Spectral Action Principle Research Articles

Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator \square_{g} is known to be essentially self-adjoint. We define complex powers (\square_{g}-i\varepsilon)^{-\alpha} by functional calculus, and show that the trace density exists as a meromorphic function of \alpha . We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine–Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments work in Lorentzian signature directly and do not rely on transition from the Euclidean setting.

Gravitation, and quantum theory, as emergent phenomena

There must exist a reformulation of quantum field theory, even at low energies, which does not depend on classical time. The octonionic theory proposes such a reformulation, leading to a pre-quantum pre-spacetime theory. The ingredients for constructing such a theory, which is also a unification of the standard model with gravitation, are : (i) the pre-quantum theory of trace dynamics – a matrix-valued Lagrangian dynamics, (ii) the spectral action principle of non-commutative geometry, (iii) the number system known as the octonions, for constructing a non-commutative manifold and for defining elementary particles via Clifford algebras, (iv) a Lagrangian with E 8 × E 8 symmetry. The split bioctonions define a sixteen dimensional space (with left-right symmetry) whose geometry (evolving in Connes time) relates to the four known fundamental forces, while predicting two new forces, SU(3) grav and U(1) grav . This latter interaction is possibly the theoretical origin of MOND. Coupling constants of the standard model result from left-right symmetry breaking, and their values are theoretically determined by the characteristic equation of the exceptional Jordan algebra of the octonions. The quantum-to-classical transition, precipitated by the entanglement of a critical number of fermions, is responsible for the emergence of classical spacetime, and also for the familiar formulation of quantum theory on a spacetime background.

Type-II two-Higgs-doublet model in noncommutative geometry

In noncommutative geometry (NCG) the spectral action principle predicts the standard model (SM) particle masses by constraining the scalar and Yukawa couplings at some heavy scale, but gives an inconsistent value for the Higgs mass. Nevertheless, the scalar sector in the NCG approach to the standard model, is in general composed of two Higgs doublets and its phenomenology remains unexplored. In this work, we present a type-II two-Higgs-doublet model in NCG, with a SM-like Higgs mass compatible with the 125 GeV experimental value and extra scalars within the alignment limit without decoupling with masses from 350 GeV.

Spectral interaction between universes

We derive a perturbative formula for the direct interaction between two four-dimensional geometries. Based on the spectral action principle we give an explicit potential up to the third order perturbation around the flat vacua. Wepresent the leading terms of the interaction as polynomials of the invariants of the two metrics and compare the expansionto the models of bimetric gravity.

Towards the signs of new physics through the spectral action

We review recent results obtained by using the spectral action applied to some simple physical models built in the noncommutative geometry framework. The first result, based on the generalization of the spectral action principle to the case of the fermionic action leads to the lowest order corrections that can possibly explain the huge difference between the electron and neutrino masses. In the second case, the extension of the family of Dirac operators for the product of finite geometry with the Riemannian manifold but without a product metric leads to a theory with two metrics that is similar to the models of bimetric gravity.

Clifford-based spectral action and renormalization group analysis of the gauge couplings

The Spectral Action Principle in noncommutative geometry derives the actions of the Standard Model and General Relativity (along with several other gravitational terms) by reconciling them in a geometric setting, and hence offers an explanation for their common origin. However, one of the requirements in the minimal formalism, unification of the gauge coupling constants, is not satisfied, since the basic construction does not introduce anything new that can change the renormalization group (RG) running of the Standard Model. On the other hand, it has been recently argued that incorporating structure of the Clifford algebra into the finite part of the spectral triple, the main object that encodes the complete information of a noncommutative space, gives rise to five additional scalar fields in the basic framework. We investigate whether these scalars can help to achieve unification. We perform a RG analysis at the one-loop level, allowing possible mass values of these scalars to float from the electroweak scale to the putative unification scale. We show that out of twenty configurations of mass hierarchy in total, there does not exist even a single case that can lead to unification. In consequence, we confirm that the spectral action formalism requires a model-construction scheme beyond the (modified) minimal framework.

