Lorentzian fermionic action by twisting euclidean spectral triples

We show how the twisting of spectral triples induces a transition from an euclidean to a lorentzian noncommutative geometry, at the level of the fermionic action. More specifically, we compute the fermionic action for the twisting of a closed euclidean manifold, then that of a two-sheet euclidean manifold, and finally the twisting of the spectral triple of electrodynamics in euclidean signature. We obtain the Weyl and the Dirac equations in lorentzian signature (and in the temporal gauge). The twisted fermionic action is then shown to be invariant under an action of the Lorentz group. This permits to interprete the field of 1-form that parametrizes the twisted fluctuation of a manifold as the (dual) of the energy momentum 4-vector.


Introduction
Noncommutative geometry [15] offers various ways to build models beyond the standard model of elementary particles (SM), recently reviewed in [12,25]. One of them [26,28] consists in twisting the spectral triple of SM by an algebra automorphism, in the sense of Connes, Moscovici [20]. This provides a mathematical justification to the extra scalar field introduced in [8] to both fit the mass of the Higgs and stabilise the electroweak vacuum. A significant difference from the construction based on spectral triples without first order condition [10,11] is that the twist does not only yield an extra scalar field, but also a supplementary 1-form field 1 , whose meaning was rather unclear so far.
Connes' theory of spectral triples provides a spectral characterization of compact riemannian manifolds [18] along with the tools for their noncommutative generalisation [17]. Extending this program to the pseudo-riemannian case is notoriously difficult. Although several interesting results in this context have been obtained recently, see e.g. [2,30,32,33], there is no reconstruction theorem for pseudo-riemannian manifolds in view, and it is still unclear how the spectral action should be handled in a pseudo-riemannian signature.
Quite unexpectedly, the twist of the SM, which has been introduced in a purely riemannian context, has something to do with the transition from the euclidean signature to the lorentzian one. In fact, the inner product induced by the twist on the Hilbert space of euclidean spinors on a four-dimensional manifold M, coincides with the Krein product of lorentzian spinors [24]. This is not so surprising, for the twist ρ coincides with the automorphism that exchanges the two eigenspaces of the grading operator (in physicist's words: that exchanges the left and the right components of spinors). And this is nothing but the inner automorphism induced by the first Dirac matrix γ 0 = c(dx 0 ). This explains why, by twisting, one is somehow able to single out the x 0 direction among the four riemannian dimensions of M. However, the promotion of this x 0 to a "time direction" is not fully accomplished, at least not in the sense of Wick rotation [22]. Indeed, regarding the Dirac matrices, the inner automorphism induced by γ 0 does not implement the Wick rotation (which maps the spatial Dirac matrices γ j to W (γ j ) := iγ j ) but actually its square: ρ(γ j ) = γ 0 γ j γ 0 = −γ j = W 2 (γ j ), for j = 1, 2, 3. (1.1) Nevertheless, a transition from the euclidean to the lorentzian does occur, and the x 0 direction gets promoted to a time direction, but this happens at the level of the fermionic action. This is the main result of this paper, summarised in propositions 4.5, 5.13, and their lorentz invariant version propositions 6.7 and 6.11.
More specifically, starting with the twisting of an euclidean manifold, then that of a twosheet euclidean manifold, and finally the twisting of the spectral triple of electrodynamics in euclidean signature [31]; we show how the fermionic action for twisted spectral triples, proposed in [24], actually yields the Weyl and the Dirac equations in lorentzian signature. In addition, the extra 1-form field acquires a clear interpretation as the dual of the energy-momentum 4-vector.
The following three aspects of the twisted fermionic action explain the change of signature: • First, in order to guarantee that the fermionic action is symmetric when evaluated on Graßmann variables (which is an important requirement for the whole physical interpretation of the action formula, also in the non-twisted case [9]), one restricts the bilinear form that defines the action to the +1-eigenspace H R of the unitary operator R that implements the twist; whereas in the non-twisted case, the restriction is to the +1-eigenspace of the grading, in order to solve the fermion doubling problem. This different choice of eigenspace had been noticed in [24] but the physical consequences were not drawn. As already emphasised above, in the models relevant for physics, R = γ 0 , and once restricted to H R , the bilinear form no longer involves derivative in the x 0 direction. In other words, the restriction to H R projects the euclidean fermionic action to what will constitute its spatial part in lorentzian signature.
• Second, the twisted fluctuations of the Dirac operator of a four-dimensional riemannian manifold are not necessarily zero [28,36], in contrast with the non-twisted case where those fluctuations always vanish. These are parametrised by the above-mentioned 1-form field. By interpreting the zeroth component of this field as an energy, one recovers a derivative in the x 0 direction, but now in a lorentzian signature.
• Third, we show that the twisted fermionic action is invariant under an actin of the Lorentz group. From that follows the interpretation of the whole 1-form field (not only its zeroth component) as the dual of the energy-momentum 4-vector.
All this is detailed as follows. In section 2, we review known material regarding twisted spectral triples, their compatibility with the real structure ( §2.1) and the new inner product they induce on the initial Hilbert space ( §2. 2). We discuss what a covariant Dirac operator is in the twisted context, and the corresponding gauge invariant fermionic action it defines ( §2.3). We finally recall how to associate a twisted partner to graded spectral triples ( §2.4).
In section 3, we investigate the fermionic action for the minimal twist of a closed euclidean manifold, that is, the twisted spectral triple having the same Hilbert space and Dirac operator as the canonical triple of the manifold, but whose algebra is doubled in order to make the twisting possible ( §3.1). In §3.2, we show that twisted fluctuations of the Dirac operator are parametrised by a 1-form field of components X µ , first discovered in [28]. In §3.3, we recall how to deal with gauge transformations in a twisted context, along the lines of [37]. We then compute the twisted fermionic action in §3.4 and show that it yields a lagrangian density similar to that of the Weyl equations in lorentzian signature, as soon as one interprets the zeroth component of X µ as the time component of the energy-momentum 4-vector of fermions. However, there are not enough spinor degrees-of-freedom to deduce the Weyl equations for this lagrangian density.
That is why in section 4 we double the twisted manifold ( §4.1), compute the twisted-covariant Dirac operator ( §4.2), and obtain Weyl equations from the fermionic action ( §4.3).
In section 5, we apply the same construction to the spectral triple of electrodynamics proposed in [31]. Its minimal twist is written in §5.1, the twisted fluctuations are calculated in §5.2, for both the free part and the finite parts of the Dirac operator. The gauge transformations are studied in §5.3 and, finally, the Dirac equation in lorentzian signature (and in the temporal gauge) is obtained in §5.4. Section 6 deals with Lorentz invariance. We conclude with some outlook and perspective. The appendices contain all the required notations for the Dirac matrices and for the Weyl & Dirac equations.

