The Cosmological Constant of One-Dimensional Matter Coupled Quantum Gravity is Quantised

TL;DR

This paper demonstrates that in one-dimensional matter-coupled quantum gravity, the cosmological constant must be quantized and physical states are restricted to matter states matching this constant, highlighting unique quantum features.

Contribution

It shows that coupling matter to one-dimensional quantum gravity enforces cosmological constant quantization and constrains physical states, with implications for quantization methods and higher-dimensional theories.

Findings

Cosmological constant is quantized and linked to matter energy spectrum.

Physical states are confined to matter states with energy equal to the cosmological constant.

Gauge fixing approaches face Gribov problems, while projector quantization yields correct results.

Abstract

Coupling any interacting quantum mechanical system to gravity in one (time) dimension requires the cosmological constant to belong to the matter energy spectrum and thus to be quantised, even though the gravity sector is free of any quantum dynamics. Furthermore, physical states are also confined to the subspace of the matter quantum states for which the energy coincides with the value of the cosmological constant. These general facts are illustrated through some simple examples. The physical projector quantisation approach readily leads to the correct representation of such systems, whereas other approaches relying on gauge fixing methods are often plagued by Gribov problems in which case the quantisation rule is not properly recovered. Whether such a quantisation of the cosmological constant as well as the other ensuing consequences in terms of physical states extend to higher dimensional matter-gravity coupled quantum systems is clearly a fascinating open issue.