Spectral action in matrix form
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Regular Article - Theoretical Physics
Spectral action in matrix form Ali H. Chamseddine1,a , John Iliopoulos2,b , Walter D. van Suijlekom3,c 1
Physics Department, American University of Beirut, Beirut, Lebanon Laboratoire de Physique-ENS-CNRS, PSL Research University and Sorbonne Universités, 24 rue Lhomond, 75231 Paris Cedex 05, France 3 Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
2
Received: 2 October 2020 / Accepted: 28 October 2020 © The Author(s) 2020
Abstract Quantization of the noncommutative geometric spectral action has so far been performed on the final component form of the action where all traces over the Dirac matrices and symmetry algebra are carried out. In this work, in order to preserve the noncommutative geometric structure of the formalism, we derive the quantization rules for propagators and vertices in matrix form. We show that the results in the case of a product of a four-dimensional Euclidean manifold by a finite space, could be cast in the form of that of a Yang–Mills theory. We illustrate the procedure for the toy electroweak model.
1 Introduction Active research during the last decade on the noncommutative nature of space-time has enhanced our understanding of the fundamental particles in nature and their interactions (we refer to [3,4,6,9,10,13] for an overview of the literature). The emergent picture is that of a space-time which is a product of continuous four-dimensional manifold with a finite space, determined uniquely from very few physical requirements [2]. In particular, when the order one condition is imposed on the Dirac operator, we obtained all the spectrum of the Standard Model, in addition to the right-handed neutrinos and a scalar singlet whose vev generate Majorana mass to the right-handed neutrinos The fermionic action is of the Dirac type while the classical bosonic action is determined from the spectral action principle. As all gauge and Higgs fields are unified within the same Dirac operator, this resulted in certain relations between the three gauge couplings and also the Higgs coupling. These relations were taken to hold only at some unification scale and the action considered as an a e-mail:
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effective action from the Wilsonian point of view (cf. [6] and references therein). The renormalization program was then carried out according to the rules of perturbative quantum field theory to the final derived form of the action. One immediately recognizes that following this method a vast amount of information is lost because the action is obtained after taking all traces of the matrices of the Clifford and symmetry algebras. Moreover, looking at the component form of the propagators of all the dynamical fields it is not possible to see the unified picture they were derived from. In particular, it is known [8] that in the traditional framework of the
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