Fermionic action for twisted spectral geometry
After an introduction to twisted spectral triples ( §2.1), we recall how the inner product induced by the twist on the Hilbert space ( §2.2) permits building a fermionic action ( §2.3). The key difference with the usual (i.e. non twisted) case is that one no longer restricts to the positive eigenspace of the grading Γ, but rather to that of the unitary R implementing the twist. Finally, we emphasise the twist-by-grading procedure, that associates a twisted partner to any graded spectral triple whose representation is sufficiently faithful ( §2.4).

Real twisted spectral triples
Twisted spectral triples have been introduced to build noncommutative geometries from type III algebras [20]. Later, they found applications in high energy physics describing extensions of Standard Model, such as the Grand symmetry model [26,28]. [20]). A twisted spectral triple (A, H, D) ρ is a unital *-algebra A that acts faithfully on a Hilbert space H as bounded operators, 2 along with a self-adjoint operator D on H with compact resolvent, called the Dirac operator, and an automorphism ρ of A such that the twisted commutator, defined as is bounded for any a ∈ A (that is [D, a] ρ is well defined on the domain of D, and extends to a bounded operator on H).
A graded twisted spectral triple is one endowed with a self-adjoint operator Γ on H such that The real structure [16] easily adapts to the twisted case [36]: as in the non-twisted case, one considers an antilinear isometry J : H → H, such that where the signs , , ∈ {±1} determine the KO-dimension of the twisted spectral triple. In addition, J is required to implement an isomorphism between A and its opposite algebra One requires this action of A • on H to commute with that of A (the order zero condition), in order to define a right representation of A on H: The part of the real structure that is modified is the first order condition. In the non-twisted case, it reads: [[D, a], b • ] = 0, ∀a, b ∈ A; while in the twisted case, it becomes [28,36] where ρ • is the automorphism induced by ρ on the opposite algebra: [36]). A real twisted spectral triple is a graded twisted spectral triple, along with a real structure J satisfying (2.3), the zeroth and the first order conditions (2.5), (2.7).
In case the automorphism ρ coincides with an inner automorphism of B(H), that is π(ρ(a)) = Rπ(a)R † , ∀a ∈ A, (2.9) where R ∈ B(H) is unitary, then ρ is said compatible with the real structure J, as soon as (2.10) The inner automorphism, hence the unitary R, are not necessarily unique. In that case, ρ is compatible with the real structure if there exists at least one R satisfying the above conditions.
In the original definition [20, (3.4)], the automorphism is not required to be a * -automorphism, but rather to satisfy the regularity condition ρ(a * ) = ρ −1 (a) * . If, however, one requires ρ to be a * -automorphism, then the regularity condition implies that (2.11) Other modifications of spectral triples by twisting the real structure have been proposed [4]. Interesting relations with the above real twisted spectral triples have been worked out in [5].

Twisted inner product
Given a Hilbert space (H, ·, · ) and an automorphism ρ of B(H), a ρ-product ·, · ρ is an inner product satisfying ∀ O ∈ B(H) and φ, ξ ∈ H, (2.12) where † is the Hermitian adjoint with respect to the inner product ·, · . One calls the ρ-adjoint of the operator O. If ρ is inner and implemented by a unitary operator R on Hthat is, ρ(O) = ROR † for any O ∈ B(H) -then, a canonical ρ-product is φ, ξ ρ = φ, Rξ . (2.14) The ρ-adjointness is not necessarily an involution. If ρ is a * -automorphism (for instance, when ρ is inner), then + is an involution iff (2.11) holds, for is equivalent to the ρ-adjointness a + := ρ(a) * being an involution, for Given a twisted spectral triple (A, H, D) ρ whose twisting-automorphism ρ coincides with an automorphism of B(H), any choice of the unitary R implementing this automorphism induces a natural twisted inner product (2.14) on H. These products are useful to define a gauge invariant fermionic action.

