Wheeler–DeWitt equation

Wheeler–DeWitt equation. The Wheeler–DeWitt equation is a foundational equation in the field of canonical quantum gravity, representing an attempt to apply the principles of quantum mechanics to the entire universe. Formulated in the 1960s by John Archibald Wheeler and Bryce DeWitt, it arises from the quantization of the Einstein field equations within the framework of the Hamiltonian formulation of general relativity. The equation is notable for its lack of an explicit time parameter, leading to profound questions about the nature of time and dynamics in a quantum theory of gravity.

Overview and motivation

The primary motivation for the Wheeler–DeWitt equation was to construct a quantum theory of the gravitational field, unifying the conceptual frameworks of general relativity and quantum mechanics. This effort was part of the broader quantum gravity program pursued by figures like Wheeler at Princeton University and DeWitt at the University of North Carolina. The approach treats the spacetime metric itself as a quantum variable, applying canonical quantization procedures developed for other field theories. Key inspirations included earlier work on the Hamiltonian structure of general relativity by Paul Dirac and the Arnowitt–Deser–Misner formalism. The equation aims to provide the quantum state of the universe, often called the wave function of the universe, a concept later explored in depth by Stephen Hawking and James Hartle.

Mathematical formulation

Mathematically, the Wheeler–DeWitt equation is derived by imposing the constraints of general relativity as operators annihilating the wave function. In the Arnowitt–Deser–Misner decomposition, the Hamiltonian constraint of the classical theory becomes the central operator. The equation is typically written as $\backslash hat\{H\}\; \backslash Psi\; [h\_\{ij\}]\; =\; 0$, where $\backslash hat\{H\}$ is the quantum Hamiltonian constraint operator and $\backslash Psi$ is the wave function of the spacetime geometry, dependent on the three-metric $h\_\{ij\}$ on a spatial hypersurface. The operator involves functional derivatives with respect to the metric and includes a potential term from the Ricci scalar of the three-geometry. A significant technical challenge, known as the factor ordering problem, arises due to the non-commutativity of the metric and momentum operators. The equation is defined on superspace, the space of all possible three-geometries, a concept developed by Wheeler and Charles Misner.

Physical interpretation and challenges

The most striking feature of the Wheeler–DeWitt equation is the absence of an external time parameter, a consequence of the general covariance of general relativity. This leads to the problem of time in quantum gravity: how to recover the familiar notion of time evolution from a seemingly static equation. Interpretations vary, with some suggesting time is an emergent property from correlations between variables within the wave function. Another major challenge is the Hilbert space problem—defining an appropriate inner product on the space of solutions to extract probabilistic predictions. Furthermore, the equation is plagued by mathematical difficulties, including the precise definition of the constraint operator and the handling of ultraviolet divergences. These issues were highlighted in critiques by Steven Weinberg and remain central to research in loop quantum gravity and other approaches.

Relation to other approaches in quantum gravity

The Wheeler–DeWitt equation is the cornerstone of the canonical approach to quantum gravity and is directly related to several modern research programs. It serves as the starting point for loop quantum gravity, developed by Abhay Ashtekar, Lee Smolin, and Carlo Rovelli, where the connection formulation leads to a discrete version of the constraint. The equation also connects to the Hartle–Hawking state in Euclidean quantum gravity and the no-boundary proposal. It stands in contrast to perturbative approaches like string theory, which embeds gravity within a broader quantum framework. The semiclassical approximation of the equation can recover the Einstein field equations, linking it to studies of quantum cosmology and the early universe, as explored by Alexander Vilenkin and others at institutions like the University of Cambridge.

Applications and implications

Despite its formal difficulties, the Wheeler–DeWitt equation has been fruitfully applied in simplified models, particularly in quantum cosmology. The minisuperspace approximation, which restricts the infinite degrees of freedom of superspace, allows for the study of the quantum state of homogeneous universes, such as the Friedmann–Lemaître–Robertson–Walker metric. This has led to insights into the possible quantum avoidance of the Big Bang singularity. The equation underpins discussions of the wave function of the universe and proposals like the no-boundary condition. Its implications extend to fundamental questions in theoretical physics, including the nature of quantum mechanics in closed systems, the origin of the arrow of time, and the information paradox in black hole thermodynamics, topics advanced by researchers like Roger Penrose and Leonard Susskind.

Category:Theoretical physics Category:Equations Category:Quantum gravity