Spectral action in matrix form

Ali H Chamseddine,John Iliopoulos,Walter D Van Suijlekom

doi:10.1140/epjc/s10052-020-08618-z

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.

Highlights

Consider a noncommutative space (A, H, D, J, γ ) defined as the product of a four-dimensional manifold M with the spectral data C∞ (M), L2 (S), iγ μ∂μ, C, γ5 where C is the charge conjugation operator, times a finite space (AF, HF, DF, JF, γF )
We have reconsidered the renormalization program of the perturbative quantum field theory resulting from the spectral action of a noncommutative space formed as a product of a continuous four-dimensional manifold times a finite space
The basic fields are the Fermi fields defining the Hilbert space and the bosonic fields represented as inner fluctuations of the Dirac operator of the noncommutative space

Summary

Action in matrix form

Consider a noncommutative space (A, H, D, J, γ ) defined as the product of a four-dimensional manifold M with the spectral data C∞ (M) , L2 (S) , iγ μ∂μ, C, γ5 where C is the charge conjugation operator, times a finite space (AF , HF , DF , JF , γF ). The Dirac operator including inner fluctuations is DA = i γ μ∂μ + A where A = aa D, bb , a, b ∈ A, a, b ∈ A0, and where A0 is the opposite algebra formed from elements J a∗ J −1, a ∈ A. We use the Gilkey formulas used in all our previous spectral action calculations (in flat space) a0. The spectral action f D2 gives [1]. We will use a cutoff function and truncate all higher order terms. There is one set of cubic terms that do not vanish, but combine to form a total divergence. We turn our attention to gauge fixing and the Feynman rules that this Lagrangian gives rise to

Gauge fixing and Feynman rules

Feynman rules

The electroweak toy model

Conclusions