The Uncanny Precision of the Spectral Action

TL;DR

This paper demonstrates that the spectral action in noncommutative geometry accurately encodes gravitational dynamics, with the leading terms providing near-perfect approximations and higher-order corrections vanishing due to cancellations.

Contribution

It shows that the spectral action's asymptotic expansion effectively captures gravity and the Higgs potential, with higher-order terms negligible or vanishing, confirming its suitability as a fundamental action functional.

Findings

The cosmological constant and scalar curvature terms suffice for accurate spectral action approximation.

Higher order terms in the spectral action vanish or are negligible due to cancellations.

The Higgs potential emerges as an exact perturbation with a smooth cutoff function.

Abstract

Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as action functional for the theory. To demonstrate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic expansion in the case of the round three sphere. We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of the product of the three sphere by a circle where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a4 and a6 both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function.