In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is ${\displaystyle \gamma _{ij}}$ and given by

${\displaystyle g_{\mu \nu }\,\mathrm {d} x^{\mu }\,\mathrm {d} x^{\nu }=(-\,N^{2}+\beta _{k}\beta ^{k})\,\mathrm {d} t^{2}+2\beta _{k}\,\mathrm {d} x^{k}\,\mathrm {d} t+\gamma _{ij}\,\mathrm {d} x^{i}\,\mathrm {d} x^{j}.}$

In that equation the Latin indices run over the values 1, 2, 3 and the Greek indices run over the values 1, 2, 3, 4. The three-metric ${\displaystyle \gamma _{ij}}$ is the field, and we denote its conjugate momenta as ${\displaystyle \pi ^{ij}}$. The Hamiltonian is a constraint (characteristic of most relativistic systems)

${\displaystyle \mathcal{H}=\frac{1}{2\sqrt{\mathrm{\gamma <!--\; \gamma \; -->}}}{G}_{ijkl}{\mathrm{\pi <!--\; \pi \; -->}}^{ij}{\mathrm{\pi <!--\; \pi \; -->}}^{kl}-<!--\; -\; -->\sqrt{\mathrm{\gamma <!--\; \gamma \; -->}}{}^{}}$

( 3 ) R = 0 {\displaystyle {\mathcal {H}}={\frac {1}{2{\sqrt {\gamma }}}}G_{ijkl}\pi ^{ij}\pi ^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!R=0}

where ${\displaystyle \gamma =\det(\gamma _{ij})}$ and ${\displaystyle G_{ijkl}=(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})}$ is the Wheeler–DeWitt metric. In index-free notation, the Wheeler–DeWitt metric on the space of positive definite quadratic forms g in three dimensions is

${\displaystyle \operatorname {tr} ((g^{-1}dg)^{2})-(\operatorname {tr} (g^{-1}dg))^{2}.}$

Quantization “puts hats” on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case. Thus we obtain the operator

${\displaystyle \stackrel{^<!--\; ^\; -->}{\mathcal{H}}=\frac{1}{2\sqrt{\mathrm{\gamma <!--\; \gamma \; -->}}}{\stackrel{^<!--\; ^\; -->}{G}}_{ijkl}{\stackrel{^<!--\; ^\; -->}{\mathrm{\pi <!--\; \pi \; -->}}}^{ij}{\stackrel{^<!--\; ^\; -->}{\mathrm{\pi <!--\; \pi \; -->}}}^{kl}-<!--\; -\; -->\sqrt{\mathrm{\gamma <!--\; \gamma \; -->}}{}^{}}$

( 3 ) R ^ . {\displaystyle {\widehat {\mathcal {H}}}={\frac {1}{2{\sqrt {\gamma }}}}{\widehat {G}}_{ijkl}{\widehat {\pi }}^{ij}{\widehat {\pi }}^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!{\widehat {R}}.}

Working in “position space”, these operators are

${\displaystyle {\hat {\gamma }}_{ij}(t,x^{k})\to \gamma _{ij}(t,x^{k})}$

${\displaystyle {\hat {\pi }}^{ij}(t,x^{k})\to -i{\frac {\delta }{\delta \gamma _{ij}(t,x^{k})}}.}$

One can apply the operator to a general wave functional of the metric ${\displaystyle {\widehat {\mathcal {H}}}\Psi [\gamma ]=0}$ where:

${\displaystyle \Psi [\gamma ]=a+\int \psi (x)\gamma (x)dx^{3}+\int \int \psi (x,y)\gamma (x)\gamma (y)dx^{3}dy^{3}+…}$

which would give a set of constraints amongst the coefficients ${\displaystyle \psi (x,y,…)}$. This means the amplitudes for ${\displaystyle N}$ gravitons at certain positions is related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating ${\displaystyle \omega (g)}$ as an independent field so that the wave function is ${\displaystyle \Psi [\gamma ,\omega ]}$.