Canonical quantum gravity

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In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.

The quantization is based on decomposing the metric tensor as follows,

$g_{{\mu \nu }}dx^{\mu }\,dx^{\nu }=(-\,N^{2}+\beta _{k}\beta ^{k})dt^{2}+2\beta _{k}\,dx^{k}\,dt+\gamma _{{ij}}\,dx^{i}\,dx^{j}$

where the summation over repeated indices is implied, the index 0 denotes time $\tau =x^{0}$ , Greek indices run over all values 0, . . ., ,3 and Latin indices run over spatial values 1, . . ., 3. The function $N$ is called the lapse function and the functions $\beta _{k}$ are called the shift functions. The spatial indices are raised and lowered using the spatial metric $\gamma _{{ij}}$ and its inverse $\gamma ^{{ij}}$ : $\gamma _{{ij}}\gamma ^{{jk}}=\delta _{i}{}^{k}$ and $\beta ^{i}=\gamma ^{{ij}}\beta _{j}$ , $\gamma =\det \gamma _{{ij}}$ , where $\delta$ is the Kronecker delta. Under this decomposition the Einstein–Hilbert Lagrangian becomes, up to total derivatives,

$L=\int d^{3}x\,N\gamma ^{{1/2}}(K_{{ij}}K^{{ij}}-K^{2}+{}^{{(3)}}R)$

where ${}^{{(3)}}R$ is the spatial scalar curvature computed with respect to the Riemannian metric $\gamma _{{ij}}$ and $K_{{ij}}$ is the extrinsic curvature,

$K_{{ij}}=-{\frac {1}{2}}({\mathcal {L}}_{{n}}\gamma )_{{ij}}={\frac {1}{2}}N^{{-1}}\left(\nabla _{j}\beta _{i}+\nabla _{i}\beta _{j}-{\frac {\partial \gamma _{{ij}}}{\partial t}}\right),$

where ${\mathcal {L}}$ denotes Lie-differentiation, $n$ is the unit normal to surfaces of constant $t$ and $\nabla _{i}$ denotes covariant differentiation with respect to the metric $\gamma _{{ij}}$ . Note that $\gamma _{{\mu \nu }}=g_{{\mu \nu }}+n_{{\mu }}n_{{\nu }}$ . DeWitt writes that the Lagrangian "has the classic form 'kinetic energy minus potential energy,' with the extrinsic curvature playing the role of kinetic energy and the negative of the intrinsic curvature that of potential energy." While this form of the Lagrangian is manifestly invariant under redefinition of the spatial coordinates, it makes general covariance opaque.

Since the lapse function and shift functions may be eliminated by a gauge transformation, they do not represent physical degrees of freedom. This is indicated in moving to the Hamiltonian formalism by the fact that their conjugate momenta, respectively $\pi$ and $\pi ^{i}$ , vanish identically (on shell and off shell). These are called primary constraints by Dirac. A popular choice of gauge, called synchronous gauge, is $N=1$ and $\beta _{i}=0$ , although they can, in principle, be chosen to be any function of the coordinates. In this case, the Hamiltonian takes the form

$H=\int d^{3}x{\mathcal {H}},$

where

${\mathcal {H}}={\frac {1}{2}}\gamma ^{{-1/2}}(\gamma _{{ik}}\gamma _{{jl}}+\gamma _{{il}}\gamma _{{jk}}-\gamma _{{ij}}\gamma _{{kl}})\pi ^{{ij}}\pi ^{{kl}}-\gamma ^{{1/2}}{}^{{(3)}}R$

and $\pi ^{{ij}}$ is the momentum conjugate to $\gamma _{{ij}}$ . Einstein's equations may be recovered by taking Poisson brackets with the Hamiltonian. Additional on-shell constraints, called secondary constraints by Dirac, arise from the consistency of the Poisson bracket algebra. These are ${\mathcal {H}}=0$ and $\nabla _{j}\pi ^{{ij}}=0$ . This is the theory which is being quantized in approaches to canonical quantum gravity.

