Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics

Connes' notion of non-commutative geometry (NCG) generalizes Riemannian geometry and yields a striking reinterepretation of the standard model of particle physics, coupled to Einstein gravity. We suggest a simple reformulation with two key mathematical advantages: (i) it unifies many of the traditional NCG axioms into a single one; and (ii) it immediately generalizes from non-commutative to non-associative geometry. Remarkably, it also resolves a long-standing problem plaguing the NCG construction of the standard model, by precisely eliminating from the action the collection of 7 unwanted terms that previously had to be removed by an extra, non-geometric, assumption. With this problem solved, the NCG algorithm for constructing the standard model action is tighter and more explanatory than the traditional one based on effective field theory.


INTRODUCTION
Since the early 1980's, Connes and colleagues have been developing a branch of mathematics known as noncommutative geometry ("NCG") [1]. Its mathematical interest stems from the fact that it provides a natural generalization of Riemannian geometry (much as Riemannian geometry in turn provides a natural generalization of Euclidean geometry). Its physical interest stems from the fact that it provides a framework in which the standard model of particle physics (coupled to gravity) takes on an elegant geometric interpretation, while its action condenses to a more concise form [2][3][4][5][6][7][8][9][10]. For an introduction, see Refs. [11,12] and Sec. 1 in Ref. [13].
There are both theoretical and observational reasons to think that the current standard model of particle physics is not the final story [14]. If the NCG reformulation of the standard model is a step in the right direction, what is the next step in this direction? In a previous paper [13], we argued that Connes' non-commutative geometry (or at least the part of it that plays a role in the reformulation of the standard model) could be naturally generalized to non-associative geometry (for earlier work on non-associative geometry, see [15][16][17][18][19]), and we explained why there are good mathematical [27] and physical [28] motivations for such a generalization.
We are now able to go much further: we give a simple and compact reformulation of Connes' non-commutative framework that immediately generalizes from the associative to the non-associative case. In the process, several of the standard NCG axioms and formulae are neatly reinterpreted. The idea, in brief, is as follows.
In the traditional presentation of Connes' approach, a geometry is described by a so-called real spectral triple {A, H, D, J, γ} (again, see e.g. Refs. [11,12] for an introduction). The essence of our reformulation is the observation that these elements naturally fuse to form a new algebra B. This yields new insight, even in the associative case. In particular, we will see that it neatly resolves a longstanding puzzle (highlighted by Chamseddine and Connes e.g. in the concluding section of Ref. [9]) about the traditional NCG formulation of the standard model. It does so by precisely eliminating from the action the collection of 7 unwanted terms that previously had to be removed by an extra (empirically-motivated) assumption (called the "massless photon" assumption in [9]), in order to obtain precise agreement between the standard model action and the spectral action [8][9][10]12].

REFORMULATION OF CONNES' FRAMEWORK
In this section, we present our formulation in three steps. In the first step, we explain how to extend A (a *algebra) to ΩA (the differential graded * -algebra of forms over A), even when A is non-associative. In the second step, we explain how H may be promoted to a bimodule over A by defining a new algebra B 0 = A ⊕ H; and similarly, H may be promoted to a bimodule over ΩA by defining a new algebra B = ΩA ⊕ H. In the process, we obtain a new view of the operator J, and Connes' "orderzero" and "order-one" axioms. Again, a key virtue of our reformulation is that it naturally extends to the case where A is non-associative. In the third step, we express the inner fluctuations of D in our language.