Clifford structures in noncommutative geometry and the extended scalar sector

We consider aspects of the noncommutative approach to the standard model based on the spectral action principle. We show that as a consequence of the incorporation of the Clifford structures in the formalism, the spectral action contains an extended scalar sector, with respect to the minimal standard model. This may have interesting phenomenological consequences. Some of these new scalar fields carry both weak isospin and color indices. We calculate the new terms in spectral action due to the presence of these fields. Our analysis demonstrates that the fermionic doubling in the noncommutative geometry is not just a presence of spurious degrees of freedom, but it is an interesting and peculiar property of the formalism, which leads to physically valuable conclusions. Some of the new fields do not contribute to the physical fermionic action, but they appear in the bosonic spectral action. Their contributions to the Dirac operator correspond to couplings with the spurious fermions, which are projected out.

The superspace representation of super Yang–Mills theory on noncommutative geometry

A few years ago, we found the supersymmetric(SUSY) counterpart of the spectral triple which specified noncommutative geometry(NCG). Based on "the triple", we considered the SUSY version of the spectral action principle and had derived the action of super Yang--Mills theory, minimal supersymmetric standard model, and supergravity. In these theories, we used vector notation in order to express a chiral or an anti-chiral matter superfield. We also represented the NCG algebra and the Dirac operator by matrices which operated on the space of matter field. In this paper, we represent the triple in the superspace coordinate system $(x^\mu,\theta,\bar{\theta})$. We also introduce "extracting operators" and the new definition of the supertrace so that we can also investigate the square of the Dirac operator on the Minkowskian manifold in the superspace. We finally re-construct the super Yang--Mills theory on NCG in the superspace coordinate to which we are familiar to describe SUSY theories.

Supergravity on the noncommutative geometry

Two years ago, we found the supersymmetric counterpart of the spectral triple which specified noncommutative geometry. Based on the triple, we derived gauge vector supermultiplets, Higgs supermultiplets of the minimum supersymmetric standard model and its action. However, unlike the famous theories of Connes and his co-workers, the action does not couple to gravity. In this paper, we obtain the supersymmetric Dirac operator $\mathcal{D}_M^{(SG)}$ on the Riemann-Cartan curved space replacing derivatives which appear in that of the triple with the covariant derivatives of general coordinate transformation. We apply the supersymmetric version of the spectral action principle and investigate the heat kernel expansion on the square of the Dirac operator. As a result, we obtain a new supergravity action which does not include the Ricci curvature tensor.

Infrared Horava–Lifshitz gravity coupled to Lorentz violating matter: a spectral action approach

We continue our study of Horava–Lifshitz (HL) type theories using the methods of spectral geometry. In this work we construct the infrared action of gravity and matter coupled to gravity in the most general way respecting the foliation preserving diffeomorphisms. This is done with the help of the spectral action principle based on some generalized Dirac operators. The gravity part reproduces the infrared limit of the HL gravity, while the matter part gives the generalization of the earlier suggested models. Due to the fact that the same Dirac operator is used in the construction of both sectors, the parameters of the gravity and matter parts are related. We expect that this potentially could naturally exclude fine tunings needed to get some desired properties as well as open new possibilities for the experimental tests of the model.

Can Spectral Action be a Window to Very High Energies?

The principles of noncommutative geometry impose severe restrictions on the structure of (almost) commutative field theories. The Standard Model fits surprisingly well into the noncommutative framework. Here we overview some universal predictions of the spectral action principle for the behavior of bosonic theories at very high energies.

Non-associative geometry and the spectral action principle

Chamseddine and Connes have argued that the action for Einstein gravity, coupled to the SU(3) × SU(2) × U(1) standard model of particle physics, may be elegantly recast as the “spectral action” on a certain “non-commutative geometry.” In this paper, we show how this formalism may be extended to “non-associative geometries,” and explain the motivations for doing so. As a guiding illustration, we present the simplest non-associative geometry (based on the octonions) and evaluate its spectral action: it describes Einstein gravity coupled to a G2 gauge theory, with 8 Dirac fermions (which transform as a singlet and a septuplet under G2). This is just the simplest example: in a forthcoming paper we show how to construct more realistic models that include Higgs fields, spontaneous symmetry breaking and fermion masses.

