Space and Time Dimensions of Algebras with Applications To

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Sceaux, 0 J 2 = b χaχ ǫ , matter. χ H = . (i.e. ± = 1. − H χ η a 2 hc a be can which , orentzian 6 rafunda- a or [6] o example, For . 8Otbr2017 October 18 a )defining 1) = on H J seven is such tions between χ, η and J defined by three signs V which satisfies (v,u) = (u, v). It is indefinite be- (ǫ′′,κ,κ′′) introduced in Eqs. (3) to (5). cause we do not assume that (v, v) > 0 if v 6= 0. It In a Clifford algebra Cℓ(p, q) such that p+q is even, was shown [8–10] that S can be equipped with two ′′ ′′ the four signs (ǫ,ǫ ,κ,κ ) determine a pair of space indefinite inner products (·, ·)+ and (·, ·)− such that and time dimensions (s,t) modulo 8 in a unique way. We propose to assign the same dimensions (s,t) to (γj φ, ψ)± = ±(φ, γj ψ)±, (1) any algebra satisfying the same relations between χ, η and J. This is similar to the way Atiyah related for every γj, and every elements φ and ψ of S. These the KO-dimension to p − q mod 8 in Clifford al- two inner products are unique up to multiplication by a real factor and they are invariant under the ac- gebras [3]. These two dimensions solve the question + whether indefinite spectral triples have a notion cor- tion of Spin(p, q) , the connected component of the responding to the KO-dimension [7]. identity in Spin(p, q). Moreover, S can be equipped Such an assignment is meaningful because it is with two charge conjugations J±, which are anti- compatible with the tensor product: the dimensions linear maps such that J±γj = ±γj J±. corresponding to the signs of the (graded) tensor To give a concrete representation of the indefinite inner products and the charge conjugations, we product A1⊗ˆ A2 of two such algebras are the sum of equip the complex vector space S with its standard the dimensions of A1 and A2 mod 8. When we apply this to a spectral triple of noncom- (positive definite) scalar product h·, ·i. By using re- mutative geometry, χ is the usual chirality operator, sults scattered in the literature [8,11–15], we can state the following. In all representations used in η is a fundamental symmetry defining a Krein-space † structure and J is the usual charge conjugation. Our practice, the gamma matrices satisfy γj = ǫj γj = −1 approach allows us to assign a space-time dimen- γj , where † denotes the adjoint with respect to the sion to the finite algebra of the almost-commutative scalar product, and γj = ζj γj , where the overline spectral triple of models of particles. denotes complex conjugation and ζj is +1 (γj is real) The paper starts with a description of χ, η and J or −1 (γj is imaginary). Shirokov showed [15] that in a Clifford algebra, which sets up the correspon- these representations can be classified by the num- dence between commutation relations and space- ber of their symmetric gamma matrices (i.e. such T time dimensions. Then, this correspondence is that γj = γj ) modulo 4, which is equal to ℓ + ζ shown to hold for more general algebras by proving mod 4, where ζ can take the value 0 or 1. Both cases that it is compatible with the graded tensor product are useful in practice. For example, the most com- of algebras. In section 4, we introduce Krein spaces, mon representations of Cℓ(1, 3) are the Dirac, Ma- which are the natural generalizations of Hilbert jorana and chiral gamma matrices [16]. The number spaces associated to spinors on pseudo-Riemannian of symmetric matrices of these representations is 2,3 manifolds. Section 5 defines the corresponding gen- and 2, corresponding to ζ =0, 1 and 0. eralized spectral triples, that we call indefinite Then, it can be shown that the chirality matrix (p−q)/2 spectral triples. This framework is then applied to χ = i γ1 ...