# 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[1] in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann[2] using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac.[3] 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,

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

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

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

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.