Abstract
A rationality result previously proved for Robertson-Walker metrics is extended to a homogeneous anisotropic cosmological model, namely the Bianchi type-IX minisuperspace. It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors w_1(t), w_2(t), w)3(t), and their higher derivates with respect to time. We begin with the computation of the Dirac operator of this geometry and calculate the coefficients a_0 ,a_2 ,a_4 of the spectral action by using heat kernel methods and parametric pseudodifferential calculus. An efficient method is devised for computing the Seeley-de Witt coefficients of a geometry by making use of Wodzicki’s noncommutative residue, and it is confirmed that the method checks out for the cosmological model studied in this article. The advantages of the new method are discussed, which combined with symmetries of the Bianchi type-IX metric, yield an elegant proof of the rationality result.
Highlights
In quantum cosmology, according to the Hartle-Hawking proposal, a path integral approach is formulated in terms of a sum over 4-dimensional geometries, with an action functional determined by the Einstein-Hilbert action for Euclidean gravity on the 4-dimensional geometries
It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors w1(t), w2(t), w3(t), and their higher derivates with respect to time
An efficient method is devised for computing the Seeley-de Witt coefficients of a geometry by making use of Wodzicki’s noncommutative residue, and it is confirmed that the method checks out for the cosmological model studied in this article
Summary
The heat kernel method that uses pseudodifferential calculus for computing the Seeleyde Witt coefficients of an elliptic positive operator on a compact manifold relies on the pseudodifferential symbol of the operator in local charts, see chapter 1 of the book [30]. Since ∇S is the lift of the Levi-Civita connection ∇ to the spin bundle, one starts with computing the matrix of 1-forms ω = (ωba) such that ∇ = d + ω in terms of θa in a local chart x = (xμ) ∈ U , which can be lifted to the matrix of the spin connection 1-forms by making use of the Lie algebra isomorphism μ : so(m) → spin(m) given by μ(A). The Dirac operator is a differential operator of order 1, and using the Fourier inversion formula, its action on a spinor s written in the chosen local chart U can be expressed by. Ξm) ∈ Rm. In general, the effect of change of coordinates on pseudodifferential symbols suggests that it is natural to consider Rm as the cotangent fibre at point x, see for example Lemma 1.3.2 on page 24 of [30]. The specific expressions for the pk(x, ξ) are given in the appendix A
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