The Uncanny Precision of the Spectral Action |
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Authors: | Ali H Chamseddine Alain Connes |
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Institution: | 1. Physics Department, American University of Beirut, Beirut, Lebanon 3. I.H.E.S., F-91440, Bures-sur-Yvette, France 2. College de France, 3 rue Ulm, F75005, Paris, France 4. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA
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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 an 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 sphere S
3. 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 S
3 × S
1, 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 a
2n
vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the
higher order terms a
4 and a
6 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. |
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Keywords: | |
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