Twisted fermionic action
The fermionic action for a real spectral triple (A, H, D; J, Γ) is [9,1] is a bilinear form defined by the covariant Dirac operator D ω := D + ω + JωJ −1 [17], with ω is a self-adjoint element of the set of generalised one-forms whileψ is a Graßmann vector in the Fock spaceH + of classical fermions, corresponding to the positive eigenspace H + ⊂ H of the grading Γ, that is, The fermionic action is invariant under a gauge transformation, that is the simultaneous adjoint action of the group U(A) of unitaries of A, both on H ψ, and on the covariant Dirac operator: D ω → (Ad u)D ω (Ad u) † . In twisted spectral geometry, the fermionic action is constructed [24] substituting D ω with a twisted covariant Dirac operator where ω ρ is an element of the set of twisted one-forms [20], such that D ωρ is self-adjoint; 3 and by replacing the inner product with the ρ-product (2.12). Instead of (2.17), one thus considers the bilinear form A gauge transformation is given by the same action (2.20) of U(A) on H, but the Dirac operator transforms in the following 'twisted' manner [37]: (2.24) The r.h.s. of (2.24) is still a twisted covariant Dirac operator The transformation ω ρ → ω u ρ is the twisted version of the law of transformation of the gauge potential in noncommutative geometry [17].
In case the twist ρ is compatible with the real structure in the sense of (2.10) for some unitary R, the bilinear form (2.23) is invariant under the simultaneous transformation (2.20)-(2.24) [24,Prop. 4.1]. However, the antisymmetry of the form A ρ Dω ρ is not guaranteed, unless one restricts to the positive eigenspace of R, that is (2. 26) This has been discussed in [24,Prop. 4.2] and led to the following: Definition 2.6. For a real twisted spectral triple (A, H, D; J) ρ , the fermionic action is whereξ is the Graßmann vector associated to ξ ∈ H R .
In the spectral triple of SM, the restriction to H + is there to solve the fermion doubling problem [38]. It also selects out the physically meaningful elements of H = L 2 (M, S) ⊗ H F , that is, those spinors whose chirality in L 2 (M, S) coincides with their chirality as elements of the finite-dimensional Hilbert space H F . In the twisted case, the restriction to H R is there to guarantee the antisymmetry of the bilinear form A ρ Dω ρ . However, the eigenvectors of R may not have a well-defined chirality. If fact, they cannot have it when the twist comes from the grading (see §2.4 below), since the unitary R implementing the twist (given in (2.35)) anticommutes with the chirality Γ = diag (I H + , −I H − ), so that (2.28) From a physical standpoint, by restricting to H R rather than H + , one loses a clear interpretation of the elements of the Hilbert space: a priori, an element of H R is not physically meaningful, since its chirality is ill-defined. However, we show in what follows that -at least in two examples: a manifold and the almost-commutative geometry of electrodynamics -the restriction to H R is actually meaningful, for it allows to obtain the Weyl and Dirac equations in the lorentzian signature, even though one starts with a riemannian manifold.
Before that, we conclude this section with two easy but useful lemmas. The first recalls how the symmetry properties of the bilinear form A D = J·, D· do not depend on the explicit form of the Dirac operator, but solely on the signs , in (2.3). The second stresses that once restricted to H R , the bilinear forms (2.17) and (2.23) differ only by a sign. Proof. By definition, an antilinear isometry satisfies Jφ, Jξ = φ, ξ = ξ, φ . Thus, In particular, for KO-dimension 2, 4 one has = −1, = 1, so A D is antisymmetric. The same is true for A Dω in (2.17), because the covariant operator D ω satisfies the same rules of sign (2.3) as D.
Lemma 2.8. Given D, and a unitary R compatible with J in the sense of (2.10), one has Proof. For any φ, ξ ∈ H R , we have where we used (2.10) as R † J = JR † and (2.26) as R † φ = φ.

Minimal twist by grading
The twisted spectral triples recently employed in physics are built by minimally twisting a usual spectral triple (A, H, D). The idea is to substitute the commutator [D, ·] with a twisted one [D, ·] ρ , while keeping the Hilbert space and the Dirac operator intact, because they encode the fermionic content of the theory and there is, so far, no experimental indications of extra fermions beyond those of the SM. However, for the spectral triples relevant for physics, [D, ·] and [D, ·] ρ cannot be simultaneously bounded [36, §3.1]. So in order to be able to twist the commutator, one needs to play with the only object that remaining available, namely the algebra. where I B is the identity of the algebra B.
If the initial spectral triple is graded, a natural minimal twist may be obtained as follows. The grading Γ commutes with the representation of A, so the latter is actually a direct sum of two representations on the positive and negative eigenspaces H + , H − of Γ (see (2.19)). Therefore, one has enough space on H = H + ⊕ H − to represent twice the algebra A. It is tantamount to where p ± := 1 2 (I H ± Γ) and π ± (a) := π 0 (a) |H ± are respectively the projections on H ± and the restrictions on H ± of π 0 . If π ± are faithful, then (A ⊗ C 2 , H, D) ρ with ρ the flip automorphism ρ(a, a ) := (a , a), is indeed a twisted spectral triple, with grading Γ. Furthermore, if the initial spectral triple is real, then so is this minimal twist, with the same real structure [36]. 4 The flip ρ is a * -automorphism that satisfies (2.11), and coincides on π(A ⊗ C 2 ) with the inner automorphism of B(H) implemented by the unitary As recalled in the next section, the canonical ρ-product (2.14) associated to the minimal twist of a closed riemannian spin manifold of dimension 4 turns out to coincide with the lorentzian Krein product on the space of lorentzian spinors [24]. The aim of this paper is to show that a similar transition from the euclidean to the lorentzian also occurs for the fermionic action.
We first investigate how this idea comes about, by studying in the next section the simplest example of the minimal twist of a manifold. Then, in the following sections, we show how to obtain the Weyl equations in the lorentzian signature by doubling the twisted manifold and, finally, the Dirac equation by minimally twisting the spectral triple of electrodynamics in [31].

