Unimodular JT gravity and de Sitter quantum cosmology

In this work, we show that a gauge-theoretic description of Jackiw-Teitelboim (JT) gravity naturally yields a Henneaux-Teitelboim (HT) unimodular gravity via a central extension of its isometry group, valid for both flat and curved two-dimensional spacetimes. HT gravity introduces a unimodular time canonically conjugate to the cosmological constant, serving as a physical time in quantum cosmology. By studying the mini-superspace reduction of $\text{HT}_2$ gravity, the Wheeler-DeWitt equation becomes a Schr\"odinger-like equation, giving a consistent and unitary quantum theory. Analysis of the wavefunction's probability density reveals a quantum distribution for the scale factor $a$, offering a quantum perspective on the expansion and contraction of the universe. In this perspective, the possibility of reaching the singular point $a=0$ signals that topology change could occur. Finally, we give a consistent quantum description of unimodular time that aligns seamlessly with Page-Wootters formulation of quantum mechanics, where quantum correlations between unimodular time and JT gravity are studied in $\text{HT}_2$ quantum cosmology.