The minimum supersymmetric standard model on noncommutative geometry

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’.

We review a gravitational model based on noncommutative geometry and the spectral action principle. The space-time geometry is described by the tensor product of a four-dimensional Riemanian manifold by a discrete noncommutative space consisting of only two points. With a specific choice of the finite dimensional involutive algebra, the noncommutative spectral action leads to the standard model of electroweak and strong interactions minimally coupled to Einstein and Weyl gravity. We present the main mathematical ingredients of this model and discuss their physical implications. We argue that the doubling of the algebra is intimately related to dissipation and the gauge field structure. We then show how this noncommutative spectral geometry model, a purely classical construction, carries implicit in the doubling of the algebra the seeds of quantization. After a short review on the phenomenological consequences of this geometric model as an approach to unification, we discuss some of its cosmological consequences. In particular, we study deviations of the Friedmann equation, propagation of gravitational waves, and investigate whether any of the scalar fields in this model could play the role of the inflaton.

Noncommutative geometric spaces with boundary: Spectral action

On the definition of spacetimes in noncommutative geometry: Part I

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Caratheodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes's distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator ? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Caratheodory distance dh defined by A. In this paper we precise this link, showing that the equality of d and dh strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).

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.

I give a summary review of the research program using noncommutative geometry as a framework to determine the structure of space-time. Classification of finite noncommutative spaces under few assumptions reveals why nature chose the Standard Model and the reasons behind the particular form of gauge fields, Higgs fields and fermions as well as the origin of symmetry breaking. It also points that at high energies the Standard Model is a truncation of Pati-Salam unified model of leptons and quarks. The same conclusions are arrived at uniquely without making any assumptions except for an axiom which is a higher form of Heisenberg commutation relations quantizing the volume of space-time. We establish the existence of two kinds of quanta of geometry in the form of unit spheres of Planck length. We provide answers to many of the questions which are not answered by other approaches, however, more research is needed to answer the remaining challenging questions.

Multiplication operators on non-commutative spaces

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.

It is shown that the Dirac operator becomes 2 ×2 matrix acting on the space of chiral fermions in the previously-proposed quaternionic formulation of the Dirac theory. The interaction part of thus-obtained Dirac operator formally corresponds to the 2 ×2 matrix of the generalized one-form of Coquereaux et al. in the non-commutative geometric approach to the electro-weak unification. It is pointed out that the mathematical formulation of Coquereaux et al. allows to introduce an arbitrary parameter which can be reduced to unity unless it vanishes and whose presence spoils a unique construction of Yang-Mills-Higgs theory in their approach. For instance, one may choose the parameter to vanish, resulting in only pure Yang-Mills theory. It is also pointed out that the original model construction of Coquereaux et al. corresponds to employ a quaternion algebra which is isomorphic to but totally independent of that used in the Dirac theory. According to this model construction the Higgs boson mass is predicted to be the same as that of the gauge boson in a toy model with U(1) ×U(1) down to U(1). Bosonic sector of the Weinberg-Salam theory is also constructed along the same line using quaternions with components being 2 ×2 matrices. The construction is unique without recourse to SU(2|1) and predicts mH = √2mW and sin 2θW = 1/4 at the classical level.

A note on superfields and noncommutative geometry

We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class (HC) of quasi-normed symmetric sequence or function spaces E for which their Banach envelopes \(\widehat{E}\) are also symmetric spaces. The class of symmetric spaces satisfying (HC) contains but is not limited to order continuous spaces. Let \(\mathcal {M}\) be a non-atomic, semifinite von Neumann algebra with a faithful, normal, \(\sigma \)-finite trace \(\tau \) and E be as symmetric function space on \([0,\tau (\mathbf 1 ))\) or symmetric sequence space. We compute Banach envelope norms on \(E(\mathcal {M},\tau )\) and \(C_E\) for any quasi-normed symmetric space E. Then we show under assumption that \(E\in (HC)\) that the Banach envelope \(\widehat{E(\mathcal {M},\tau )}\) of \(E(\mathcal {M},\tau )\) is equal to \(\widehat{E}\left( \mathcal {M},\tau \right) \) isometrically. We also prove the analogous result for unitary matrix spaces \(C_E\).

