Highlights of non-commutative spectral geometry

A summary of non-commutative spectral geometry as an approach to unification has been presented. The role of the doubling of the algebra, the seeds of quantization and some cosmological implications have been briefly discussed.


Introduction
In the various attempts to quantize gravity, it is either assumed a purely gravitational theory without any matter fields, or it is considered that the interaction between gravity and matter is the most important element of the dynamics. We have adopted here the second approach.
Much below the Planck energy scale, gravity can be considered as a classical theory, and the laws of physics can be described to a good approximation by an effective action and continuum fields. As the energies, however, approach the Planck scale, the quantum nature of spacetime becomes apparent, and the simple prescription, dictating that physics can be described by the sum of the Einstein-Hilbert, and the Standard Model (SM) action ceases to be valid. In the framework of non-commutative spectral geometry (NCSG) [1], gravity and the SM fields were put together into matter and geometry on a non-commutative space made from the product of a four-dimensional standard commutative manifold by a non-commutative internal space.
In what follows, we have briefly presented the elements of NCSG as an approach to unification [2], and highlight the relation between the doubling of the algebra and the gauge fields [3], an essential element to make the link with the SM. We have then discussed how the doubling of the algebra is related to dissipation, which incorporates the seeds of quantization [3], and finally discussed briefly some cosmological consequences [4,5,6,7,8], since the model lives by construction in high energy scales, offering a natural framework to address early universe cosmology.

Elements of non-commutative spectral geometry
NCSG [9,10] is based on a two-sheeted space, made from the product of a four-dimensional smooth compact Riemannian manifold M (a continuous geometry for spacetime), by a discrete non-commutative space F (an internal geometry for the SM) composed by only two points. The non-commutative nature of the discrete space F is given by a spectral triple (A, H, D), where A is an involution of operators on the finite-dimensional Hilbert space, H of Euclidean Fermions, and D is a self-adjoint unbounded operator in H. All information about space is encoded in the algebra of coordinates A, which is related to the gauge group of local gauge transformations.
Assuming the algebra A to be symplectic-unitary, then A = M a (H) ⊕ M k (C), with k = 2a and H denoting the algebra of quaternions [11]. The choice k = 4 is the first value that produces the correct number (k 2 = 16) of Fermions in each of the three generations, with the number of generations being a physical input. While the choice of algebra A constitutes the main input of the theory, the choice of Hilbert space H is irrelevant. The operator D corresponds to the inverse of the Euclidean propagator of Fermions, and is given by the Yukawa coupling matrix, which encodes the masses of the elementary Fermions and the Kobayashi-Maskawa mixing parameters. The SM Fermions provide the Hilbert space H of a spectral triple for the algebra A, while the Bosons are obtained through inner fluctuations of the Dirac operator of the product geometry.
One applies the spectral action principle [12], stating that the bare Bosonic-Euclidean action is the trace of the heat kernel associated with the square of the Dirac operator, and is of the form Tr(f (D/Λ)), where f is a cut-off function, and Λ fixes the energy scale. This action can be seenà la Wilson as the bare action at scale Λ. The Fermionic term can be included by adding (1/2) Jψ, Dψ , where J is the real structure on the spectral triple, and ψ is a spinor in the Hilbert space of the quarks and leptons. For the four-dimensional Riemannian geometry, the trace is expressed perturbatively in terms of the geometrical Seeley-deWitt coefficients a n [13] as: Since, its Taylor expansion at zero vanishes, it reduces to: where f plays a rôle only through its momenta f 0 , f 2 , f 4 , which are three real parameters, related to the coupling constants at unification, the gravitational constant, and the cosmological constant respectively. The computation of this asymptotic expression results to the full Lagrangian for the SM minimally coupled to gravity, with neutrino mixing and Majorana mass terms. This purely geometric approach to the SM leads to the correct representations of the Fermions with respect to the gauge group of the SM, the Higgs doublet appears as part of the inner fluctuations of the metric, and spontaneous symmetry breaking mechanism arises naturally with the negative mass term without any tuning [1]. The see-saw mechanism is obtained, the 16 fundamental Fermions are recovered, and a top quark mass of ∼ 179 GeV is predicted [1]. The model also predicts the correct order of magnitude for the Higgs mass. Strictly speaking, the predicted Higgs mass of approximately 170 GeV is ruled out from the experimental data, nevertheless it is rather remarkable that the order of magnitude is correct, given that the NCGS approach based on the particular choice for the algebra A must be seen as an effective theory.