Step 1: Promoting A → ΩA Let A be a unital * -algebra over a field F . (A may be non-commutative, or even non-associative.) We introduce, for each element a ∈ A, a corresponding formal symbol δ[a]. ΩA is the algebra generated by A and these differentials δ[a], modulo the relations ] (with f ∈ F , and a, a ′ ∈ A) and, in addition, modulo appropriate associativity relations. For example, in the usual case where A is associative, we take ΩA to be associative as well, and impose relations like (aδ[b])c = a(δ[b]c), etc; in this way, we recover the usual algebra ΩA defined, e.g.
in Section 6.1 of Ref. [20]. More generally, when A is non-associative, ΩA must be equipped with compatible associativity relations (examples are given in Ref. [23] 6 ]δ[a 7 ]))a 8 is an element of Ω 3 A. In particular, Ω 0 A = A; and if ω m ∈ Ω m A and ω n ∈ Ω n A then ω m ω n ∈ Ω m+n A, so ΩA is graded. If we define δ[a * ] = −δ[a] * , then ΩA is a * -algebra, with its * -operation naturally inherited from A. Note that we can interpret δ as a linear map from Ω 0 A → Ω 1 A; and this may, in turn, be promoted to a linear map d : Ω m A → Ω m+1 A which (even in the non-associative case) may be defined recursively by requiring it to satisfy a graded , so that (ΩA, d) is a differential graded * -algebra.
Step 2: Promoting Ω 0 A → B0 and ΩA → B Usually, in non-commutative geometry, one starts by defining (in two steps) a bi-representation of A on H, so that H becomes a bi-module over A. In the first step, one defines a left action of A on Hi.e. a bilinear product ah = L a h ∈ H between elements a ∈ A and h ∈ H. In the second step one uses J, an anti-unitary operator on H, to define a corresponding right-action of A on Hi.e. another bilinear product ha ≡ R a h ∈ H given by R a ≡ JL a * J * . The left and right action are required to satisfy the so-called order-zero condition (ah)b = a(hb) or, equivalently, [L a , R b ] = 0 (∀a, b ∈ A).
We would like to reformulate this construction in a way that makes sense even when A is non-associative. Fortunately, the natural definition of a "bi-representation" of a non-associative algebra (or, equivalently, a "bimodule" over a non-associative algebra) was found long ago (perhaps by Samuel Eilenberg [21]), and is explained simply and succinctly in Ch. II.4 of Ref. [22]. The idea is that a bimodule H over A is nothing but a new algebra with the product between two elements of B 0 where aa ′ ∈ A is the product inherited from A, while ah ′ ∈ H and ha ′ ∈ H are precisely the left-and rightactions defined above. In this language, two familiar axioms of non-commutative geometry -namely, (i) the associativity of A and (ii) the order-zero condition -are condensed into the single assumption that B 0 is an associative algebra. Furthermore, the familiar definition of right-action in terms of left action, R a = JL a * J * , is reinterpreted as the statement that the map is an anti-automorphism of B 0 , with period 2 when the KO dimension is 0, 1, 6 or 7 mod 8 (i.e. when J 2 = 1) and period 4 when the K0 dimension is 2, 3, 4 or 5 mod 8 (i.e. when J 2 = −1). In particular, when J 2 = 1, B 0 is a *algebra, with * operation given by (3). The advantage of this reformulation is that it continues to make sense when A is non-associative: in this case, B 0 is non-associative, too, and the familiar order zero condition is replaced by a compatible restriction on the associativity properties of B 0 . For example, if A is an alternative algebra, like the octonions, we can require B 0 to be an alternative algebra, too. The interpretation of J in terms of the anti-automorphism (3) is unaffected. Just as we give a bi-representation of A on H by defining a new algebra B 0 = A⊕H, we give a bi-representation of ΩA on H by defining a new algebra with the product between two elements of B where ωω ′ ∈ ΩA is the product inherited from ΩA, while ωh ′ ∈ H and hω ′ ∈ H are bilinear products that define In the case where A is non-associative, the familiar order-zero and order-one conditions are replaced by compatible associativity constraints on B.