Preliminary: minimally twisted manifold
We compute the fermionic action for the minimal twist of a closed euclidean spin manifold M.
Since we aim at finding back the Weyl and Dirac equations, we work in dimension 4, assuming gravity is negligible (hence the flat metric). This is tantamount to choosing in (2.3)

Minimal twist of a riemannian manifold
The minimal twist of M is the real, graded, twisted spectral triple where C ∞ (M) is the algebra of smooth functions on M, L 2 (M, S) is the Hilbert space of square integrable spinors with inner product (dµ the volume form) and ð := −iγ µ ∂ µ is the euclidean Dirac operator with γ µ the self-adjoint euclidean Dirac matrices (see (A.2)). The real structure and grading are (cc denotes complex conjugation) where each of the two copies of C ∞ (M) acts independently and faithfully by point-wise multiplication on the eigenspaces L 2 (M, S) ± of γ 5 . The automorphism ρ of C ∞ (M) ⊗ C 2 is the flip It coincides with the inner automorphism of B(H) implemented by the unitary which is nothing but the Dirac matrix γ 0 (this choice is not unique, as will be investigated in [3]). It is compatible with the real structure (2.10) with Proof. The first equation is checked by direct calculation, using the explicit form of γ µ , along with (3.5) and (writing ρ(a) for π M (ρ(a))): (3.10) The second follows from (2.11), the third from (2.3), noticing that J commutes with ∂ µ , having constant components: Corollary 3.1.1. The boundedness of the twisted commutator follows immediately:

Twisted fluctuation for a manifold
Substituting, in a twisted spectral triple, D with the twisted covariant D ωρ (2.21) is called a twisted fluctuation. The minimally twisted manifold (3.2) has non-vanishing self-adjoint twisted fluctuations (2.21) of the form ð X := ð + X, (3.12) where This has been shown in [36,Prop. 5.3]; in contrast with the non-twisted case, where the selfadjoint fluctuation of ð always vanishes, irrespective of the dimension of the manifold M [17].
In [36] the self-adjointness of ð X was guaranteed imposing the selfadjointness of ω ρ +J ω ρ J −1 , but not necessarily the one of ω ρ . One may worry that the non-vanishing of X is an artefact of this choice, and that X might actually vanish as soon as ω ρ = ω † ρ . The following lemma clarifies this point.
Proof. By Lem. 3.1 and its corollary, one obtains for a i : which is of the form (3.14) with h µ := i g i (∂ µ f i ) and h µ : where the last equality follows from (3.9), applied to W µ viewed as an element of Multiplying the first equation by σ λ and using Tr(σ λ σ µ ) = 2δ µλ yields the first part of (3.16).
Consequently, imposing that ω ρ = 0 be self-adjoint, that is imposing (3.16) with h µ = 0, does not imply that X µ vanishes (it does vanish only if h µ is purely imaginary). In other words, as long as h µ / ∈ iR, the self-adjointness of ω ρ does not forbid a non-zero twisted fluctuation.

Gauge transformation
. It (and its twist) acts on H according to (3.5) as (we omit the symbol of representation) Under a gauge transformation with unitary u ∈ C ∞ (M) ⊗ C 2 , the fields h µ and h µ parametrising the twisted one-form ω ρ in (3.14) transform as Proof. Under a gauge transformation, the twisted one-form ω ρ is mapped to (see (2.25)) where we used (3.11) for a = u * , namely as well as (3.9) for a = u, together with uW µ u * = W µ since u commutes with W µ . Therefore, W µ → W µ + u∂ µ u * , which with the explicit representation of W µ (3.14) and u (3.21) reads Although h µ , h µ transform in a nontrivial manner, their real parts 1 2 f µ , 1 2 f µ remain invariant. This explains why the fluctuation X in (3.14) is invariant under a gauge transformation (2.24). Furthermore, by simultaneously transforming spinors according to (2.20), the twisted fermionic action is invariant, by construction. So one expects that any ψ ∈ H R is unchanged under the adjoint action of Ad u. This is true, as one checks from (3.4) that uJ uJ −1 = I for any unitary u.

Twisted fermionic action for a manifold
Let us first work out the positive eigenspace H R (2.26) for R = γ 0 as in (3.7).
We now compute the fermionic action (2.27) for a minimally twisted manifold.
The fermionic action is then obtained by substituting φ = ξ in (3.24) and replacing the components ζ of ξ by the associated Graßmann variableζ,φ: The striking fact about (3.34) is the disappearance of the derivative in the x 0 direction, and the appearance, instead, of the zeroth component of the real field f µ parametrising the twisted fluctuation X. This derivative, however, can be restored interpreting −if 0 ζ as ∂ 0 ζ, i.e. assuming with f 0 independent of x 0 . Denoting by σ µ M = {I 2 , σ j } the upper-right components of the minkowskian Dirac matrices (see (A.4)), the integrand in the fermionic action then reads (with summation on the µ index) which reminds of the Weyl lagrangian densities (A.12): but with the σ 2 M matrix, that prevents to simultaneously identifyζ with Ψ r and −ζ † σ 2 M with iΨ † r . To make such an identification possible, one needs more spinorial degrees of freedom. They are obtained in the next section, multiplying the manifold by a two point space.

Doubled manifold and Weyl equations
In constructing a spectral triple for electrodynamics, the authors of [31, §3.2] first consider, as an intermediate step, the product of a manifold with the finite-dimensional spectral triple This model describes a U (1) gauge theory, but fails to describe classical electrodynamics for two reasons, discussed at the end of [31, §3]: first, the finite Dirac operator is zero so the electrons are massless; second, H F is not big enough to capture the required spinor degrees-of-freedom. However, none of the above arises as an issue if one wishes to obtain the Weyl lagrangian, since the Weyl fermions are massless anyway, and they only need half of the spinor degrees-offreedom as compared to the Dirac fermions.

