The classical equations of motion of a quantum field theory are obtained by minimizing an action. When we apply this procedure to a gauge theory, there is an interesting issue. The variation of the classical action along a gauge direction is zero. Thus, one of the traditional classical equations that are obtained by such a variation does not actually exist. Alternately, the quantum theory is defined by gauge fixing the classical action. This gauge fixed action is slightly different from the classical action and it can be shown that this procedure also results in some classical equations not existing. The missing equations are the constraint equations of the gauge theory - Coulomb’s law in the case of electromagnetism and the time components of Einstein’s equations in the case of gravity. What does this imply?
We show that even though the constraint equations are not obeyed, the time derivatives of the constraints are automatically satisfied by the quantum evolution. A quantum state that does not obey the constraint can be consistently time evolved. The time evolution of this quantum state would be identical to that of a classical state that contains some “dark” object i.e. dark charge in the case of electromagnetism and dark matter in the case of gravity. But, there is no new physics associated with this “dark” object - it is simply a choice of initial quantum state of the theory. In this talk, I will describe these effects in the simple model of “mini-superspace” i.e. homogeneous quantum cosmology. As a consequence, we will show that one can obtain natural time evolution in quantum cosmology and that the time evolution of the quantum state of the universe need not be described by the Wheeler deWitt equation which requires the universe to be in an energy eigenstate thus forcing the quantum state of the universe to be timeless.