Cosmological-Billiards Groups and self-adjoint BKL Transfer Operators

TL;DR

This paper explores the mathematical structure of cosmological billiards in general relativity, classifies their periodic orbits, and introduces a self-adjoint semiclassical transfer operator that differs from traditional modular group analysis, offering new insights into quantum chaos.

Contribution

It introduces a self-adjoint semiclassical transfer operator for cosmological billiards, contrasting with the modular group approach, and advances understanding of quantum BKL maps and their spectra.

Findings

Classified periodic orbits of cosmological billiards.

Established the self-adjoint nature of the semiclassical transfer operator.

Outlined differences between billiard groups and quantum spectra.

Abstract

Cosmological billiards arise as a map of the solution of the Einstein equations, when the most general symmetry for the metric tensor is hypothesized, and points are considered as spatially decoupled in the asymptotic limit towards the cosmological singularity, according to the BKL (Belinski Khalatnikov Lifshitz) paradigm. In $4=3+1$ dimensions, two kinds of cosmological billiards are considered: the so-called 'big billiard' which accounts for pure gravity, and the 'small billiard', which is a symmetry-reduced version of the previous one, and is obtained when the 'symmetry walls' are considered. The solution of Einstein field equations is this way mapped to the (discrete) Poincar\'e map of a billiard ball on the sides of a triangular billiard table, in the Upper Poincar\'e Half Plane (UPHP). The billiard modular group is the scheme within which the dynamics of classical chaotic systems on surfaces of constant negative curvature is analyzed. The periodic orbits of the two kinds of billiards are classified, according to the different symmetry-quotienting mechanisms. The differences with the description implied by the billiard modular group are investigated and outlined. In the quantum regime, the eigenvalues (i.e. the sign that wavefunctions acquire according to quantum BKL maps) for periodic phenomena of the BKL maps on the Maass wavefunctions are classified. The complete spectrum of the semiclassical operators which act as BKL map for periodic orbits is obtained. Differently form the case of the modular group, here it is shown that the semiclassical transfer operator for Cosmological Billiards is not only the adjoint operator of the one acting on the Maass waveforms, but that the two operators are the same \it{self-adjoint} operator, thus outlining a different approach to the Langlands Jaquet correspondence.