Minimal twist of a two-point almost-commutative geometry
The product -in the sense of spectral triple -of a four-dimensional closed euclidean manifold M with the two-point space (4.1) is with real structure J = J ⊗ J F and grading Γ = γ 5 ⊗ γ F , where ð, J , γ 5 are as in (3.4), while  Following §2.4, the minimal twist of (4.2) is given by the algebra A ⊗ C 2 , acting on H as π(a, a ) = for a := (f, g), a := (f , g ) ∈ A; with twist π(ρ(a, a )) = π(a , a) = In both of the equations above, we have denoted where π M is the representation (3.5) of C ∞ (M) ⊗ C 2 on L 2 (M, S).

Twisted fluctuation of of a doubled manifold
We begin with some notations and a technical lemma. Following (3.13) and (4.7), given any Notice thatZ is not the complex conjugate of Z, since in (4.8) the complex conjugation acts neither on i nor on the Dirac matrices. This guarantees that¯and commute not only for Z µ , i.e. Z µ = (Z µ ) = π M (f µ ,f µ ), but also for Z, i.e.
The notationZ is thus unambiguous, and denotes indistinctly the two members of (4.9).
With this lemma, it is easy to compute the twisted fluctuation ω ρ + J ω ρ J −1 for a generic twisted 1-form where f µ , f µ and g µ , g µ denote, respectively, the real and the imaginary parts of From (4.5)-(4.6), one gets so that, for (a, a ) as in (4.5) and using (4.10) one finds with The explicit form of the real structure and its inverse, along with the second equation (4.10), yield Summing up (4.22) and (4.25), one obtains (4.29) where Z := P +Q = −iγ µ Z µ with (the last equation follows from the explicit form (4.20) of V, W and (4.7) of F , G). By (4.19)), this reads as Similarly,Z = −iγ µZ µ withZ µ = X µ − iY µ . Hence, (4.26) yields which is nothing but (4.17).
where ð X is the twisted-covariant operator (3.12) of a manifold.
Proof. A generic twisted fluctuation (4.26) (adding a summation index i and redefining Z= i Z i ) is self-adjoint iff Z = Z † andZ =Z † . By (4.9), and the third equation in (4.10), both conditions are equivalent to Z = −Z , that is −iγ µ Z µ +Z µ = 0. As discussed below (3.18), this is equivalent to Z µ = −Z µ . From (4.27), this last condition is equivalent to z µ = −z µ , that is Substituting in (4.18), one obtains so that (4.17) gives The result follows adding ð ⊗ I 2 .
Self-adjointness directly into the bold notation: by (4.31), X ⊗ I 2 + iY ⊗ γ F is self-adjoint iff X = −X and Y = Y. Since X =X, Y =Ȳ by construction, this is equivalent by the third equation (4.10) to X = X † and Y = −Y † .

Weyl equations from the twisted fermionic action
We show that the action defined by the component ð X ⊗ I 2 of the twisted covariant Dirac operator (4.30) of the doubled manifold (i.e. we assume that g µ = 0) yields the Weyl equations. Non vanishing g µ will be taken into account in the spectral triple of electrodynamics.
Following the choice made in (3.7), we take as a unitary implementing the action of ρ on H R = γ 0 ⊗ I 2 . (4.34) It has eigenvalues ±1 and is compatible with the real structure in the sense of (2.10) with = −1. A generic element η in the +1-eigenspace H R is where φ, ξ ∈ L 2 (M, S) are Dirac the eigenspinors of γ 0 (lemma 3.4), with Weyl components ϕ, ζ.
Proposition 4.4. The twisted fermionic action induced by ð ⊗ I 2 on the doubled manifold is Proof. For η, η ∈ H R given by (4.35), remembering that J F e =ē and J Fē = e, one has where the first inner product is in H and the second in L 2 (M, S). The action is then obtained substituting η = η and promoting ζ, ϕ to Graßmann variables. The antisymmetric bilinear form A ρ ð X becomes symmetric when evaluated on Graßmann variables (as in the proof of [31,Prop. 4
Identifying the physical Weyl spinors as ψ :=ζ, ψ † := ±iφ † σ 2 (4.40) (the sign is discussed below), the lagrangian density in the action (4.36) becomes Even though the lagrangian density is lorentzian, one may argue the action is not the Weyl one, for the manifold over which one integrates is still riemannian. We come back to this in the conclusion.
In these two examples -manifold and doubled manifold -the main difference between the twisted and the usual fermionic actions does not lay so much in the twist of the inner product than in the restriction to different subspaces. Indeed, by lemma 2.8 the twist of the inner product just amounts to a global sign. As stressed in the following remark, this is the restriction to H R instead of H + that explains the change of signature.
Remark 4.6. The disappearance of ∂ 0 has no analogous in the non twisted case. In that case, ψ ∈ H + and there is no fluctuation X, so that • for a manifold, the usual fermionic action Jψ, ðψ vanishes since ðψ ∈ H − while J ψ ∈ H + ; • for a doubled manifold, By (4.40), the integrand is the euclidean version L l E := iΨ † lσ µ ∂ µ Ψ l of the Weyl lagrangian L l M . Following the result of §3.3, one expects that the field f µ remains invariant under a gauge transformation. In order not to make the paper too long, we do not check this here, but we will do it for the spectral triple of electrodynamics in §5.3. We will also give there the meaning of the other field g µ that parametrises the twisted fluctuation in Prop. 4.3. As in the non-twisted case, this will identify with the U (1) gauge field of electrodynamics.

Minimal twist of electrodynamics and Dirac equation
We first introduce the spectral triple of electrodynamics (as formalised in [31,43]), then write down its minimal twist ( §5.1) following the recipe prepared in §2.4. We compute the twisted fluctuation in §5.2. Gauge transformations are investigated in §5.3: in addition to the X µ field encountered already for the minimal twist of the (doubled) manifold, we obtain a U (1) gauge field. Finally, we compute the fermionic action in §5.4 and derive the lorentzian Dirac equation.