γ2ℓ anticommutes with all γj and † 2 ζ define the spectral triple of Lorentzian quantum satisfies χ = χ, χ = 1 and χ = (−1) χ. To con- electrodynamics (QED) and its Lagrangian. struct the indefinite inner products we define linear operators η± (called fundamental symmetries) such † 2 that (φ, ψ)± = hφ, η±ψi, η± = η± and η± = 1. From 2. Automorphisms of Clifford algebras † Eq. (1): η±γj η± = ±γj. These fundamental symmetries are unique up to a sign and are built as follows. We investigate the commutation relations of three Let Ms be the product of all self-adjoint gamma ma- operators in Clifford algebras over vector spaces of trices and Mas the product of the anti-self-adjoint even dimension 2ℓ. They are simple algebras whose ones, then ℓ irreducible representation is S ≃ C2 . q(q−1)/2 q The Clifford algebra Cℓ(p, q) with p + q = 2ℓ is η+ = i χ Mas, the algebra generated by 2ℓ matrices γ over S such p(p+1)/2 p j η− = i χ Ms. that γiγj + γj γi =2ǫj δij , where p coefficients ǫj are equal to +1 and q are equal to -1. To construct the charge conjugations J±, let ni An indefinite inner product on a complex vector be the number of imaginary gamma matrices and N space V is a non-degenerate sesquilinear form on their product. Then

2 (1−ζ)(p−q)/2 ni J+ = i χ NK, m, n 0 2 4 6

(1−ζ)(p−q)/2+ζ ni+1 J− = i χ NK, κ,ǫ 1 -1 -1 1 ′′ ′′ where K stands for complex conjugation. The rela- κ ,ǫ 1 -1 1 -1 Table 1 tion ζ = ni + (p − q)/2 mod 2 holds. The “charge ′′ ′′ † Signs ǫ,κ and ǫ ,κ in terms of the KO-dimension n and m. conjugations” J± satisfy J±γj = ±γj J±, J±J± =1 and J± commutes with complex conjugation. These Convention m n J η conditions determine J and J uniquely up to a + − − sign. West-coast p + q p q J− η+ In every Clifford algebra Cℓ(p, q) with p + q even, East-coast p + q q − p J+ η− we can choose J = J+ or J = J− and η = η+ or North-coast −p − q p − q J− η− η = η−. This makes four possible conventions for South-coast −p − q q − p J+ η+ which we have: Table 2 Value of the KO-dimension n and of m and operators as a 2 J = ǫ, (2) function of convention. The dimensions m and n are to be Jχ = ǫ′′χJ, (3) understood modulo 8. n Jη = ǫκηJ, (4) ❅ 0 2 4 6 m ❅ ′′ ′′ ηχ = ǫ κ χη, (5) 0 (0,0) (4,4) (1,7) (5,3) (2,6) (6,2) (3,5) (7,1) where ǫ = (−1)n(n+2)/8, ǫ′′ = (−1)n/2, κ = 2 (1,1) (5,5) (2,0) (6,4) (3,7) (7,3) (0,2) (4,6) (−1)m(m+2)/8 and κ′′ = (−1)m/2. Note that ǫ and 4 (2,2) (6,6) (3,1) (7,5) (4,0) (0,4) (1,3) (5,7) ′′ ′′ ǫ are the same functions of n as κ and κ are of 6 (3,3) (7,7) (4,2) (0,6) (5,1) (1,5) (6,0) (2,4) ′′ m. These signs were defined so that ǫ and ǫ agree Table 3 with Connes’ KO-dimension tables [5,17]. Related (t, s) where t is the number of time dimensions and s the tables can be found in the literature [18–20]. number of space dimensions s as a function of m and n. The The values of the KO dimension n and of the new general solution is (t+8j,s+8k), where j and k are integers. dimension m