Wheeler–DeWitt equation

In theoretical physics, the Wheeler–DeWitt equation[1] is an equation for which the solution is a wave function defined on the space of three dimensional metrics and physical matter fields. A solution, \( |\psi\rangle \), to the equation can be interpreted as providing the amplitude for a particular three dimensional geometry to form the boundary for a class of 4-space metrics in a manifold, M. The wave function \( |\psi\rangle \) depends only on the boundary 3-geometry as specified by a spatial metric \( h_{ij} \) and - optionally - on a configuration of matter fields, \( \phi \). Notably the wave function does not depend on time. Using the wave function we can furthermore calculate the probability that a given 3-metric + matter field configuration exists in the manifold as \( \langle\psi |\psi\rangle \).

I.e. the wave function that is a solution to the Wheeler–DeWitt equation can provide us with a way to test our models of the universe; for example by considering a particular physical configuration given by a h_{ij} and \phi and comparing with observations. A solution that describes our universe today, for example, should be able to accurately model the observed variation in the cosmic background radiation. Solutions to the Wheeler-DeWitt equation have therefore been interpreted as the Universal wave function.

Motivation and background

The Wheeler–DeWitt equation can be derived from a path integral using the gravitational action

\( \Psi[g_{\mu\nu},\phi] = \int\limits_{C}d[g_{\mu\nu}]d[\phi]e^{-I[g_{\mu\nu},\phi]} \)

were one integrates over a class of four-metrics and matter fields matching certain boundary conditions. Because the concept of a universal time coordinate seems unphysical, and at odds with the principles of general relativity, the action is evaluated around a 3-metric which we take as the boundary of the classes of four-metrics and on which a certain configuration of matter fields exists. This latter might for example be the current configuration of matter in our universe as we observe it today. Evaluating the action so that it only depends on the 3-metric and the matter fields is sufficient to remove the need for a time coordinate as it effectively fixes a point in the evolution of the universe.

It is also important to note that the action integral is taken over so called Euclidean metrics; i.e. metrics with signature (+,+,+,+) that are obtained from Lorentzian metrics - with signature (-,+,+,+) - by performing a Wick rotation of the time coordinate t \rightarrow it. The reason for this choice is that these, Euclidean, metrics are compact and therefore have no boundary at infinity on which we need to fix values of the wave function and its derivatives. It has been proposed that is is less likely that the universe should be asymptotically flat or have some sort of boundary at infinity since we, the observer, are unable to leave the universe to “look in” and hence consider the wave function, and action, in the same way as we would for example consider it in a scattering experiment in traditional quantum mechanics where there are definite non-interacting “in” and “out” states. This forms the basis for Hawking and Hartle’s no boundary proposal and is the motivator for choosing compact (no boundary) metrics when evaluating the action integral.
Mathematical formalism

The Wheeler–DeWitt equation[2] 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 reduces 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. Bryce DeWitt first published this equation in 1967 under the name "Einstein–Schrödinger equation"; it was later renamed the "Wheeler–DeWitt equation".[3]

Simply speaking, the Wheeler–DeWitt equation says

\( \hat{H}(x) |\psi\rangle = 0 \)

where \( \hat{H}(x) \) is the Hamiltonian constraint in quantized general relativity. Unlike ordinary quantum field theory or quantum mechanics, the Hamiltonian is a first class constraint on physical states. We also have an independent constraint for each point in space.

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. This property is known as timelessness . The reemergence of time requires the tools of decoherence and clock operators.

We also need to augment the Hamiltonian constraint with momentum constraints

\( \vec{\mathcal{P}}(x) \left| \psi \right\rangle = 0 \)

associated with spatial diffeomorphism invariance.

In minisuperspace approximations, we only have one Hamiltonian constraint (instead of infinitely many of them).

In fact, the principle of general covariance in general relativity implies that global evolution per se does not exist; the time 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.

The Wheeler-DeWitt Equation - Renate Loll