WheelerDeWitt

The Wheeler-DeWitt equation is a foundational equation in canonical quantum gravity, named after John Archibald Wheeler and Bryce DeWitt. It arises from quantizing general relativity in the Arnowitt-Deser-Milnor (ADM) formalism, where spacetime is split into spatial slices. The result is a Hamiltonian constraint that, upon quantization, becomes an operator equation acting on the wavefunctional Ψ[hij(x), φ(x)], which encodes the quantum state of the three-geometry and matter fields. The Wheeler-DeWitt equation is typically written as Ĥ Ψ = 0, expressing a timeless condition on the universal wavefunction.

A defining feature of the equation is the absence of explicit time. Because it treats all spatial configurations on equal footing, Ψ evolves without a preferred external time parameter, leading to the so-called problem of time in quantum gravity. In practice, researchers study simplified models, such as minisuperspace reductions with a finite number of degrees of freedom, to explore conceptual issues and potential cosmological implications.

The equation plays a central role in quantum cosmology and the study of the universe’s initial conditions. It has been used in conjunction with proposals for the universe’s boundary conditions, including the Hartle-Hawking no-boundary proposal and Vilenkin’s tunneling proposal. However, the Wheeler-DeWitt equation is subject to ambiguities, notably in operator ordering and regularization, which affect specific predictions. As a foundational step in the canonical approach to quantum gravity, the equation remains a focal point for discussions of how gravity and quantum mechanics might coexist, even as complete empirical validation remains elusive.