Wheeler-deWitt equation

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In theoretical physics, the Wheeler-DeWitt equation is an equation that a wave function of the Universe should satisfy in a theory of quantum gravity. An example of such a wave function is the Hartle-Hawking state.

Simply speaking, the WDW equation says

$\hat{H} |\psi\rangle = 0$

where $\hat{H}$ is the total Hamiltonian constraint in quantized general relativity.

Although symbols $\hat{H}$ and $|\psi\rangle$ may appear familiar, their interpretation in Wheeler-deWitt equation is substantially different from non-relativistic quantum mechanics. $|\psi\rangle$ is no longer a spatial wave function in traditional sense (i.e. a complex-valued function that is defined on a 3-dimensional spacelike surface and normalized to unity). Instead it is a functional of field configurations on all of spacetime. This wave function contains all information about geometry and matter content of the universe. $\hat{H}$ is still an operator that acts on the Hilbert space of wave functions, but it is not the same Hilbert space as in nonrelativistic case, and Hamiltonian no longer determines evolution of the system ( so the Schrodinger equation $\hat{H} |\psi\rangle = i \hbar \partial / \partial t |\psi\rangle$ no longer applies ).

In fact, the principle of general covariance in general relativity implies that global evolution per se does not exist; $t$ is just a label we assign to one of the coordinate axes. Thus, what we think about as time evolution of any physical system is just a gauge transformation, similar to that of QED induced by U(1) local gauge transformation $\psi \rightarrow e^{i\theta(\vec{r} )} \psi$ ( where $\theta(\vec{r})$ plays the role of local time ). The role of Hamiltonian is simply to restrict the space of "kinematical" states of the Universe to that of "physical" states - the ones that follow gauge orbits. For this reason we call it "Hamiltonian constraint." Upon quantization, physical states become wave functions that lie in the kernel of Hamiltonian operator.

In general the Hamiltonian vanishes for a theory with general covariance or time scaling invariance.

If the Eigenstate of the Hamiltonian usually depend on $n x, n y, n z,..$ for the continuous case we have the form of the 'Energy' in terms of a functional:

$E[n]=< \Psi [n] | \mathcal H | \Psi [n] >$