of resultant space-time and internal velocities, with a proportionality parameter determined by the finite-part Dirac operator DF . Certainly, it would be interesting to apply the presented results in some concrete physical model, with the standard model of fundamental interactions as an ultimate goal. Appendix In this Appendix, we show that the function a defined in the proof of Proposition 11 (and also used in the proof of Theorem 4) is a causal function along the curve γ, i.e. that the matrix given in Proposition 4(b) is positive semi-definite. Let us compute the entries of that matrix. We have: • a,0 + a,1 = d √ λ2 λ1 \|ϕ2\| \|ϕ1\| csc2 Θ, • a,0 − a,1 = d √ λ1 λ2 \|ϕ2\| \|ϕ1\| csc2 Θ, • b,0 + b,1 = d √ λ2 λ1 \|ϕ1\| \|ϕ2\| csc2 Θ, • b,0 − b,1 = d √ λ1 λ2 \|ϕ1\| \|ϕ2\| csc2 Θ, • c = − csc Θeiθc , with Θ(t) = dlγ(t) + ε and θc = sgn(θϕ − θξ)ε − θξ and where the parameters t and γ(t) have been omitted. The partial derivatives of c are defined in such a way that its directional derivative is maximal along γ and corresponds to d dt (c ◦ γ) = −ddlγ dt cos Θ csc2 Θeiθc = −d2γ̇0 √ λ1λ2 cos Θ csc2 Θeiθc . This can be done by setting: • c,0 + c,1 = −d √ λ2 λ1 cos Θ csc2 Θeiθc , • c,0 − c,1 = −d √ λ1 λ2 cos Θ csc2 Θeiθc , since d dt(c ◦ γ) = γ̇0 (λ1(c,0 + c,1) + λ2(c,0 − c,1)) . 22 N. Franco and M. Eckstein So we must prove that the following matrix is positive semi-definite: A = d csc2 Θ  √ λ2 λ1 \|ϕ2\| \|ϕ1\| 0 √ λ2 λ1 cos Θeiθc ± sin Θeiθc 0 √ λ1 λ2 \|ϕ2\| \|ϕ1\| ∓ sin Θeiθc √ λ1 λ2 cos Θeiθc√ λ2 λ1 cos Θe−iθc ∓ sin Θe−iθc √ λ2 λ1 \|ϕ1\| \|ϕ2\| 0 ± sin Θe−iθc √ λ1 λ2 cos Θe−iθc 0 √ λ1 λ2 \|ϕ1\| \|ϕ2\|  , where ± corresponds to sgn(d1 − d2). Since the eigenvalues of A are the roots of the characteristic polynomial: det(A− λ1) = λ4 − c1λ 3 + c2λ 2 − c3λ+ c4, from Vieta’s formulas it is sufficient to check that ck ≥ 0 for k = 1, . . . , 4. We can notice that this matrix is singular. Indeed, by multiplying the first line by \|ϕ1\| \|ϕ2\| cos Θe−iθc and the second line by ∓ √ λ2 λ1 \|ϕ1\| \|ϕ2\| sin Θe−iθc and adding them, we get exactly the third line. So c4 = det(A) = 0. The other coefficients can be calculated by algebraic computations from Newton’s identities using the following traces: trA = d csc2 Θ 1√ λ1 √ λ2 \|ϕ1\| \|ϕ2\| , trA2 = d2 csc4 Θ −2 \|ϕ1\|2 \|ϕ2\|2 (λ2 − λ1)2 sin2 Θ− 2λ1λ2 + 1(√ λ1 √ λ2 \|ϕ1\| \|ϕ2\| )2 , trA3 = d3 csc6 Θ −3 \|ϕ1\|2 \|ϕ2\|2 (λ2 − λ1)2 sin2 Θ− 3λ1λ2 + 1(√ λ1 √ λ2 \|ϕ1\| \|ϕ2\| )3 , c1 = trA = d csc2 Θ 1√ λ1 √ λ2 \|ϕ1\| \|ϕ2\| ≥ 0, c2 = 1 2 ( (trA)2 − trA2 ) = d2 csc4 Θ \|ϕ1\|2 \|ϕ2\|2 (λ2 − λ1)2 sin2 Θ + λ1λ2(√ λ1 √ λ2 \|ϕ1\| \|ϕ2\| )2 ≥ 0, c3 = 1 6 ( (trA)3 − 3 trA2 trA+ 2 trA3 ) = 0. So all of the coefficients of the characteristic polynomial are non-negative and the matrix is positive semi-definite. Acknowledgements This work was supported by a grant from the John Templeton Foundation. References [1] Barrett J.W., Lorentzian version of the noncommutative geometry of the standard model of particle physics, J. Math. Phys. 48 (2007), 012303, 7 pages, hep-th/0608221. 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Exploring the Causal Structures of Almost Commutative Geometries

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