Quantum Cosmology and Conformal Invariance

TL;DR

This paper links quantum cosmology near singularities to conformal mechanical models, revealing new models based on nilpotent coadjoint orbits and their relation to the conformal group SO(1,2).

Contribution

It identifies the connection between Belinsky-Khalatnikov-Lifshitz models and DFF conformal mechanics, and introduces new conformal quantum models based on nilpotent coadjoint orbits.

Findings

Gravity near singularities reduces to conformal mechanical models.

Deformation with negative cosmological constant yields discrete spectra.

New conformal models based on ADE non-compact groups are constructed.

Abstract

According to Belinsky, Khalatnikov and Lifshitz, gravity near a space-like singularity reduces to a set of decoupled one-dimensional mechanical models at each point in space. We point out that these models fall into a class of conformal mechanical models first introduced by de Alfaro, Fubini and Furlan (DFF). The deformation used by DFF to render the spectrum discrete corresponds to a negative cosmological constant. The wave function of the universe is the zero-energy eigenmode of the Hamiltonian, also known as the spherical vector of the representation of the conformal group SO(1,2). A new class of conformal quantum mechanical models is constructed, based on the quantization of nilpotent coadjoint orbits, where the conformal group is enhanced to an ADE non-compact group for which the spherical vector is known.