# Canonical general relativity

﻿
Canonical general relativity

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 DeWittref|dewitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmannref|bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac.ref|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,

where the summation over repeated indices is implied, the index 0 denotes time $au=x^0$, 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 are called the shift functions. The spatial indices are raised and lowered using the spatial metric $gamma_\left\{ij\right\}$ and its inverse $gamma^\left\{ij\right\}$: $gamma_\left\{ij\right\}gamma^\left\{jk\right\}=delta_i\left\{\right\}^k$ and , $gamma=detgamma_\left\{ij\right\}$, where $delta$ is the Kronecker delta. Under this decomposition the Einstein-Hilbert Lagrangian becomes, up to total derivatives,

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

where $\left\{\right\}^\left\{\left(3\right)\right\}R$ is the spatial scalar curvature computed with respect to the Riemannian metric $gamma_\left\{ij\right\}$and $K_\left\{ij\right\}$ is the extrinsic curvature,:where $abla_i$ denotes covariant differentiation with respect to the metric $gamma_\left\{ij\right\}$. 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 , although they can, in principle, be chosen to be any function of the coordinates. In this case, the Hamiltonian takes the form:$H=int d^3xmathcal\left\{H\right\},$where:$mathcal\left\{H\right\}=frac\left\{1\right\}\left\{2\right\}gamma^\left\{-1/2\right\}\left(gamma_\left\{ik\right\}gamma_\left\{jl\right\}+gamma_\left\{il\right\}gamma_\left\{jk\right\}-gamma_\left\{ij\right\}gamma_\left\{kl\right\}\right)pi^\left\{ij\right\}pi^\left\{kl\right\}-gamma^\left\{1/2\right\}\left\{\right\}^\left\{\left(3\right)\right\}R$and $pi^\left\{ij\right\}$ is the momentum conjugate to $gamma_\left\{ij\right\}$. 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 $mathcal\left\{H\right\}=0$ and $abla_jpi^\left\{ij\right\}=0$. This is the theory which is being quantized in approaches to canonical quantum gravity.

canonical general relativity all have to deal with the problem of time. In short, in general relativity, time is just another coordinated as a result of general covariance. In quantum field theories, especially in the hamiltonian formulation, the formulation is split between 3 dimensions of space, and 1 dimension of time.

ee also

*Loop quantum gravity is one of this family of theories.
*Loop quantum cosmology (LQC) is a finite, symmetry reduced model of loop quantum gravity.

ources and notes

# cite journal|author = B. S. DeWitt | title = Quantum theory of gravity. I. The canonical theory | journal = Phys. Rev. | volume = 160 | pages = 1113&ndash;48 | year = 1967 | doi = 10.1103/PhysRev.160.1113 ( [http://prola.aps.org/abstract/PR/v160/i5/p1113_1 pdf] )
# see, "e.g." P. G. Bergmann, [http://prola.aps.org/abstract/PR/v144/i4/p1078_1 Hamilton-Jacobi and Schrödinger Theory in Theories with First-Class Hamiltonian Constraints ] "Helv. Phys. Acta Suppl." 4, 79 (1956) and references.
# cite journal|author = P. A. M. Dirac | title = Generalized Hamiltonian dynamics | journal = Can. J. Math. | volume = 2 | pages = 129&ndash;48 | year = 1950 ( [http://www.jstor.org/stable/100496 link] ) cite book|author = P. A. M. Dirac | title = Lectures on quantum mechanics | publisher = Yeshiva University | location = New York | year = 1964
#P. A. M. Dirac [http://prola.aps.org/abstract/PR/v114/i3/p924_1 Fixation of Coordinates in the Hamiltonian Theory of Gravitation] "Phys. Rev." 114, 924 (1959)
#P. A. M. Dirac [http://www.jstor.org/view/00804630/ap000952/00a00050/0 The Theory of Gravitation in Hamiltonian Form] "Proc. Roy. Soc." (London) A246, 333 (1958)
#R. Arnowitt, S. Deser. and C, W, Misner [http://arxiv.org/abs/gr-qc/0405109 The Dynamics of General Relativity] from "Gravitation: an introduction to current research", Louis Witten ed. (Wiley 1962), chapter 7, pp 227--265.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• General relativity — For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. General relativity Introduction Mathematical formulation Resources …   Wikipedia

• Golden age of general relativity — The Golden Age of General Relativity is the period roughly from 1960 to 1975 during which the study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics. During this… …   Wikipedia

• Doubly special relativity — (DSR) also called deformed special relativity or, by some, extra special relativity is a modified theory of special relativity in which there is not only an observer independent maximum velocity (the speed of light), but an observer independent… …   Wikipedia

• Doubly-special relativity — (DSR) also called deformed special relativity or, by some, extra special relativity is a modified theory of special relativity in which there is not only an observer independent maximum velocity (the speed of light), but an observer independent… …   Wikipedia

• History of loop quantum gravity — General relativity is the theory of gravitation published by Albert Einstein in 1915. According to it, the force of gravity is a manifestation of the local geometry of spacetime. Mathematically, the theory is modelled after Bernhard Riemann s… …   Wikipedia

• Loop quantum gravity — Not to be confused with the path integral formulation of LQG, see spin foam. This article is about LQG in its Canonical formulation.. Beyond the Standard Model …   Wikipedia

• Quantum gravity — is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature (electromagnetism, weak interaction, and strong interaction), with general relativity, the theory of the fourth… …   Wikipedia

• Stress-energy tensor — The stress energy tensor (sometimes stress energy momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of …   Wikipedia

• Horace Hearne Institute — The Horace Hearne Jr. Institute for Theoretical Physics is at Louisiana State University. The Hearne Institute is funded by a donation of two endowed chairs by Horace Hearne Jr. and the State of Louisiana, as well as additional grants from a… …   Wikipedia

• Metric tensor — In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a… …   Wikipedia