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Cited by 3,385Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

Quantum Theory of Gravity. I. The Canonical Theory

DeWitt, Bryce S.·1967·Physical Review
DOI 10.1103/physrev.160.1113
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Abstract

Following an historical introduction, the conventional canonical formulation of general relativity theory is presented. The canonical Lagrangian is expressed in terms of the extrinsic and intrinsic curvatures of the hypersurface ${x}^{0}=\mathrm{constant}$, and its relation to the asymptotic field energy in an infinite world is noted. The distinction between finite and infinite worlds is emphasized. In the quantum theory the primary and secondary constraints become conditions on the state vector, and in the case of finite worlds these conditions alone govern the dynamics. A resolution of the factor-ordering problem is proposed, and the consistency of the constraints is demonstrated. A 6-dimensional hyperbolic Riemannian manifold is introduced which takes for its metric the coefficient of the momenta in the Hamiltonian constraint. The geodesic incompletability of this manifold, owing to the existence of a frontier of infinite curvature, is demonstrated. The possibility is explored of relating this manifold to an infinite-dimensional manifold of 3-geometries, and of relating the structure of the latter manifold in turn to the dynamical behavior of space-time. The problem is approached through the WKB approximation and Hamilton-Jacobi theory. Einstein's equations are revealed as geodesic equations in the manifold of 3-geometries, modified by the presence of a "force term." The classical phenomenon of gravitational collapse shows that the force term is not powerful enough to prevent the trajectory of space-time from running into the frontier. The as-yet unresolved problem of determining when the collapse phenomenon represents a real barrier to the quantum-state functional is briefly discussed, and a boundary condition at the barrier is proposed. The state functional of a finite world can depend only on the 3-geometry of the hypersurface ${x}^{0}=\mathrm{constant}$. The label ${x}^{0}$ itself is irrelevant, and "time" must be determined intrinsically. A natural definition for the inner product of two such state functionals is introduced which, however, encounters difficulties with negative probabilities owing to the barrier boundary condition. In order to resolve these difficulties, a simplified model, the quantized Friedmann universe, is studied in detail. In order to obtain nonstatic wave functions which resemble a universe evolving, it is necessary to introduce a clock. In order that the combined wave functions of universe-cum-clock be normalizable, it turns out that the periods of universe and clock must be commensurable. Wave packets exhibiting quasiclassical behavior are constructed, and attention is called to the phenomenological character of "time." The innerproduct definition is rescued from its negative-probability difficulties by making use of the fact that probability flows in a closed finite circuit in configuration space. The article ends with some speculations on the uniqueness of the state functional of the actual universe. It is suggested that a viewpoint due to Everett should be adopted in its interpretation.

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Crossref · 10.1103/physrev.160.1113

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  • The wave function of the universe satisfies the Wheeler-DeWitt equation.

    “This wave function satisfies a sort of Schrödinger equation, known as the WheelerDeWitt equation [DeWitt (1967); Wheeler (1968)]:”

    From The cosmological constant problem (1989) · cites this paper as DeWitt (1967)

  • The Wheeler-DeWitt equation is the quantized Hamiltonian constraint of general relativity.

    “the Wheeler-DeWitt equation [1], which is the quantized Hamiltonian constraint of general relativity.”

    From Loop quantum cosmology: I. Kinematics (2000) · cites this paper as [1]

  • Homogeneous minisuperspace models reduce the infinite number of degrees of freedom of general relativity to finitely many.

    “This symmetry condition reduces the infinite number of degrees of freedom of general relativity to finitely many ones for these homogeneous minisuperspace models [1,2]”

    From Loop quantum cosmology: I. Kinematics (2000) · cites this paper as [1]

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Authors
DeWitt, B. S.
Affiliations
University of North Carolina at Chapel Hill (US)
Corresponding author
DeWitt, Bryce S.
Venue
Physical Review
Publisher
American Physical Society (APS)
Published
1967-08-25
Volume
160
Issue
5
Pages
1113-1148
ISSN
0031-899X (ISSN-L)
DOI
10.1103/physrev.160.1113
OpenAlex
W2064185701
Cited by
3,385
References
1
Citations by year
2026: 132 · 2025: 161 · 2024: 171 · 2023: 151 · 2022: 134 · 2021: 134 · 2020: 114 · 2019: 128 · 2018: 135 · 2017: 112 · 2016: 92 · 2015: 98 · 2014: 100 · 2013: 78 · 2012: 93
FWCI
16.91
Type
journal article
Language
en
Open access
closed
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
Physical Review (publishedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
quantum gravity, canonical quantum gravity, Hamiltonian constraint, geodesic incompleteness, WKB approximation, Hamilton-Jacobi theory, gravitational collapse, quantum state of the universe, negative probabilities, intrinsic time, extrinsic curvature

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