Step 3: Inner Fluctuations of D Since D arises as the representation on H of the derivation δ : Ω 0 A → Ω 1 A, our approach suggests that the inner fluctuations D → D A are simply given by D A = D + ∆, where ∆ denotes the representation on H of a certain subset of the inner derivations of ΩA. In particular, since the map δ is order one with respect to the grading on ΩA, and satisfies δ[a * ] = −δ[a] * (which is equivalent to D † = D), ∆ must refer to the subset of inner derivations that respect these extra conditions. For example, consider the case where ΩA is an associative algebra: the general inner derivations of such an algebra are given by ∆ ω = L ω − R ω , ∀ ω ∈ ΩA; the condition that ∆ ω should be order one further restricts us to ∆ ω1 = L ω1 − R ω1 where ω 1 ∈ Ω 1 A; and hermiticity of D further restricts us to ω * 1 = ω 1 . This exactly reproduces Connes' familiar formula for the fluctuations of D in the associative case.
But, again, our formulation extends to the case where A and ΩA are non-associative algebras, since these also have inner derivations that are perfectly well defined and unambiguous. For example, if ΩA is a Lie, Jordan, or alternative algebra, its inner derivations are given by ∆ , respectively (where ω and ω ′ are arbitrary elements of ΩA). In fact, in our new formulation, it seems natural to go a step further and suggest that the natural fluctuations of D, rather than being given by (an appropriate subset of) the derivations of ΩA, are instead given by (an appropriate subset of) the derivations of B itself. This issue is discussed in [23].

APPLICATION TO THE STANDARD MODEL
In this section, we first review the traditional formulation of the standard model in non-commutative geometry, and then explain how our reformulation naturally yields a new constraint that resolves a well-known puzzle that arises in the traditional formulation. For clarity, we will deal in this section with a single-generation of standard model fermions; the extension to the full set of three generations is straightforward.
The standard model is described by a finitedimensional real spectral triple {A, H, D, J, γ} of K0 dimension 6. A is a * -algebra given by C ⊕ H ⊕ M 3 (C), where C is the algebra of complex numbers, H is the algebra of quaternions, and M 3 (C) is the algebra of 3 × 3 complex matrices. H is a 32-dimensional complex Hilbert space (32 is the number of fermionic degrees of freedom in a standard model generation, including the right-handed neutrino). To describe the action of γ and J on H, it is convenient to split H into four 8-dimensional subspaces H = H R ⊕ H L ⊕H R ⊕H L . Here H R and H L contain the right-handed and left-handed particles, whileH R andH L contain the corresponding anti-particles. If h R ∈ H R is a right-handed particle (withh R ∈H R the corresponding anti-particle) and h L ∈ H L is a left-handed particle (withh L ∈H L the corresponding anti-particle), then the helicity operator γ and the anti-linear charge conjugation operator J act as follows: To describe the action of A on H, it is convenient to further split each of the four spaces (H R , H L ,H R ,H L ) into a lepton and quark subspace: Each of the four lepton spaces {L R , L L ,L R ,L L } is a copy of C 2 ; an element of any of these four spaces correspondingly carries a doublet (neutrino vs. electron) index. Each of the four quark spaces {Q R , Q L ,Q R ,Q L } is a copy of C 2 ⊗ C 3 ; an element of any one of these four spaces correspondingly carries two indices: a doublet (up quark vs. down quark) index and a triplet (color) index. Now consider an element a = {λ, q, m} ∈ A F , where λ ∈ C is a complex number, q ∈ H is a quaternion, and m ∈ M 3 (C) is a 3 × 3 complex matrix, and write where α and β are complex numbers. Here q is the standard 2×2 complex matrix representation of a quaternion, and q λ is the corresponding diagonal embedding of C in H. Then L a (the left action of a on H) is given by where q, q λ and λI 2×2 act on the doublet index, while m acts on the color index.