Minimal twist of electrodynamics
The spectral triple of electrodynamics is the product of a riemannian manifold M ( still assumed to be four-dimensional) by a two-point space like (4.1), except that D F is no longer zero (since fermions are massive). In order to satisfy the axioms of noncommutative geometry, this forces to enlarge H F from C 2 to C 4 (see [31,43] for details). Hence where ð, J , γ 5 are as in (3.4), d ∈ C is a constant parameter, and carrying an adjoint action of a unitary u := e iθ ∈ C ∞ (M, U (1)) on D ω , implemented by Computing the action (fermionic and bosonic, via the spectral action formula), one gets that this fields is the U (1) gauge potential of electrodynamics.
A minimal twist is obtained by replacing A ED by A = A ED ⊗ C 2 along with its flip automorphism ρ (2.34), with the representation π 0 of A defined by (2.33). Explicitly, where F, F , G and G are as in (4.7). The image of (a, a ) ∈ A under the flip ρ is represented by π(ρ(a, a )) = π(a , a) = In agreement with (3.7), we choose as unitary R ∈ B(H) implementing the twist It is compatible with the real structure in the sense of (2.10) with = −1, as before.

Twisted fluctuation of the Dirac operator
The twisted commutator [D, a] ρ being linear in D, we treat separately the free part ð ⊗ I 4 and the finite part γ 5 ⊗ D F of the Dirac operator. The results are summarised in Prop. 5.6.

The free part
We show (Prop. 5.3 below) that self-adjoint twisted fluctuations of ð ⊗ I 4 are parametrised by two real fields: X µ arising from the minimal twist of a manifold (3.12) and the U (1) gauge field Y µ of electrodynamics. To arrive there, we need a couple of lemmas.
where we use the notation (4.8) for with F, F , G, G as in (4.7), and V, V , W, W as in (4.20).
From (4.9) and the third equation (4.10), these four conditions are equivalent to Z = −Z , i.e.
By lemma 5.2, one knows that with z µ = f ∂ µ v +ḡ∂ µw and z µ = f ∂ µ v +ḡ ∂ µw . Denoting f µ , g µ the real and imaginary part of z µ and (similarly for z µ ), then (5.17) is equivalent to f µ = −f µ and g µ = g µ , that is In other terms, Going back to (5.13), one obtains This implies -but is not equivalent -to imposing the self-adjointness of ω ρM + J ω ρM J −1 , As discussed below Lem. 3.2 for the minimal twist of a manifold, the relevant point is that the stronger condition (5.21) does not imply that the twisted fluctuation Z be zero. The final form of the twist-fluctuated operator is the same, whether one requires (5.21) or (5.22).

The finite part
In the spectral triple of electrodynamics, the finite part γ 5 ⊗ D F of the Dirac operator D (5.1) does not fluctuate [31], for it commutes with the representation π 0 (5.2) of A ED . The same is true for the minimal twist of electrodynamics.
Proposition 5.5. The finite Dirac operator γ 5 ⊗ D F has no twisted fluctuation.

23)
where Z is given by Prop. 5.3.
Remark 5.7. Expectedly, substituting ρ = Id, one returns to the non-twisted case: the triviality of ρ is tantamount to equating (5.7) with (5.8), that is to identify the 'primed' functions with their 'un-primed' partners. Hence, Z = Z. Imposing self-adjointness, the third eq. (4.10) gives Z = −Z. Going back to (5.19), this yields f µ = 0. Therefore, X µ vanishes and remains only the U (1) gauge field Y. The latter is

24)
and coincides with the gauge potential γ µ ⊗ B µ of the spectral triple of electrodynamics(5.3) in the non-twisted case.

Gauge transformation
We discuss the transformation of the fields X and Y parametrizing the twisted fluctuation Z, along the lines of §3.3. A unitary u of A ED ⊗ C 2 is of the form u = (v, v ), where v := (e iα , e iβ ), v := (e iα , e iβ ) are unitaries of A ED , with α, α , β, β ∈ C ∞ (M, R). It (and its twist) acts on L 2 (M, S) ⊗ C 4 as Proof. Since γ F ⊗ D F twist-commutes with the algebra, in the transformation (2.25) of the gauge potential it is enough to consider ð ⊗ I 4 . So ω ρ M in (5.13) transforms to where we used [ð⊗I 4 , u * ] ρ = (ð⊗I 4 )u * as in (3.23). By (5.25) and Lem. 5.1, this transformation writes  Since A , B twist-commute with γ µ and A commutes with P µ (and B with Q µ ), one has that P µ is mapped to P µ + A∂ µĀ and Q µ to Q µ + B ∂ µB . Thus Z µ = P µ +Q µ in (5.17) is mapped to Z µ + A∂ µĀ +B ∂ µ B . With the representations (5.18) of Z µ and (5.26) of A, B, this means The result follows remembering that X µ and Y µ are the real and imaginary parts of Z µ .
By imposing that both Z and its gauge transform are self-adjoint, that is by lemma 4.1: z µ = −z µ and z µ − i∂ µ θ = −z µ − i∂ µ θ, one is forced to identify θ = θ + constant. Then (5.27) means that Y µ = g µ I 4 undergoes the transformation This is a U (1) gauge field, formally similar to the one (5.4) of the (euclidean) non-twisted case. By computing the twisted fermionic action, we show that this actually identifies with the U (1) of electromagnetism, but now in lorentzian signture.
Remark 5.9. For θ − θ a non-zero constant, the gauge transformation preserves the selfadjointness of the twisted fluctuation, even though u is not invariant by the twist. This is because such an u satisfies the weaker condition for preserving selfadjointness -pointed out in [37, §5.1] -namely ρ(u) * u twist-commutes with D.