The minimum supersymmetric standard model on noncommutative geometry

We have obtained the supersymmetric extension of a spectral triple that specifies a noncommutative geometry. We assume that the functional space |$\mathcal {H}$| consists of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model (MSSM). We introduce the internal fluctuations of the Dirac operator on the finite space as well as on the manifold by elements of the algebra |$\mathcal {A}$| in the triple. So, we obtain not only the vector supermultiplets that mediate |${\rm SU}(3)\otimes {\rm SU}(2)\otimes {\rm U}(1)_Y$| gauge degrees of freedom but also Higgs supermultiplets that appear in the MSSM from the same standpoint. According to the supersymmetric version of the spectral action principle, we calculate the square of the fluctuated total Dirac operator and verify that the Seeley–DeWitt coefficients give the correct action of the vector and Higgs supermultiplets. We also verify that the relation between the coupling constants of |${\rm SU}(3)$|⁠, |${\rm SU}(2)$|⁠, and |${\rm U}(1)_Y$| is same as that of SU(5) unification theory.

The supersymmetric Yang-Mills theory on noncommutative geometry

Recently, we found the supersymmetric counterpart of the spectral triple. When we restrict the representation space to the fermionic functions of matter fields, the counterpart which we name "the triple" reduces to the original spectral triple which defines noncommutative geometry. We see that the fluctuation to the supersymmetric Dirac operator induced by algebra in the triple generates vector supermultiplet which mediates gauge interaction. Following the supersymmetric version of spectral action principle, we calculate the heat kernel expansion of the square of fluctuated Dirac operator and obtain the correct supersymmetric Yang-Mills action with U(N) gauge symmetry.

The supersymmetric Dirac operator on noncommutative geometry

We extend naturally the spectral triple which define noncommutative geometry (NCG) in order to incorporate supersymmetry and obtain supersymmetric Dirac operator D_M which acts on Minkowskian manifold. Inversely, we can consider the projection which reducts D_M to $\not{D}$, the Dirac operator of the original spectral triple. We investigate properties of the Dirac operator, some of which are inherited from the original Dirac operator. Z/2 grading and real structure are also supersymmetrically extended. Using supersymmetric invariant product, the kinematic terms of chiral and antichiral supermultiplets which represent the wave functions of matter particles and their superpartners are provided by D_M. Considering the fluctuation given by elements of the algebra to the extended Dirac operator, we can expect to obtain vector supermultiplet which includes gauge field and to obtain super Yang-Mills theory according to the supersymmetric version of spectral action principle.

Noncommutative spectral geometry: a short review

We review the noncommutative spectral geometry, a gravitational model that combines noncommutative geometry with the spectral action principle, in an attempt to unify General Relativity and the Standard Model of electroweak and strong interactions. Despite the phenomenological successes of the model, the discrepancy between the predicted Higgs mass and the current experimental data indicate that one may have to go beyond the simple model considered at first. We review the current status of the phenomenological consequences and their implications. Since this model lives by construction at high energy scales, namely at the Grand Unified Theories scale, it provides a natural framework to investigate early universe cosmology. We briefly review some of its cosmological consequences.

Global and local aspects of spectral actions

The principal object in noncommutative geometry is the spectral triple consisting of an algebra , a Hilbert space and a Dirac operator . Field theories are incorporated in this approach by the spectral action principle, which sets the field theory action to , where f is a real function such that the trace exists and Λ is a cutoff scale. In the low-energy (weak-field) limit, the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

Erratum to: The Holst Action by the Spectral Action Principle

Erratum to: The Holst Action by the Spectral Action Principle

On gravity, torsion and the spectral action principle

We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine–Connes spectral action, deduce the equations of motions and discuss critical points.

The Holst Action by the Spectral Action Principle

We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields.