- 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 byBryce DeWitt ref|dewitt in a seminal 1967 paper, and based on earlier work byPeter G. Bergmann ref|bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented byPaul 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 theHartle-Hawking state ,Regge calculus , theWheeler-DeWitt equation andloop quantum gravity .The quantization is based on decomposing the

metric tensor as follows,$g\_\{mu\; u\}dx^\{mu\}dx^\{\; u\}=(-,N^2+eta\_keta^k)dt^2+2eta\_kdx^kdt+gamma\_\{ij\}dx^idx^j$

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 $eta\_k$ 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 $eta^i=gamma^\{ij\}eta\_j$, $gamma=detgamma\_\{ij\}$, where $delta$ is theKronecker delta . Under this decomposition the Einstein-Hilbert Lagrangian becomes, up tototal derivative s,$L=int\; d^3x,Ngamma^\{1/2\}(K\_\{ij\}K^\{ij\}-K^2+\{\}^\{(3)\}R)$

where $\{\}^\{(3)\}R$ is the spatial

scalar curvature computed with respect to the Riemannian metric $gamma\_\{ij\}$and $K\_\{ij\}$ is the extrinsic curvature,:$K\_\{ij\}=frac\{1\}\{2\}N^\{-1\}left(\; abla\_jeta\_i+\; abla\_ieta\_j-frac\{partialgamma\_\{ij\{partial\; au\}\; ight),$where $abla\_i$ denotes covariant differentiation with respect to the metric $gamma\_\{ij\}$. 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 makesgeneral 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, calledsynchronous gauge , is $N=1$ and $eta\_i=0$, although they can, in principle, be chosen to be any function of the coordinates. In this case, the Hamiltonian takes the form:$H=int\; d^3xmathcal\{H\},$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 takingPoisson bracket s 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 $abla\_jpi^\{ij\}=0$. 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 ofgeneral covariance . In quantum field theories, especially in thehamiltonian 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 = 1113–48 | 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 = 129–48 | 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.

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