in terms of the conventions are given The relation between (p,q) and (t, s) is: p = t, q = s (West coast), p = s, q = t (East coast), p = −s, q = −t (North in Table 2. Note that m and n are defined modulo 8. coast) and p = −t, q = −s (South coast). In physics the Dirac operator is written D = µ iγ ∇µ in the so-called West-coast convention [21] In Lorentzian spacetime we need plane waves to µ and D = γ ∇µ in the East-coast one. Charge- describe scattering experiments. Therefore, only the conjugation symmetry requires JD = DJ and the West and East coast conventions are allowed. It is reality of the fermionic Lagrangian implies that D interesting to note that, for both conventions, the is self-adjoint with respect to the indefinite inner dimensions n and m are the same. Indeed, in the product. Thus, the West-coast convention corre- West coast convention (p, q) = (1, 3) and n = p − sponds to J = J− and η = η+ while the East-coast q = 6 mod 8, while for the East coast convention one to J = J+ and η = η− [22]. This is related to the (p, q) = (3, 1) and n = q − p = 6 mod 8. In other signature of the metric. Indeed, in Minkowski space words, the KO dimension n is the number of time time we want plane wave solutions of the (massive) dimensions t minus the number of space dimensions µ Dirac equation: ψ(x)= ueikµx . Compatibility with s while m is their sum t+s whatever the convention. 2 2 the dispersion relation k = me, where me is the It is therefore tempting to intepret the dimension fermion mass, implies the metric (+, −, −, −) for m and n in terms of time and space dimensions by the West-coast convention and (−, +, +, +) for the solving n = t−s mod 8 and m = t+s mod 8. This East-coast one [22]. associates a space dimension and a time dimension In Euclidean space, we can be interested in to any Clifford algebra. µ the real solutions ψ(x) = uekµx . The metric By inverting the relation between (s,t) and (−, −, −, −) corresponding to J = J− and η = η− is (m,n), we can associate two pairs of space and time often used and we call it the North-coast convention dimensions (j, k) modulo 8 to every pair (m,n) (see because Euclid lived on the coast of North Africa. Table 3). Indeed, if (j, k) is a solution of j − k = n The remaining possibility is J = J+ and η = η+ mod 8 and j + k = m mod 8, then (j +4, k + 4) that we call the South-coast convention. is also a solution. This corresponds to the Clifford

3 algebra isomorphism Cℓ(s,t + 8) ≃ Cℓ(s +8,t) ≃ χ = χ1⊗ˆ χ2, (7) Cℓ(s +4,t + 4) [23]. The relation between (s,t) and J = J χ|J2|⊗ˆ J χ|J1|, (8) (p, q) for the four conventions is given in the caption 1 1 2 2 |η1||η2| |η2| |η1| of Table 3. η = i η1χ1 ⊗ˆ η2χ2 . (9) To derive these equations, we first define an indefinite inner product (·, ·) on H in terms of the indefi- 3. Generalization nite inner products (·, ·)1 and (·, ·)2 on H1 and H2. The adjoint of an operator T with respect to an in- We generalize the previous results by defining a definite inner product is denoted by T ×. mod-8-spacetime representation to be a quadruple We consider the example where H1 and H2 are S = (H,χ,η,J), where H is a complex Hilbert space the spinor spaces of two Clifford algebras. Then, we equipped with two self-adjoint involutions χ and η use Robinson’s theorem [9], which states that there 2 2 (i.e. χ = η = 1) and an anti-unitary operator J