Wheeler–deWitt equation
The Wheeler–DeWitt equation is a field equation. It is part of a theory that attempts to combine mathematically the ideas of quantum mechanics and general relativity, a step toward a theory of quantum gravity. In this approach, time plays no role in the equation, leading to the problem of time. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergmann-Komar "group" (which is the diffeomorphism group on-shell, but differs off-shell).
All defined and understood descriptions of string/M-theory deal with fixed asymptotic conditions on the background spacetime. At infinity, the "right" choice of the time coordinate "t" is determined (because the space-time is asymptotic to some fixed space-time) in every description, so there is a preferred definition of the Hamiltonian (with nonzero eigenvalues) to evolve states of the system forward in time. This avoids all the need to dynamically generate a time dimension using the Wheeler-de Witt equation. Thus, the equation has not played a role in string theory thus far.
There could exist a Wheeler-de Witt style manner to describe the bulk dynamics of quantum theory of gravity. Some experts believe that this equation still holds the potential for understanding quantum gravity; however, decades after the equation was published, completely different approaches, such as string theory, have brought physicists as clear results about quantum gravity.
In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is and given by
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