What does the Wheeler-DeWitt equation imply about the Schrödinger equation concerning the wave function? The WDW equation is: $\hat{H}(x)|\psi \rangle=0.$ Schrödinger’s time dependent wave function equation says: $$i\hbar \frac{\mathrm d}{\mathrm dt} | \Psi(t)\rangle=\hat{H}|\Psi(t)\rangle.$$
Does it make any sense at all to equate these two wave functions? If Wheeler-DeWitt is true, does that imply $$i\hbar \frac{\mathrm d}{\mathrm dt}|\Psi(t)\rangle=0?$$
I doubt this is important, but the question does intrigue me because I am curious about how the wave function works mathematically and I only recently learned about the Wheeler-DeWitt equation.
Ultimately, what would this mean physically?
 A: The Wheeler deWitt equation (WDE) describes a quantum system that doesn't evolve with respect to a parameter $t$. In quantum theory real physical systems are described by quantum observables, not by parameters, so the parameter time is unobservable. So the fact that the WDE state $|\Psi\rangle$ doesn't evolve with respect to parameter time highlights a problem that already existed. The WDE state is a state in which a clock system $C$ with a time observable $\hat{T}$ is entangled with the rest of the universe:
$$|\Psi\rangle=\sum_t|t\rangle|\psi(t)\rangle,$$
where the $|t\rangle$ states are eigenstates of the time observable, see
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.27.2885
https://arxiv.org/abs/1610.04773
The relative state of the rest of the universe $|\psi(t)\rangle$ evolves over time according to some relevant equation of motion such as the Schrodinger equation. There is a corresponding treatment for the Heisenberg picture:
https://arxiv.org/abs/2108.02771
This understanding of time has been experimentally tested too
https://arxiv.org/abs/1310.4691
https://arxiv.org/abs/1710.00707