is a unique indefinite inner product (up to a real ′′ that satisfy Eqs. (2) to (5) for some signs ǫ, ǫ , κ scalar factor) on the space of spinors such that the ′′ and κ . The KO-dimension n is compatible with γµ matrices generating the Clifford algebra satisfy µ × µ the tensor product, in the sense that the dimension (γ ) = γ . If we impose that η+ was chosen for of the tensor product of two algebras is the sum of both the first and second Clifford algebras,we obtain their dimensions. It is physically important that the the following indefinite inner product on H: same holds for the other dimension m, because the state space of a many-body problem is built from |η1||η2| |η1| (φ1 ⊗ φ2, ψ1 ⊗ ψ2)= i (φ1, ψ1)1(φ2,χ2 ψ2)2. the tensor product of one-particle states. By using the chirality operator χ, we can write Since this definition depends only on |η1|, |η2| and H = H+ ⊕ H−, where χv = ±v for v ∈ H±. An χ2, we can extend it to any mod-8-spacetime repre- element v of H± is said to be homogeneous and its sentation. Formula (9) expresses the compatibility parity is |v| = 0 if v ∈ H+ and |v| = 1 if v ∈ H−. of η with this indefinite inner product. This ensures The parity of a linear or antilinear map T on H is |T | = 0 if χTχ = T and |T | = 1 if χTχ = −T . From hφ1 ⊗ φ2, ψ1 ⊗ ψ2i = hφ1, ψ1i1hφ2, ψ2i2, relations (3) and (5) we see that ǫ′′ = (−1)|J| and κ′′ = (−1)|η|+|J|. and implies the Kasparov identities [25] ˆ The graded tensor product ⊗ of operators is de- ˆ × |T1||T2| × ˆ × |T2||φ1| (T1⊗T2) = (−1) T1 ⊗T2 , fined by (T1⊗ˆ T2)(φ1 ⊗φ2) = (−1) T1φ1 ⊗T2φ2 ˆ † |T1||T2| † ˆ † when φ1 and φ2 are homogeneous. It is the natural (T1⊗T2) = (−1) T1 ⊗T2 , tensor product of Clifford algebra theory thanks to Chevalley’s relation [24]: for the tensor product of two linear operators and

× |η1||η2|+|T1||T2| × × (T1⊗ˆ T2) = (−1) T ⊗ˆ T , ˆ 1 2 Cℓ(p1, q1)⊗Cℓ(p2, q2)= Cℓ(p1 + p2, q1 + q2), (6) † |T1||T2| † † (T1⊗ˆ T2) = (−1) T1 ⊗ˆ T2 , which shows that the graded tensor product is in- for the tensor product of two antilinear operators. deed compatible with space and time dimensions. The tensor product of mod-8-spacetime represen- ˆ The graded tensor product T1⊗T2 is the same op- tations is associative and symmetric [26]. It can be |T2| erator as the non-graded tensor product T1χ1 ⊗T2. checked that χ and η are self-adjoint involutions and For example, the graded Dirac operator of the tensor J is an anti-unitary map, which satisfy Eqs (2) to (5) product: D = D1⊗ˆ 1+1⊗ˆD2 is the same operator for the signs of some dimensions (m,n). Indeed, we as the one given by Connes: D = D1 ⊗ 1+ χ1 ⊗ D2. first observe through an explicit calculation that the Let us consider two mod-8-spacetime representa- signs associated to S1⊗ˆ S2 only depend on the signs |J1||J2| tions S1 = (H1,χ1, η1,J1) and S2 = (H2,χ2, η2,J2) associated to S1 and S2: ǫ = (−1) ǫ1ǫ2 (where with signs determined by (m ,n ) and (m ,n ), re- |J1||J2| ′′ ′′ ′′ ′′ ′′ ′′ ′′ 1 1 2 2 (−1) = (1+ǫ1 +ǫ2 −ǫ1 ǫ2 )/2), ǫ = ǫ1 ǫ2 , κ = spectively. Then, the graded tensor product S = (|η1|+|J1|)(|η2|+|J2|) ′′ ′′ ′′ (−1) κ1κ2 and κ = κ1 κ2 . Then, S1⊗ˆ S2 is the mod-8-spacetime representation de- the additivity of the dimensions in mod-8-spacetime fined by the Hilbert space H = H1 ⊗ H2 and the representations follows from the fact that it holds operators for Clifford algebras (see Eq. (6)).

