In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein’s equations. We shall investigate this problem in the de-Broglie-Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein’s equations are derived.

The problem of quantum friction in the framework of Bohmian quantum mechanics is studied. The appropriate equations for such a system is written and solved exactly for some cases. Also two approximate solutions are found which represent the transition of a system from an upper state to the ground state caused by the friction. The physical nature of these solutions are examined.

Brans–Dicke scalar–tensor theory provides a conformal coupling of the scalar field with gravity in Einstein’s frame. This model is equivalent to an interacting quintessence in which dark matter is coupled to dark energy. This provides a natural mechanism to alleviate the coincidence problem. We investigate the dynamics of this model and show that it leads to comparable dark energy and dark matter densities today.