Spectral Action Beyond the Weak-Field Approximation

Abstract

The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality. In the ultraviolet regime, p → ∞, the action decays as 1/p 4 in any even dimension.Recent advances [12,14] in explaining some key features of gravity and Standard Model through the spectral action of noncommutative geometry brought this subject to a focus of interest in theoretical physics. In noncommutative geometry, all information is encoded in a spectral triple (A, H, D), where A is an algebra acting on a Hilbert space H and D is a selfadjoint operator on H which plays the role of a Dirac operator [13,14,19]. In this approach, the action is the so-called spectral action introduced by Chamseddine and Connes [9-11]and defined for Λ ∈ R + which plays the role of a cut-off (and needed to make D/Λ dimensionless) and for a function f such that, of course, f (D 2 /Λ 2 ) is trace-class. In general, one chooses f ≥ 0 since the action Tr f (D 2 /Λ 2 ) ≥ 0 will have the correct sign for an Euclidean action. This action is the appropriate one in the framework of noncommutative geometry to reproduce several physical situations like the Einstein-Hilbert action in gravitation or the Yang-Mills-Higgs action in the standard model of particle physics [14], and the positivity of the function f implies positivity of actions for gravity, Yang-Mills or Higgs couplings, and the Higgs mass term is negative. Till the end of this Section we shall present a non-technical summary of our results to give a more physics-oriented reader a chance to appreciate them without going through the mathematics of the rest of this paper.Let M = R 2m be an even dimensional real plane, d = 2m ≥ 2, endowed with a spin structure given by the spinor bundle S = C 2 m . We denote by D the free Dirac operator and by D A the standard Dirac operator with a gauge connection A acting on the Hilbert space H := L 2 (M, S). We will use the notations and conventions from [25, eq. (3.26)], namely in local coordinates

Spectral dimensions from the spectral action

“…Comparing (10) with the stand expansion of g µν used in perturbation theory, g µν = δ µν + √ 16πG N h µν with G N being Newton's constant, identifies the natural scale for Λ as the Planck mass m Pl = (8πG N ) −1/2 (also see [64] for a related discussion). A lengthy but in principle straightforward calculation [31,32,[60][61][62] then yields the expression for the inverse propagators of the physical fields including the full-momentum dependence…”

“…[31,32] and study the properties of the spectral action beyond the framework of effective field theory. More specifically, we construct the generalized spectral dimension D S (T ) [33][34][35] resulting from the spectral action principle and compare our findings with other approaches to quantum gravity.…”

Spectral action with zeta function regularization

“…Although in the low momentum regime the expansion (2.5) recovers the Standard Model action, the high momentum regime does not contain positive powers of the field derivatives [5,19], exhibiting the structure…”

“…The A i with i ¼ 1; 2; 3 are constants which depend on the details of the function χ, and for typical choices of that function, the three constants are not too different from unity. In the case of the cutoff being the characteristic function of the unit interval, the OðΛ −2 Þ are not present in the asymptotic expansion; however, careful analysis of the situation shows [19] that the anzatz (2.6) coincides with the left-hand side only for momenta smaller than the cutoff Λ.…”

Asymptotic and Exact Expansions of Heat Traces

“…In noncommutative geometry, the heat trace is the cornerstone of bosonic spectral action computations [9,19,[46][47][48]. The large energies expansion of the latter is based on the asymptotic expansion of the heat trace associated with the relevant Dirac operator.…”