4 These space and time dimensions can be used to that T (αx + βy)= αT x + βTy) has a Krein-adjoint classify topological insulators and superconductors T × defined by (y,T ×x) = (x,Ty). It is Krein-anti- with symmetries, as well as for the investigation of unitary if, furthermore, T ×T = TT × = 1. PT -symmetric Hamiltonians, because of the pres- Any fundamental symmetry η is Krein self- ence of a Krein-space structure [27–29], to which we adjoint. The relation between the Krein adjoint now turn because it is crucial to generalize spectral T × and the adjoint T † with respect to the scalar triples to pseudo-Riemannian manifolds. product of H is T † = ηT ×η. In physical applications the Krein adjoint is the most natural. For example, the Dirac operator on a 4. Krein spaces pseudo-Riemannian manifold is Krein-self-adjoint. In Gupta-Bleuler quantization the Krein adjoint is We already met Hilbert spaces H, with scalar covariant. In gauge field theory, the BRST charge product denoted h·, ·i, which are equipped with a is Krein self-adjoint. The Hilbert adjoint depends fundamental symmetry (i.e. a self-adjoint operator on the choice of a fundamental symmetry, it is not η such that η2 = 1) defining an indefinite inner covariant in Gupta-Bleuler quantization and the product (φ, ψ)= hφ, ηψi. In the mathematical liter- BRST approach. We can now define an indefinite ature, such a space is called a Krein space. It was no- spectral triple. ticed by Helga Baum [8] that the spinor bundle of a pseudo-Riemannian manifold is naturally equipped with the structure of a Krein space. This was gener- 5. Indefinite spectral triple alized by Alexander Strohmaier to noncommutative geometry [30]. After some pioneering works [41–43], many pa- Substituting an indefinite inner product for a pers were devoted to the extension of noncommuta- scalar product has a striking physical consequence: tive geometry to Lorentzian geometry [7,30,33,37– the possible existence of states with negative “prob- 39,44–67]. Inspired by these references, we define an abilities” (i.e. such that (ψ, ψ) < 0). These states even-dimensional real indefinite spectral triple to be: were first met in physics by Dirac in 1942 in his (i) A ∗-algebra A represented on a Krein space quantization of electrodynamics [31]. He interpreted K equipped with a Hermitian form (·, ·) and a negative-norm states as describing a hypothetical fundamental symmetry η. We assume that the world [32]. Negative-norm states have now become representation satisfies π(a∗)= π(a)×. familiar in physics through their role in the Gupta- (ii) A chirality operator χ, i.e. a linear map on K Bleuler and Becchi-Rouet-Stora-Tyutin (BRST) such that χ2 = 1 and χ† = χ (where the ad- quantizations of gauge fields. joint χ† is defined by χ† = ηχ×η). The algebra In most applications, the indefinite inner prod- commutes with χ. uct (·, ·) is natural (i.e. uniquely defined, up to a (iii) An antilinear charge conjugation J, such that scalar factor, by some symmetry condition) and J †J = 1. the scalar product h·, ·i is somewhat arbitrary. In (iv) A set of signs (ǫ,ǫ′′,κ,κ′′) describing relations a Lorentzian manifold, the scalar product corre- (2) to (5) between χ, η and J. sponds to the Wick rotation following some choice (v) A Krein-self-adjoint Dirac operator D, which of a time-like direction (see Ref. [33] for a precise satisfies JD = DJ and χD = −Dχ. definition). Krein spaces are a natural framework The hypothesis π(a∗) = π(a)× is a simplifying as- for gauge field theories [34–36] and Lorentzian spec- sumption, which is well-adapted to particle physics tral triples [7,30,33,37–40]. applications, but which needs not be longer true We present now some essential properties of op- when discrete structures replace manifolds [68]. We erators on Krein spaces but, true to the physics tra- refer the reader to Refs. [30,33,58,63] for the func- dition, we do not describe their functional analytic tional analytic aspects of indefinite spectral triples. properties. If K is a Krein space, a linear opera- If we compare with Connes’ spectral triples, we see tor T : K → K has a Krein-adjoint T × defined by that we have an additional object (the fundamen- (T ×x, y) = (x,Ty) for every x and y in K. A linear tal symmetry η) and two additional signs: κ and operator is Krein-self-adjoint if T × = T and Krein- κ′′. Because of this more complex structure, the unitary if T ×T = TT × = 1. KO-dimension n is no longer enough to classify An anti-linear map (i.e. a map T : K → K such indefinite spectral triples and we need both m and