All canonical theories of general relativity have to deal with the problem of time. In short, in general relativity, time is just another coordinate as a result of general covariance. In quantum field theories, especially in the Hamiltonian formulation, the formulation is split between three dimensions of space, and one dimension of time.

Sources and notes

Arnowitt, R.; Deser, S.; Misner, C. W. (2008). "The Dynamics of General Relativity". General Relativity and Gravitation 40 (9): 1997–2027. arXiv:gr-qc/0405109. Bibcode:2008GReGr..40.1997A. doi:10.1007/s10714-008-0661-1.
- Originally from Witten, L. (1962). Gravitation: An Introduction to Current Research. John Wiley & Sons. pp. 227–265.
^ Bergmann, P. (1966). "Hamilton–Jacobi and Schrödinger Theory in Theories with First-Class Hamiltonian Constraints". Physical Review 144 (4): 1078–1080. Bibcode:1966PhRv..144.1078B. doi:10.1103/PhysRev.144.1078.
^ Dewitt, B. (1967). "Quantum Theory of Gravity. I. The Canonical Theory". Physical Review 160 (5): 1113–1148. Bibcode:1967PhRv..160.1113D. doi:10.1103/PhysRev.160.1113.
^ Dirac, P. A. M. (1958). "Generalized Hamiltonian Dynamics". Proceedings of the Royal Society of London A 246 (1246): 326–332. Bibcode:1958RSPSA.246..326D. doi:10.1098/rspa.1958.0141. JSTOR 100496.
Dirac, P. A. M. (1958). "The Theory of Gravitation in Hamiltonian Form". Proceedings of the Royal Society of London A 246 (1246): 333–343. Bibcode:1958RSPSA.246..333D. doi:10.1098/rspa.1958.0142. JSTOR 100497.
Dirac, P. A. M. (1959). "Fixation of Coordinates in the Hamiltonian Theory of Gravitation". Physical Review 114 (3): 924–930. Bibcode:1959PhRv..114..924D. doi:10.1103/PhysRev.114.924.
Dirac, P. A. M. (1964). Lectures on quantum mechanics. Yeshiva University. ISBN 0-387-51916-5.

Theories of gravitation

Standard

Newtonian gravity (NG)	Newton's law of universal gravitation History of gravitational theory

General relativity (GR)	Introduction History Mathematics Resources Tests Post-Newtonian formalism Linearized gravity ADM formalism

Alternatives to
general relativity

Paradigms	Classical theories of gravitation Quantum gravity Theory of everything

Early modifications	Einstein–Cartan Bimetric theories Gauge theory gravity Teleparallelism Composite gravity f(R) gravity Massive gravity Modified Newtonian dynamics (MOND) Nonsymmetric gravitation Scalar–tensor theories Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories Nordström Whitehead Geometrodynamics Induced gravity Tensor–vector–scalar

Quantisation attempts	Euclidean quantum gravity Canonical Quantum Gravity Wheeler–DeWitt equation Loop quantum gravity Causal dynamical triangulation Causal sets DGP model

Unification attempts	Kaluza–Klein theory Dilaton Supergravity

Unification and quantisation attempts	Noncommutative geometry Self-creation cosmology Semiclassical gravity Superfluid vacuum theory Logarithmic BEC vacuum String theory M-theory F-theory Heterotic string Type I string theory Type 0 string theory Bosonic string theory Type II string theory Little string theory Twistor theory Twistor String theory

Generalisations/Extensions of GR	Liouville gravity Lovelock theory 2+1D topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

Pre-Newtonian theories
and Toy models

Quantum gravity

Central concepts

Black holes

Quantum field theory
in curved spacetime

Approaches

String theory	Bosonic string theory M-theory Supergravity Superstring theory

Canonical quantum gravity	Loop quantum gravity Wheeler–DeWitt equation

Euclidean quantum gravity	Hartle–Hawking state

Others	Causal dynamical triangulation Causal sets Noncommutative geometry Spin foam Group field theory Superfluid vacuum theory Twistor theory

Toy models

Applications

Quantum cosmology	Eternal inflation Multiverse FRW/CFT duality

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Canonical quantum gravity

See also

Sources and notes