Wheeler-deWitt equation

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In theoretical physics, the Wheeler-DeWitt equation is a functional differential equation. It is ill defined in the general case, but very important in theoretical physics, especially in quantum gravity. It is a functional differential equation on the space of three dimensional spatial metrics. The Wheeler-deWitt equation has the form of an operator acting on a wave functional, the functional reduce to a function in cosmology. Contrary to the general case, the Wheeler-deWitt equation is well defined in mini-superspaces like the configuration space of cosmological theories. An example of such a wave function is the Hartle-Hawking state.

Simply speaking, the Wheeler-DeWitt equation says

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

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

Although the symbols $\hat{H}$ and $|\psi\rangle$ may appear familiar, their interpretation in the Wheeler-deWitt equation is substantially different from non-relativistic quantum mechanics. $|\psi\rangle$ is no longer a spatial wave function in the traditional sense of a complex-valued function that is defined on a 3-dimensional space-like surface and normalized to unity. Instead it is a functional of field configurations on all of spacetime. This wave function contains all of the information about the 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 the nonrelativistic case, and the Hamiltonian no longer determines evolution of the system, so the Schrödinger 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 a Hamiltonian is simply to restrict the space of the "kinematic" states of the Universe to that of "physical" states - the ones that follow gauge orbits. For this reason we call it a "Hamiltonian constraint." Upon quantization, physical states become wave functions that lie in the kernel of the Hamiltonian operator.

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

If the eigenstate of the Hamiltonian usually depends on n_x , n_y, n_z, ..., for the continuous case we have the form of the energy in terms of a functional:

$E[n]= \langle \Psi [n] | \mathcal H | \Psi [n] \rangle$