5 n. The classification carried out in section 3 holds Tr(M1 ⊗ M2)=Tr1(M1)Tr2(M2), also for indefinite spectral triples because they are particular cases of mod-8-spacetime representa- where Tr1 is the trace over the spinor fibre over M and Tr2 the trace over the finite dimensional Krein tions. More precisely, Let (A1, H1,D1,J1,χ1, η1) space K2. and (A2, H2,D2,J2,χ2, η2) be two real even- dimensional indefinite spectral triples. Supplement The family of open sets ensures that, for any com- the tensor products defined in section 3 with pactly supported variation, there is an n0 such that Un contains the support of the variation for every n>n0 and the variation of An is well defined. This A = A1⊗Aˆ 2, is compatible with the NCG point of view because ˆ ˆ D = D1⊗1+1⊗D2, Suijlekom proved the conceptually important fact π = π1⊗ˆ π2. that any noncommutative geometry can be considered as an algebra bundle over a Hausdorff base It can be checked that this tensor product is space [78]. We calculate Lb for QED in this paper indeed a real even-dimensional indefinite spectral and for the electroweak and standard models in a × triple (i.e. D = D and D commutes with J and forthcoming publication. We use the fermionic La- anticommutes with χ). grangian Lf = (Ψ,DΨ) [7,43,44]. Note that in the The extension of this tensor product to odd- Riemannian NCG Lagrangian the first Ψ is replaced dimensional indefinite spectral triples seems dif- by JΨ [17]. ficult, if one considers the complexity of the Rie- An important and long-standing problem is that mannian case [69–73]. Note that Farnsworth also only physical states Ψ must be used in the fermionic advocates the use of a graded tensor product [73]. Lagrangian, while the trace Tr is over the whole We define now the indefinite spectral triple en- space K1 ⊗ K2. Lizzi and coll. [41] pointed out that coding models of particles physics. the bosonic Lagrangian over physical states is different from Lb and not physically valid (it contains 6. Particle physics models CPT -symmetry violating terms). We discuss now this so-called fermion doubling problem. Particle physics models (QED, electroweak, standard model) are described by an almost commuta- 7. Fermion doubling problem tive spectral triple, i.e. the tensor product of the spectral triple S1 of a manifold M and a finite di- The name of the fermion doubling (or quadru- mensional spectral triple S2. pling) problem comes from the fact that a state Connes and Lott [74] derived the fermionic and in K1 ⊗ K2 corresponding to a particle p can be gauge Lagrangians of the standard model in Rie- written as a linear combination of ψ ⊗ pL, ψ ⊗ pR, c c mannian space. In Lorentzian spacetime Dungen ψ ⊗ pL and ψ ⊗ pR. Since ψ ∈ K1 = Γ(M,S) is a found the fermionic Lagrangian but the gauge La- four-dimensional (Dirac) spinor, each particle is de- grangian is considered to be an open problem [7]. scribed by a 16-dimensional vector instead of a four- Surprisingly, the problem was already solved in El- dimensional one. sner’s outstanding Master’s thesis [43] (see also [75– The solution proposed by Elsner and coll. [43,77] 77]), where many aspects of noncommutative ge- (which is different from the one of Ref. [41]) is to ometry (e. g. Connes differential algebra, curvature, consider as physical the linear combinations of ψL ⊗ c c c c bosonic and fermionic Lagrangians) are generalized pL, ψR ⊗ pR, ψL ⊗ pL and ψR ⊗ pR, where ψL and to the case where the Hilbert space is replaced by ψR are two-dimensional Weyl fermions (obtained as ± ± c a vector space equipped with a sesquilinear form. χ1 ψ where χ1 = (1 ± χ1)/2), ψL/R = J1ψL/R in c Indefinite spectral triples clearly fit into that frame- the fermionic space K1 and pL/R = J2pL/R in the work. When the base manifold M is not compact, gauge space K2. The degrees of freedom are reduced Elsner defines a family of bosonic Lagrangians An = to four (two for ψL ⊗ pL and two for ψR ⊗ pR). The |g|dxLb(x), where the relatively compact antiparticles states do not correspond to additional RUn p open sets Un form an exhausting family covering M degrees of freedom because they are obtained by × and Lb = −Tr(θ θ) is the Lagrangian density. The applying J to the particle states [43,79]. two-form θ in Connes’ differential algebra is the We propose two justifications for this choice. Lizzi curvature of the gauge potential ρ and the trace is and coll. [41] noticed that left-handed particle (for

