The Cosmological constant of one-dimensional matter coupled quantum gravity is quantised

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. Comment: Dedicated to John R. Klauder on the occasion of his 70th birthday. To appear in the Proceedings of the Third International Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3), 1st-7th November 2003, Cotonou (Republic of Benin), 29 pages