Dissipation and the origin of quantization
The central ingredient in the NCSG, namely the doubling of the algebra A = A 1 ⊗ A 2 acting on the space H = H 1 ⊗ H 2 is related to dissipation and to the gauge field structure [3].
To highlight the justification of this claim, let us consider the equation of the classical one-dimensional damped harmonic oscillator mẍ + γẋ + kx = 0, with time independent m, γ and k, which is a simple prototype of open systems. In the canonical formalism for open systems, the doubling of the degrees of freedom is required in such a way as to complement the given open system with its time-reversed image, thus, obtaining a globally closed system for which the Lagrangian formalism is well defined. Considering the oscillator in the doubled y-coordinate, mÿ − γẏ + ky = 0, and then using the coordinates x 1 (t) = x(t) + y(t) / √ 2 and x 2 (t) = x(t) − y(t) / √ 2, the Lagrangian of this closed system takes the form: where we have introduced the vector potential as: A i = (B/2) ij x j for i, j = 1, 2 with B ≡ γ c/e and ii = 0, 12 = − 21 = 1. It describes two particles with opposite charges e 1 = −e 2 = e in the (oscillator) potential Φ ≡ (k/2e)(x 1 2 − x 2 2 ) ≡ Φ 1 − Φ 2 , with Φ i ≡ (k/2/e)x i 2 and in the constant magnetic field B defined as B = ∇ × A.
The doubled coordinate, e.g., x 2 acts as the gauge field component A 1 to which the x 1 coordinate is coupled, and vice versa. The energy dissipated by one of the two systems is gained by the other. The gauge field acts as the bath or reservoir in which the system is embedded [3].
The NCSG classical construction carries implicit in the doubling of the algebra seeds of quantization [3]. 't Hooft has conjectured that, provided some specific energy conditions are met and some constraints are imposed, loss of information might lead to a quantum evolution [14]. By considering the classical damped harmonic oscillator and its time-reversed image, we have shown [3] that the obtained Hamiltonian belongs to the class of Hamiltonians considered by 't Hooft. We have shown [3] that the dissipation term in the Hamiltonian is responsible for the zero point contribution to the energy, which is the signature of quantization.

Cosmological consequences
In the low-energy limit, the corrections to the background Einstein's equations do not occur at the level of a Friedmann-Lemaître-Robertson-Walker (FLRW) background [4]. One may have naively claimed that this was expected, arguing that in a spatially homogeneous spacetime, the spatial points are equivalent and any non-commutative effects are then expected to vanish. However, this argument does not apply here; the non-commutativity is incorporated in the internal manifold F and the spacetime is a commutative four-dimensional manifold. The coupling between the Higgs field and the background geometry can no longer be neglected once the energies reach the Higgs scale, in which case, the non-minimal coupling of Higgs field to curvature leads to corrections to Einstein's equations even for homogeneous and isotropic cosmological models [4]. The effect of the non-minimal coupling of the Higgs field can be seen in two ways: It leads to an effective gravitational constant, or it increases the Higgs mass [4].
The non-minimal coupling between the Higgs field and the Ricci curvature may turn out to be crucial in early universe cosmology [5,6]. Such a coupling has been introduced ad hoc in the literature, in an attempt to drive inflation through the Higgs field. However, the value of the coupling constant between the scalar field and the background geometry should be dictated by the underlying theory. Actually, even if classically, the coupling between the Higgs field and the Ricci curvature could be set equal to zero, a non-minimal coupling would be induced once quantum corrections in the classical field theory were considered.
We have thoroughly investigated [6] where the Higgs field could play the role of the inflaton leading to a sufficient period of inflation with induced temperature anisotropies, which are in agreement with the current measurements. The Higgs potential is V (H) = λ 0 H 4 − μ 2 0 H 2 , with μ 0 and λ 0 subject to radiative corrections as functions of energy. For large enough values of the Higgs field, the normalized value of μ 0 and λ 0 must be calculated. We have shown [6] that for each value of the top quark mass, there is a value of the Higgs mass, where the effective potential is about to develop a metastable minimum at large values of the Higgs field, and the Higgs potential is locally flattened. Calculating [6] the renormalization of the Higgs self-coupling up to two-loops, we have constructed an effective potential, which fits the renormalization group improved potential around the flat region. There is a very good analytic fit to the Higgs potential around the minimum of the potential [6], given as: The parameter c encodes the appearance of an extremum and depends on the values for top quark mass and Higgs mass. A search in the parameter space using a Monte Carlo chain has shown [6] that even though slow-roll inflation can be realized -a result which does not hold for minimally coupled Higgs field -the resulting ratio of perturbation amplitudes is too large for any experimentally allowed values for the masses of the top quark and the Higgs Boson.
Finally, by considering the energy lost to gravitational radiation by orbiting binaries and requiring the magnitude of deviations from general relativity (GR) obtained within the NCSG context, to be less than the allowed uncertainty in the data, we have imposed [7] an upper limit to the moment f 0 , which is used to specify the initial conditions on the gauge couplings. In particular, by setting β 2 = (5π)/(48Gf 0 ), we have imposed the lower limit as: β > 7.55 × 10 −13 m −1 . This observational constraint may seem weak, however, it is comparable to the existing constraints on similar, ad hoc, additions to GR. Moreover, since the strongest constraint comes from systems with high orbital frequencies, future observations of rapidly orbiting binaries, relatively close to the Earth, could improve it by many orders of magnitude. Thus, by purely astrophysical observations, we were able to constrain the natural length, defined through the f 0 momentum of the cut-off function f , at which the non-commutative effects become dominant.

Conclusions
NCSG offers an elegant and purely geometric interpretation of the SM of electroweak and strong interactions. The doubling of the algebra is an essential element in order to get the gauge fields of the SM. Moreover, the NCSG classical construction carries implicit in its feature of the doubling of the algebra seeds of quantization. The NCSG model lives at unification scale; thus, it provides an excellent framework to address early universe cosmological questions, while to study astrophysical consequences one will have to go beyond the perturbative approach.