6 × ′ × the Lorentz group) ψL must also be left-handed par- generally not proportional, Trθ θ = 2Tr θ θ for the ticles pL for the gauge group to be physically mean- indefinite spectral triples of QED, the electroweak ingful. We can complete this by pointing that on- and standard models. This will be clear for QED shell particles and anti-particles are solutions of dif- and will be discussed for the other models in a forth- ferent (i.e. charge conjugate) Dirac equations in the coming paper. presence of an external field. Therefore, particles and antiparticles are distinct and there is no reason c 8. Indefinite spectral triple of QED to couple ψL/R with pR/L. The second justification comes from Grand Uni- fied Theories (GUT), where each particle p also ap- The NCG model of QED in Riemannian space c c time was described by Dungen and Suijlekom [82] pears in four varieties: pL/pR and pL/pR [80]. In GUT, the identification of the spinor and gauge vari- and in Lorentzian space by Dungen [7] who did not ables is so obvious that it is implicit and a state like include antiparticles and the charge conjugation op- ψL ⊗ pL is simply denoted by pL. erator. The manifold spectral triple consists of a All the physical states satisfy the Weyl property Lorentzian 4D manifold M, a Dirac operator D1 = µ χΨ = Ψ proposed long ago by Connes [81]. It was iγ ∇µ, where ∇µ is the spin connection and the also proposed to supplement the Weyl condition Krein space Γ(M,S) [8]. In the chiral representation with the Majorana condition JΨ = Ψ [44]. However, of the gamma matrices [16] the operators are c c this would allow states of the form ψR⊗pL+ψR⊗pL, 2 which are not physical from our point of view, and −10 01  0 σ  the selected space of states would not be a complex χ1 = , η1 = ,J1 = i K, 0 1 10 −σ2 0 vector space because J is antilinear.      

We can now define the trace of M = M1 ⊗ M2 2 over physical states as where K means complex conjugation and σ is the Pauli matrix. We use the same algebra A = C ⊕ C ′ as Dungen [7] and the Krein space K = C4 with Tr M = hψP ,M1ψP ihpP ,M2pP i 2 X X basis states (e ,e ,ec ,ec ). The representation and P =L/R ψP ,pP L R R L operators are, in terms of Pauli matrix σ1, σ2 and σ3 +hψc ,M ψc ihpc ,M pc i . P 1 P P 2 P ′ 3 Note that Tr and Tr are different because Tr has  a1 0   σ 0  c π2(a,b)= , χ2 = − , no sum over ψP and has sums over non physical 3 c  0 b1   0 σ  states such as ψL ⊗ pR or ψL ⊗ pR. To rewrite this in a more convenient way, we can use the operator 3 1  −σ 0   0 ǫ2σ  ̟ defined by Connes [5] (who calls it ǫ) and iden- η2 = , J2 = K, 3 1 tified by Elsner as a particle/antiparticle operator.  0 σ   σ 0  In our framework, ̟ commutes with A2, χ2 and η2 1 0 −σ2 0 and anticommutes with J2. Connes also requires ̟     ̟ = ,D2 = im . 2 to commute with D2, but we shall not use this prop-  0 −1   0 σ  erty because it would cancel the Majorana mass of the neutrinos. We define ̟ by ̟pL/R = pL/R and To compare the fermionic Lagrangian on particles c c ̟pL/R = −pL/R. By using the duality property of and antiparticles we use the relation (JΨ,DJΨ) = † ǫκ(Ψ,DΨ). We determine ǫκ by computing the di- anti-linear operators hJiu, vi = hJi v,ui for u and v ′ in Ki, we can transform Tr into mensions of the total spectral triple. For the manifold spectral triple (m1,n1) = (4, 6) and for the fi- ′ σ σ + Tr M = Tr (M χ )Tr (M χ ̟ ) nite spectral triple (m2,n2)= (2, 2) if ǫ2 = −1 and X 1 1 1 2 2 2 σ=± (m2,n2) = (6, 6) if ǫ2 = 1. In the literature it is † −1 σ † −1 σ + generally assumed that n2 = 6 but n2 = 2 is also +Tr1(J1M J χ1 )Tr2(J2M J χ2 ̟ ) , 1 1 2 2 valid for QED. The total spectral triple has now di- + where ̟ = (1+ ̟)/2 projects K2 onto particle mensions (m,n) = (6, 0) if ǫ2 = −1 and (m,n) = states. (2, 4) if ǫ2 = 1. In both cases ǫκ = 1, which implies This choice of physical states solves the fermion (JΨ,DJΨ) = (Ψ,DΨ). Thus, we can also define the doubling problem because, although Tr and Tr′ are fermionic Lagrangian over the particle states only.