where y l = y l,11 y l,12 y l,21 y l, 22 and y q = y q,11 y q,12 y q,21 y q,22 are arbitrary 2×2 matrices that act on the doublet indices in the lepton and quark sectors, respectively, while m = a b b 0 and n = c d 0 0 (11) are 2 × 2 and 6 × 2 matrices, respectively; and in n we have used vector notation to emphasize that c, d and 0 are 3 × 1 columns. Of the 8 complex parameters {a, b, c, d}, only a is present in the standard model (where it corresponds to the right-handed neutrino's majorana mass). The remaining 7 parameters {b, c, d} present a puzzlethey are an unwanted blemish that must be removed in order to match observations. Traditionally, they are removed by introducing an extra assumption (namely, that D commutes with L a for a = {λ, q λ , 0} ∈ A); but, as emphasized by Chamseddine and Connes (see e.g. Sec. 5 of Ref. [9]), this ad hoc solution is unsatisfying, and cries out for a better understanding. Our reformulation yields a simple and satisfying solution to this puzzle. We have seen that the associativ- (ii) y q,11 = y q,21 = b = 0; (iii) y l,11 = y l,21 = c = d = 0; or (iv) y l,11 = y l,21 = y q,11 = y q,21 = c = 0. Note, in particular, that solution (i) precisely corresponds to setting the 7 unwanted parameters (b, c, d) to zero, without the additional ad hoc assumption described above! We can go further by noting that the general embedding of C in H is given by q λ (n) = Re(λ)I 2×2 +Im(λ)n· σ, where σ are the three Pauli matrices, andn is a unit 3-vector specifying the embedding direction. Since all of these embeddings are equivalent, the diagonal embedding q λ = q λ (ẑ) in Eq. (8) was arbitrary, and may be replaced by the more general possibility L a L R = q λ (n l )L R and L a Q R = q λ (n q )Q R . If we redo the preceding analysis with this modification, the four solutions for D are modified accordingly: in particular, in solution (i), D is given by Eq. (9), where the 2 × 2 matrices y l and y q are arbitrary, the 6 × 2 matrix n vanishes, and the 2 × 2 matrix m is given by m = P T M P , with M an arbitrary 2 × 2 symmetric matrix and P = (I 2×2 +n l · σ)/ √ 2 a projection operator. Then, one can check the following result: given the arbitrary 2 × 2 matrices y l , y q and M , there is a preferred choice for the embedding directionsn l and n q such that, after a change of basis on H, L a is given by Eq. (8) [with the diagonal embedding q λ = q λ (ẑ)], D is given by Eq. (9), m and n are given by Eq. (11) with b = c = d = 0, while y l and y q are given by: with {y ν , y e , y u , y d , ϕ 1 , ϕ 2 } ∈ C. This is precisely the finite geometry that (after fluctuation and substitution into the spectral action) generates the standard model of particle physics (see Refs. [8][9][10]12])! This strikingly successful match between the geometric structure on the one hand, and the standard model La-grangian on the other, appears to provide significant additional evidence: (i) for the suitability of Connes' NCG framework for describing the standard model, and (ii) for the appropriateness of the reformulation presented here. DISCUSSION We end with a few brief remarks. (i) In this paper, although we have developed a formalism suited to nonassociative geometry, our main application has been to the associative finite geometry that describes the standard model of particle physics. In a forthcoming paper [23] we present a family of non-trivial non-associative geometries (and their associated spectral actions) that provide a nice illustration of our formalism in the fully non-associative case. (ii) Refs. [9,10] observe that the standard model algebra A = C ⊕ H ⊕ M 3 (C) analyzed above can be understood more deeply as a subalgebra of A ′ = M 2 (H) ⊕ M 4 (C) (see also [24]). What new light does our formalism shed on this observation? (iii) We have seen that the finite geometry K that encodes the standard model corresponds to an algebra B that is associative. But, to evaluate the spectral action, one then tensors this finite geometry with a continuous geometry to form a new geometry K ′ , and one can check that the corresponding algebra B ′ is not associative (when one goes beyond the order one associators). In this sense, non-associativity already appears in the traditional NCG embedding of the standard model. It is interesting to consider whether this non-associativity might be connected to the generalized inner fluctuations considered in [25,26], which bear a striking resemblance to the inner derivations of a non-associative (and in particular, an alternative) algebra [13].
Part of this work was carried out at the "Noncommutative Geometry and Particle Physics" Workshop at the Lorentz Center, in Leiden; we thank the organizers and participants, and particularly Ali Chamseddine and Alain Connes for helpful input. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. LB also acknowledges support from an NSERC Discovery Grant.