Superselection rule for the cosmological constant in three-dimensional spacetime

Efforts to understand the origin of the cosmological constant $\mathrm{\Lambda }$ and its observed value have led to consider it as a dynamical field rather than as a universal constant. The possibility then arises that the universe, or regions of it, might be in a superposition of quantum states with different values of $\mathrm{\Lambda }$ 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.