pseudo-Riemannian space, conformal Fedosov structure, recurrent space, symmetric space, quantum key distribution, single-photon regime, special relativity, fuzzy space, additive quantum number, conservation laws, electroweak theory, symmetry, unified field theories, Lie-Poisson algebra, quantization, matrix regularization, noncommutative geometry

To generalize the Fisher Discriminant Analysis (FDA) algorithm to the case of discriminant functions belonging to a nonlinear, finite dimensional function space F (Nonlinear FDA or NFDA), it is sufficient to expand the input data by computing the output of a basis of F when applied to it [1,2,3,4]. The solution to NFDA can then be found like in the linear case by solving a generalized eigenvalue problem on the between- and within-classes covariance matrices (see e.g. [5]). The goal of NFDA is to find linear projections of the expanded data (i.e., nonlinear transformations of the original data) that minimize the variance within a class and maximize the variance between different classes. Such a representation is of course ideal to perform classification. The application of NFDA to pattern recognition is particularly appealing, because for a given input signal and a fixed function space it has no parameters and it is easy to implement and apply. Moreover, given C classes only C - 1 projections are relevant [5]. As a consequence, the feature space is very small and the algorithm has low memory requirements and high speed during recognition. Here we apply NFDA to a handwritten digit recognition problem using the MNIST database, a standard and freely available set of 70,000 handwritten digits (28 × 28 pixels large), divided into a training set (60,000 digits) and a test set (10,000 digits). Several established pattern recognition methods have been applied to this database by Le Cun et al. [6]. Their paper provides a standard reference work to benchmark new algorithms. We perform NFDA on spaces of polynomials of a given degree d, whose corresponding basis functions include all monomials up to order d in all input variables. It is clear that the problem quickly becomes intractable because of the high memory requirements. For this reason, the input dimensionality is first reduced by principal component analysis. On the preprocessed data we then apply NFDA by expanding the training patterns in the polynomial space and solving the linear FDA eigenvalue problem. As mentioned above, since we have 10 classes we only need to compute the first 9 eigenvectors. Since the within-class variance is minimized, the patterns belonging to different classes tend to cluster in the feature space when projected on the eigenvectors. For this reason we classify the digits with a simple method such as Gaussian classifiers. We perform simulations with polynomials of degree 2 to 5. With polynomials of degree 2 the explosion in the dimensionality of the expanded space with increasing number of input dimensions is relatively restricted, so that it is possible to use up to 140 dimensions. With higher order polynomials one has to rely on a smaller number of input dimensions, but since the function space gets larger and includes new nonlinearities, one obtains a remarkable improvement in performance. The best performance is achieved with polynomials of degree 3 and 35 input dimensions, with an error rate of 1.5% on test data. This error rate is comparable to but does not outperform that of the most elaborate algorithms (Table 1). The performance of NFDA is however remarkable considering the simplicity of the method and the fact that it has no a priori knowledge on the problem, in contrast for example to the LeNet-5 algorithm [6] which has been designed specifically for handwritten character recognition. In addition, for recognition, NFDA has to store and compute only 9 functions and has thus small memory requirements and a high recognition speed. It is also possible to formulate NFDA using the kernel trick, in which case one can in principle use function spaces of infinite dimensionality [1,2,3,4]. However, the limiting factor in that formulation is the number of training patterns, which makes it not realistic for this application. The performance of NFDA could be further improved using for example a more problem-specific preprocessing of the patterns (e.g., by increasing the size of the training set with new patterns generated by artificial distortion of the original one), boosting techniques, or mixture of experts with other algorithms [5,6].

A general class of infinite dimensional Dirac operators and path integral representation of their index