Vers une extension non associative du formalisme spectral non commutatif pour aborder la physique de plus haute énergie?
De l'espace-temps riemannien à l'espacetemps non commutatif
Often, noncommutative geometry is taken to mean the noncommutativity of the 4-dimensional spacetime coordinates themselves; it is regarded as a property of quantum gravity that presumably becomes manifest at the Planck energy scale (i.e. the exceedingly high energy scale of 1019 GeV). By contrast, from the perspective of the spectral reformulation of the standard model, all of the non-gravitational ﬁelds in nature at low energies are reinterpreted as the direct manifestations of noncommutative geometry, right in front of our nose, staring us in the face! What is the motivation for reformulating the familiar action for the standard model (coupled to gravity) in the unfamiliar langage of spectral triples and spectral action? ... The spectral action ... packages all of the complexity of gravity and the standard model of particle physics into two simple and elegant terms which, in turn, follow from a simple principle (the spectral action principle described above). The compactness and tautness of this formulation suggest that it may be a step in the right direction. To give a provocative analogy: much as Minkowski “discovered” that the rather cumbersome Lorentz transformations (which formed the basis of Einstein’s original formulation of special relativity) could be elegantly re-interpreted as the geometrical statement that we live in a 4-dimensional Minkowski spacetime, Chamseddine and Connes seem to have discovered that the rather cumbersome action for the standard model coupled to gravity can be elegantly re-interpreted as the geometric statement that we live in a certain type of noncommutative geometry.
Des défauts des théories de Kaluza-Klein aux avantages des théories spectrales non commutatives
To see what is compelling about this picture, let us contrast it with Kaluza-Klein (KK) theory. To see the contrast clearly, it is enough to consider the original and simplest KK model. ... one starts from the simple and purely gravitational action for Einstein gravity [on a 5D manifold ... the product of a 4D manifold and a circle ( M4 × S1)] and obtains something tantalizingly close to 4D Einstein gravity plus 4D gauge theory – but it also captures what is unappealing about KK theory. For one thing, one typically obtains extra, unwanted ﬁelds with unwanted couplings ... , and one must explain why these extra ﬁelds and couplings are not observed in nature. For another thing, the reduction from the initial 5D action, which has the huge symmetry group Diﬀ(M4 × S1), to the ﬁnal 4D action, which has [a] much smaller symmetry group ..., fundamentally relies on the assumption that the 5D metric ... only depends on the 4D coordinates ... This assumption is supposed to be justiﬁed, in turn, by the fact that the compactiﬁed direction is so small; but this justiﬁcation assumes that one has stabilized the extra dimension – i.e. found a way to make it small and keep it small, without letting it shrink down to a singularity or blow up to macroscopic size. The problem of stabilizing extra dimensions in KK theory is a famously thorny one and, furthermore, is ultimately at the root of the so-called landscape problem in string theory. Thus the spectral and KK approaches share a similar spirit: in both cases the goal is to reinterpret the action describing ordinary 4-dimensional physics as arising from a simpler action formulated on an “extension” of 4-dimensional spacetime. But the spectral action seems to achieve this goal more elegantly and directly. In KK theory, the starting point is an action with too many ﬁelds and too much symmetry, and one must then jump through many hoops to explain why these extra ﬁelds and symmetries are unobserved in nature. By contrast, in the spectral approach, the ﬁeld content and symmetries of the standard model are obtained directly.
Des groupes non abéliens aux algèbres non associatives
The fundamental point is that, in the ordinary approach to physics, the basic input is a symmetry group: this is the starting point for specifying a model (like the standard model of particle physics). By contrast, in the spectral approach, the fundamental input is an algebra, and the symmetry group then emerges as the automorphism group of that algebra. Symmetry groups are associative by nature, but algebras are not. Just as some of the most beautiful and important groups are noncommutative, some of the most beautiful and important algebras (including Lie algebras, Jordan algebras and the Octonions) are nonassociative. Just as it would be unnatural to restrict our attention to commutative groups, it is unnatural to restrict our attention to associative algebras. In either case, imposing such an unnatural restriction amounts to blinding ourselves to something essential that the formalism is trying to tell us. From this standpoint, our task is to formulate the spectral approach to physics in such a way that the incorporation of nonassociative algebras becomes obvious and natural.
De l'invariance par difféomorphismes et symétrie de jauges internes à la covariance par ∗-automorphismes
Usually, the automorphism invariance of the spectral action is presented as a consequence of the “spectral action principle” (the principle that the action only depends on the spectrum of D which is, itself, invariant under automorphism). But in extending to the nonassociative case, we have found it important and clarifying to make a change of perspective, in which we elevate the ∗-automorphism covariance to a fundamental underlying principle which then provides guidance about all subsequent steps... The idea is that, in the spectral approach, the principle of ∗-automorphism covariance subsumes and replaces the traditional covariance principles of physics: diﬀeomorphism covariance (in Einstein gravity) and gauge covariance (in gauge theory).
Du Modèle Standard aux modèles de grande unification
We would like to reformulate the most successful Grand Uniﬁed Theories (GUTs) – e.g. those based on SU(5), SO(10) ... – in terms of the spectral action, but in order to do this, we are forced to use nonassociative input algebras. To appreciate this point, ﬁrst note that the represen- tation theory of associative ∗-algebras is much more restricted than the representation theory of Lie groups: Lie groups (like SU(5)) have an inﬁnite number of irreducible representations, but associative algebras (like the corresponding ∗-algebra M5(C) of 5×5 complex matrices, whose automorphism group is SU(5)) only have a ﬁnite number. In particular, if we ask whether key fermionic representations needed in GUT model building – such as the 10 of SU(5), the 16 of SO(10) ... – are available as the irreps of algebras with the correct corresponding automorphism groups, the answer is “no” for associative algebras, and “yes” for nonassociative algebras.
De l'ancien modèle spectral presque commutatif aux récents modèles spectraux non commutatifs: des modèles déjà tous non associatifs!
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 ... 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 .
Des outils mathématiques plutôt originaux pour une phénoménologie physique qui n'est pas exotique pour autant
It is also worth stressing again that, although the input algebra is nonassociative, the resulting physical model is not: at the end of the day, the resulting action functional is an ordinary gauge theory, build from ordinary (associative) scalar, spinor, gauge and metric ﬁelds.
//Ajout du 18 octobre 2014