Lorentzian Dirac equation from twisted fermionic action
To calculate the action, we first identify the eigenvectors of the unitary R implementing the twist.
Proof. R has eigenvalues ±1 and its eigenvectors corresponding to the eigenvalue +1 are: where The following lemma is useful to compute the contribution of γ 5 ⊗ D F and Y to the action.
Proof. Using (5.16) for Y µ and (A.2) for the Dirac matrices, one gets Along with (3.25), recalling that σ 2 † = iσ 2 andσ 2 † = −iσ 2 , yields where we used (3.28) and obtained the first equation of (5.31). The second one follows from Proposition 5.12. The fermionic action of the minimal twist of electrodynamics is the integral of the lagrangian density with D µ := ∂ µ −ig µ the covariant derivative associated to the electromagnetic four-potential (5.29).
Proof. Let A ρ D Z be the antisymmetric bilinear form (2.23) defined by the twisted-covariant Dirac operator (5.23). It breaks down into four terms: For η, η ∈ H R as in (5.30) one gets where the first and last equations come from the explicit forms (5.1) of J F and D F , while the third and fourth follow from the explicit form (5.15) of X and Y. These equations allow to reduce each of the four terms in (5.33) to a bilinear form on L 2 (M, S) rather than on the tensor product L 2 (M, S) ⊗ C 4 . More precisely, recalling Lem. 2.8 with = −1 (and noticing that ð ⊗ I 4 , X ⊗ I , iY ⊗ I , γ 5 ⊗ D F are all selfadjoint), one computes: Substituting η = η , then going to Graßmann variables, the sum of (5.35), (5.36), and (5.38) is where we used that A ð , A X and A γ 5 are antisymmetric on vectors (by Lem. 2.7, since ð, X, γ 5 all commute with J : ð and γ 5 by (2.3) in KO-dim 4, X by (4.10)), and so symmetric when evaluated on Graßmann variables. On the other hand, (5.37) is symmetric on vectors (since iY anticommutes with J ), while antisymmetric in Graßmann variables, so that (5.37) is equal to and  Remark 5.14. The physical interpretation of f 0 , g µ is gauge invariant. From (5.25), one gets where Θ := diag(e iθ e iθ ), Θ := diag(e iθ e iθ ) with θ, θ as in (5.27). Imposing the gauge transformation to preserve selfadjointness, that is θ = θ (disregarding the constant), then U is simply the multiplication by a phase. This means that U η is still in H R , so that the computation of the fermionic action A ρ D ρ(U )ZU ( U η, U η) is similar as above.

Identification of the physical degrees of freedom
The relation between the components ξ := ζ ζ , φ := ϕ ϕ of the eigenvector η of R and In any case, the physical spinors are completely determined by the projection η + of η on the +1 eigenspace H + of the grading operator, that is This is similar to the non-twisted case, where the physical spinors are determined by an eigenvector in H + .

Lorentz invariance
So far, our results do not say anything on the components f i of the twisted fluctuation for i = 1, 2, 3, because they do not appear in the lagrangian (5.32). Since f 0 identifies with an energy, it is tempting to identify f i with a momentum. This is actually achieved by acting with Lorentz transformations on the twisted fermionic action. More precisely, we first define in ( §6.1) a action of Lorentz boosts on the twisted spectral triple which leaves the twisted fermionic action invariant. We then investigate the action from the point of view of a boosted observers, both for the double manifold in §6.2 and for electrodynamics in §6.3. In both cases, we obtain equations of motion in which the components f i of the twisted fluctuation gets interpreted as a momenta.

Lorentz invariance of the twisted fermionic action
As recalled in appendix A.3, the Dirac equation on Minkowski spacetime is invariant under the action (A.17) of boosts simultaneously on spinors and on the Dirac operator. From a mathematical point of view, this action makes sense on an euclidean spin manifold M as well: although this might seem physically non-relevant at first sight, we let boosts act on euclidean spinors and on the euclidean Dirac operator as As an element of B(L 2 (M, S)), the boost operator S[Λ] is acted upon by the inner autormorphism ρ induced by R = γ 0 given in (3.7), namely Proof. Since σ 2 anticommutes with σ 1 , σ 3 and commutes with itself, one has The inner product on L 2 (M, S) is not invariant by (6.1), the twisted product is: for any ψ, φ ∈ L 2 (M, S). This is not a surprise, being the twisted product the Krein product of lorentzian spinors (see §2). Yet, the bilinear form A ρ ð is not invariant: . This can be corrected by making boosts act on the physical spinors Ψ, Ψ † (5.50). Namely, Consequently, in order to "boost the fermionic action", instead of φ Λ one should consider As a matter of fact, one checks that ξ), (6.11) and the same holds true for the operator obtained by the action of boosts on the twisted-covariant Dirac operator ð X . Therefore

Weyl equations for boosted observers
In agreement with (6.8) and (6.10), we define the action of a boost on L 2 (M, S) ⊗ C 2 as φ ⊗ e + ψ ⊗ē → (S[Λ] −1 φ) ⊗ e + (S[Λ]ψ) ⊗ē, in such a way that η ∈ H R in (4.35) is mapped to Prop. 6.7 is the boosted version of Prop. 4.5: now the whole field f µ dx µ (and not only its 0 th component) identifies with the dual p of the energy-momentum 4-vector. Nevertheless, the interpretation of the lagrangian density (6.21) is delicate, because of the sign difference in (6.23): Substituting the first (resp. second) of these equations in the left (resp. right) handed part of (6.21), one obtains This agrees with the equations of motion (6.27, 6.28) (remembering that ∂ ν = −ip ν and the factor −2 that was ignored from (6.21) to (6.22)), thus suggesting that L Λ is the sum -up to a complex factor -of the two Weyl lagrangians L l M , L r M (A.11). The point is that ψ l , ψ r comes from the action of Λ ∓ on the same Weyl spinor ϕ, and this action leaves the exponential part of the plane wave unaltered. So ψ l , ψ r should describe two plane waves with the same momenta, in contradiction with (6.29). We comment on this point in the conclusion.