7 µ The physical Lagrangian for QED with a massive Lf = (ψ, iγ (∇µ + iqAµ) − m ψ), electron is: and the CLE-Lagrangian exactly coincides with the 1 µν µ L = − F Fµν + (ψ, iγ (∇µ + iqAµ) − m ψ), physical one. 4 To complete this section, we remark that parti- where q< 0 is the electron charge and Fµν = ∂µAν − cles and antiparticles have the same mass and op- ∂ν Aµ. posite currents. In textbook QED, these properties The Dirac corresponding to this triple is are achieved through the anticommutativity of the

µ normal product of fermion operators [16]. In our D(A)= D − qγ Aµ ⊗ ̟, framework, this property follows from the funda- µ µ where mental symmetry. Indeed, by using J1γ = −γ J1 and (γµ)× = γµ we get −qA = i π(a )∂ π(b ), µ X j µ j j hJΨ,ηJΨi = hΨ, ηΨi, is self-adjoint. The curvature is hJΨ, η(γµ ⊗ 1)JΨi = −hΨ, η(γµ ⊗ 1)Ψi, iq θ = − (γµγν − gµν )F ⊗ ̟, which means that the mass is conserved and the 2 X µν µν current is reversed by charge conjugation. We did not use any anticommutation relation. and the bosonic Lagrangian is

2 µν Lb = −8q Fµν F . 9. Conclusion The prefactor of L is not correct. This problem b A particularly appealing aspect of noncommuta- was elegantly solved [83,43,75] by choosing a positive tive geometry is that the internal (fibre) and ex- definite z which commutes with π(A), Jπ(A)J −1 ternal (manifold) degrees of freedom are put into and D to redefine the trace as Tr (M) = Tr(zM). z a common geometric framework. Real Clifford al- In the case of QED a solution of the constraints is gebras can also unify spacetimes and finite objects z = ρ1 ⊗ 1, where ρ > 0. We can now define the since they describe spinors on pseudo-Riemannian Lagrangian manifolds as well as finite geometries [85]. Therefore, 1 it is not suprising that real Clifford algebras can be L = Tr (θ×θ)+ (Ψ,D(A)Ψ), CLE z 2 used to define the space and time dimensions of an which we call the Connes-Lott-Elsner Lagrangian algebra representing (in a generalized sense) a possi- since it was originally proposed by Connes and bly noncommutative spacetime. The present paper Lott in the Riemannian case, and then extended is a precise formulation of this idea and the main in- by Elsner to general signatures. Observe that, since gredient of the definition of a time dimension is the noncommutative 2-forms are defined modulo the fundamental symmetry η which allows for a kind of junk [84], the expression Tr(θ×θ) does not have an Wick rotation of spacetime. immediate meaning. In the Riemannian case, this The fermionic and gauge Lagrangian of particle problem is solved by projecting θ onto the orthog- physics models were given here for Lorentzian space- onal of the junk. This solution can be applied in times, but the derivation of the full spectral action general signature if and only if the restriction of the is still a difficult open problem. indefinite inner product to the junk is non degen- For applications to topological insulators, it would erate. Remarkably, this property holds in QED as be desirable to extend these results to the case of well as in the full standard model [43,26]. odd-dimensional algebras. The QED bosonic Lagrangian becomes Lb = 2 µν 2 −8ρq Fµν F and we can choose ρ = 1/(32q ) 10. Acknowledgments to obtain the physical bosonic Lagrangian. The 1 fermionic Lagrangian 2 (Ψ,D(A)Ψ) with We are extremely grateful to Harald Upmeier for sending us a copy of Elsner’s Master’s thesis. We Ψ=(1+ J)(χ+ψ ⊗ e + χ−ψ ⊗ e ), 1 R 1 L thank Shane Farnsworth, Johannes Kellendonk, where hψ, ψi = 1, becomes Guo Chuan Thiang, Karen Elsner, Franciscus Jozef

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