| Wheeler–DeWitt equation | |
|---|---|
| Name | Wheeler–DeWitt equation |
| Field | Quantum gravity |
| Introduced | 1960s |
| Author | John Archibald Wheeler; Bryce DeWitt |
Wheeler–DeWitt equation
The Wheeler–DeWitt equation is a formal functional differential equation intended to describe the quantum state of the entire universe by combining principles of quantum mechanics with general relativity. It arises in canonical approaches to quantum gravity and is central to attempts to construct a background-independent theory of gravitational quantum states. The equation matters because it formalizes the notion of a "wavefunction of the universe" and highlights deep conceptual issues such as the problem of time and the role of observables in a diffeomorphism-invariant theory.
The Wheeler–DeWitt equation was developed in work by John Archibald Wheeler and later formalized by Bryce DeWitt in the 1960s, building on the Hamiltonian formulation of general relativity due to Paul Dirac and work by Arnowitt, Deser and Misner (the ADM formalism). In canonical quantization one promotes the canonical variables of the gravitational field, typically the three-metric on a spatial hypersurface, to operators acting on a functional space. The resulting constraint equation, often written HΨ = 0, encodes the vanishing of the Hamiltonian constraint and is interpreted as selecting physical states invariant under time reparametrizations and spacetime diffeomorphisms. The equation links to semiclassical approximations such as the WKB approximation and to path-integral approaches like the Hartle–Hawking state.
Formally the Wheeler–DeWitt equation is a functional differential equation on the space of three-geometries (superspace): H Ψ[g_{ij}, Φ] = 0, where H is the Hamiltonian constraint operator, g_{ij} is the spatial metric on a Cauchy hypersurface, and Φ denotes matter fields (e.g., scalar fields, electromagnetism, or Yang–Mills theory). The construction uses the ADM decomposition of the metric tensor and the conjugate momentum π^{ij}, with canonical commutators promoted following Dirac quantization rules. Regularization and factor-ordering ambiguities arise; choices often reference the DeWitt supermetric on superspace and operators analogous to the Laplace–Beltrami operator. Attempts to make the equation rigorous draw on techniques from functional analysis, spectral theory, and methods used in quantization of constrained systems by Paul Dirac and in the BRST formalism.
Within canonical quantum gravity the Wheeler–DeWitt equation embodies the dynamical content of the theory while maintaining manifest spatial diffeomorphism invariance. It contrasts with covariant approaches such as perturbative quantum gravity and loop quantum gravity; the latter reforms canonical quantization using holonomies and fluxes, producing discrete spectra for geometric operators (area, volume) and modified constraint algebra implementations via the Ashtekar variables and Thiemann's constructions. The Wheeler–DeWitt framework is also compared with the path integral formulation and with effective field theory treatments of gravity developed by researchers at institutions such as Princeton University, Institute for Advanced Study, and CERN.
A central conceptual difficulty is the "problem of time": the Wheeler–DeWitt equation lacks an external time parameter and yields a static constraint HΨ = 0, apparently at odds with the manifestly time-evolving phenomena of quantum mechanics and cosmology. Proposed resolutions include: identifying an internal or relational time via matter degrees of freedom (e.g., a scalar field clock), semiclassical emergence of time through WKB separation into heavy (gravitational) and light (matter) sectors, and relational observables as advocated by Carlo Rovelli and others. These approaches intersect with debates in the foundations of philosophy of physics and with proposals from decoherence theory and the consistent histories program.
Exact solutions of the full Wheeler–DeWitt equation are rare; most progress employs symmetry-reduced models, perturbative expansions, or semiclassical approximations. In minisuperspace models (see below) and in homogeneous cosmologies like Friedmann–Lemaître–Robertson–Walker (FLRW) or Bianchi cosmologies, the functional equation reduces to a finite-dimensional differential equation solvable in many cases. The equation underlies investigations of singularity avoidance, quantum tunneling proposals for universe creation (e.g., the Hartle–Hawking no-boundary proposal and Vilenkin's tunneling proposal), and predictions for primordial cosmic microwave background imprints when combined with inflationary models and effective field methods.
Quantum cosmology uses the Wheeler–DeWitt framework to quantize cosmological degrees of freedom. In the minisuperspace approximation one truncates to a finite set of degrees of freedom (scale factor, homogeneous scalar fields), rendering the Wheeler–DeWitt equation tractable. Seminal work by DeWitt, James Hartle, and Stephen Hawking explored boundary conditions and the wavefunction of the universe. More recent studies embed minisuperspace analyses within string theory motivated landscapes, loop quantum cosmology, or use numerical quantum cosmology techniques developed at universities like Cambridge University, Harvard University, and Stanford University.
Open issues include rigorous definition of the Hamiltonian constraint operator, resolution of factor-ordering and regularization ambiguities, and establishing the semiclassical limit that reproduces classical general relativity and quantum field theory on curved spacetime. Connections to holography, the AdS/CFT correspondence, and nonperturbative approaches remain active research areas. Progress may come from cross-fertilization among canonical methods, spin foam models, asymptotic safety scenarios, and advances in numerical relativity and quantum information tools applied to gravitational systems. Institutional programs at Perimeter Institute, Max Planck Institute for Gravitational Physics (Albert Einstein Institute), and university groups continue to develop both conceptual foundations and calculational techniques related to the Wheeler–DeWitt equation.
Category:Quantum gravity Category:Foundations of quantum mechanics