Superselection rule for the cosmological constant in three-dimensional spacetime

TL;DR

The article shows that while mass admits a well-known Bargmann superselection rule in non-relativistic quantum mechanics, a parallel rule exists for the cosmological constant in three-dimensional gravity. By treating $\Lambda$ as a dynamical variable via a 2-form gauge field and analyzing the asymptotic symmetry algebra, the authors derive a Virasoro structure with central charge $c=\frac{3\ell}{2G}$; promoting $c$ to an operator and examining its action reveals that superpositions of distinct $\Lambda$ (equivalently distinct $c$) are forbidden. This provides a concrete, symmetry-based obstruction to coherent $\Lambda$ superpositions in $D=3$ with fixed $G$, linking a dynamical mechanism for $\Lambda$ to global Virasoro charges. The result highlights a deep connection between spacetime asymptotic symmetries and quantum superselection sectors, with potential implications for the role of the cosmological constant in quantum gravity.

Abstract

Efforts to understand the origin of the cosmological constant Λ and its observed value have led to consider it as a dynamical field rather than as a universal constant. Then the possibility arises that the universe, or regions of it, might be in a superposition of quantum states with different values of Λ, so that its actual value would not be definite. There appears to be no argument to rule out this possibility for a generic spacetime dimension D. However, as proved herein, for D=3 there exists a superselection rule that forbids such superpositions. The proof is based on the asymptotic symmetry algebra.