Dirac equation for boosted observers
Similarly to what has been done for the double manifold in (6.17), we make the boost acts on L 2 (M, S) ⊗ C 4 in such a way that η ∈ H R in (5.30) is mapped to (6.33) Proposition 6.9. The fermionic action from the minimal twist of electrodynamics, as seen from a boosted observer, is the integral of the lagrangian density where D µ = ∂ µ − ig µ . Proposition 6.11. For a constant fluctuation f µ , a plane wave solution of (6.39) (resp. (6.40)) coincides with a plane wave solution of the Dirac equation with mass m = −(1 + i) d 2 (resp. m = (1 + i)d 2 ), whose (dual) generalised momentum P has components P ν = Λ µ ν P µ in the boosted frame, where P 0 = −f 0 , P j = f j , resp. P 0 = f 0 , P j = −f j . (6.42) Proof. From (A. 19) and (A.20), a plane wave solution (A.8) of (6.39) satisfies and a similar equation with σ µ , inverting ψ l and ψ r . If the first part of (6.42) holds, then these equations are equivalent toσ ν M P µ ψ l = −id 1+i ψ r and a similar equation for σ µ . These coincide with the Dirac equation (A.10) with mass m = −(1 + i) d 2 . Similarly, a plane wave solution of (6.40) satisfies which becomesσ ν M P µ ψ l = id 1+i ψ r if the second part of (6.42) holds. Together with a similar equation for σ µ , these coincide with the Dirac equations (A.10), with mass m = (1 + i)d 2 .; To guarantee a positive mass, one should impose d = m(i ± 1) with m ∈ R + . Identifying the imaginary/real axis of the complex plane with the space/time directions of two dimensional Minkowski space, the set of all physically acceptable values of d is the future light-cone, while in the non boosted case it was the imaginary axis d = im, m ∈ R.

Conclusion and outlook
The twisted fermionic action associated to the minimal twist of a doubled manifold and that of the spectral triple of electrodynamics yields, respectively, the Weyl and the Dirac equations in lorentzian signature, although one started with an euclidean manifold. The 1-form field parametrizing the twisted fluctuation gets interpreted as an energy-momentum four-vector. It was known that fluctuations of the geometry generate the bosonic content of the theory (including the Higgs sector). What is new here is that they generate also the energy-momentum. In other terms, the dynamics is obtained as a fluctuation of the geometry ! It should be checked that a similar transition from the riemannian to the pseudo-riemannian also takes place for the minimal twist of the Standard Model. This will be the subject of future works, as well as the extension of these results to curved riemannian manifolds.
Some points that deserve to be better understood are: • Is the twisted fermionic action really lorentzian, since the manifold M under which one integrates remains riemannian ? Actually this is not a problem if one takes as domain of integration a local chart (as in quantum field theory: the Wick rotation is usually viewed as a local operation), up to a change of the volume form (see [22] for details). Nevertheless, one may hope that the twist actually changes the metric on the manifold, through Connes distance formula for instance (relations between causal structure and this distance have already been worked out in [41,34,35], but without taking into account the twist).
• The twisted fermionic action is invariant under an action of the Lorentz group, and the equations of motions in the boosted frame coincide with those derived from the Weyl and Dirac equations in the boosted frame as well. But the boosted Lagrangians do not agree, because of the difference of sign in the definition of the physical left/right spinors. As stated in the text, this sign difference is not compatible with the initial restriction to H R . To overcome this difficulty, one may relax this restriction. Whether this still permits to define an antisymmetric bilinear form, that yields a physically meaningful action, will be investigated elsewhere.
In any case, the results presented here suggest an alternative attack to the problem of extending the theory of spectral triples to lorentzian geometries. That the twist does not fully implement the Wick rotation (it does it only for the Hilbert space but not for the Dirac operator) is not so relevant after all. More than being able to spectrally characterise a pseudoriemannian manifold, what matters most for the physics is to obtain an action that makes sense in a lorentzian context. The present work shows that this happens for the fermionic action.
The spectral action in the twisted context is still an open problem. The interpretation of the 1-form field f µ dx µ as the energy-momentum 4-vector might be relevant in this context as well.
Contrary to most approaches in the literature (e.g. [1], [29]), we do not obtain a lorentzian action by implementing a lorentzian structure on the geometry. The latter somehow "emerges" from the riemannian one. This actually makes sense remembering that the regularity condition imposed by Connes and Moscovici (see Rem. 2.3) has its origin in Tomita's modular theory. More precisely, the automorphism ρ that defines a twisted spectral triple should be viewed as the evaluation, at some specific value t, of a one-parameter group of automorphism ρ t . For the minimal twist of spectral triples, the flip came out as the only automorphism that makes the twisted commutator bounded. It is not yet clear what would be the corresponding oneparameter group of automorphisms. Should it exist, this will indicate that the time evolution in the Standard Model has its origin in the modular group. This is precisely the content of the thermal time hypothesis of Connes and Rovelli [21]. So far, this hypothesis has been applied to algebraic quantum field theory [39,40], and for general considerations in quantum gravity [42]. Its application to the Standard Model would be a novelty.