The Conformal Primon Gas at the End of Time

TL;DR

This paper links quantum gravity near singularities to conformal quantum mechanics and automorphic L-functions, revealing a novel prime-number-based gas model that encodes zeros of these functions.

Contribution

It introduces a conformal quantum mechanics framework constrained by modular invariance, connecting automorphic L-functions to a gas of prime-labeled oscillators, extending Julia's primon gas concept.

Findings

Wavefunctions proportional to L-functions along the critical axis

L-functions along positive real axis equal to a gas partition function

Universal features derived from averaging the partition function's logarithm

Abstract

The Belinksy-Khalatnikov-Lifshitz dynamics of gravity close to a spacelike singularity can be mapped, at each point in space separately, onto the motion of a particle bouncing within half the fundamental domain of the modular group. We show that the semiclassical quantisation of this motion is a conformal quantum mechanics where the states are constrained to be modular invariant. Each such state defines an odd automorphic $L$-function. In particular, in a basis of dilatation eigenstates the wavefunction is proportional to the $L$-function along the critical axis and hence vanishes at the nontrivial zeros. We show that the $L$-function along the positive real axis is equal to the partition function of a gas of non-interacting charged oscillators labeled by prime numbers. This generalises Julia's notion of a primon gas. Each state therefore has a corresponding, dual, primon gas with a distinct nontrivial set of chemical potentials that ensure modular invariance. We extract universal features of these theories by averaging the logarithm of the partition function over the chemical potentials. The averaging produces the Witten index of a fermionic primon gas.