The Wheeler–DeWitt equation[1] for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt.
The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity.
In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called "problem of time".
[2] More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables.
In canonical gravity, spacetime is foliated into spacelike submanifolds.
In that equation the Latin indices run over the values 1, 2, 3, and the Greek indices run over the values 1, 2, 3, 4.
The Hamiltonian is a constraint (characteristic of most relativistic systems)
{\displaystyle G_{ijkl}=(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})}
In index-free notation, the Wheeler–DeWitt metric on the space of positive definite quadratic forms g in three dimensions is
Quantization "puts hats" on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case.
One can apply the operator to a general wave functional of the metric
as an independent field, so that the wave function is
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.
Contrary to the general case, the Wheeler–DeWitt equation is well defined in minisuperspaces like the configuration space of cosmological theories.
is the Hamiltonian constraint in quantized general relativity, and
Unlike ordinary quantum field theory or quantum mechanics, the Hamiltonian is a first-class constraint on physical states.
may appear familiar, their interpretation in the Wheeler–DeWitt equation is substantially different from non-relativistic quantum mechanics.
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.
This wave function contains all of the information about the geometry and matter content of the universe.
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 the evolution of the system, so the Schrödinger equation
Various attempts to incorporate time in a fully quantum framework have been made, starting with the "Page and Wootters mechanism" and other subsequent proposals.
[4][5] The reemergence of time was also proposed as arising from quantum correlations between an evolving system and a reference quantum clock system, the concept of system-time entanglement is introduced as a quantifier of the actual distinguishable evolution undergone by the system.
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
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
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.
Upon quantization, physical states become wave functions that lie in the kernel of the Hamiltonian operator.
In general, the Hamiltonian[clarification needed] vanishes for a theory with general covariance or time-scaling invariance.