# Wheeler–DeWitt equation (motivation and background)

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

$g_{\mu \nu }\,\mathrm {d} x^{\mu }\,\mathrm {d} x^{\nu }=(-\,N^{2}+\beta _{k}\beta ^{k})\,\mathrm {d} t^{2}+2\beta _{k}\,\mathrm {d} x^{k}\,\mathrm {d} t+\gamma _{ij}\,\mathrm {d} x^{i}\,\mathrm {d} x^{j}.$ 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 $\gamma _{ij}$ is the field, and we denote its conjugate momenta as $\pi ^{ij}$ . The Hamiltonian is a constraint (characteristic of most relativistic systems) where $\gamma =\det(\gamma _{ij})$ and $G_{ijkl}=(\gamma _{ik}\gamma _{jl}+\gamma _{il}\gamma _{jk}-\gamma _{ij}\gamma _{kl})$ 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

$\operatorname {tr} ((g^{-1}dg)^{2})-(\operatorname {tr} (g^{-1}dg))^{2}.$ 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 Working in “position space”, these operators are

${\hat {\gamma }}_{ij}(t,x^{k})\to \gamma _{ij}(t,x^{k})$ ${\hat {\pi }}^{ij}(t,x^{k})\to -i{\frac {\delta }{\delta \gamma _{ij}(t,x^{k})}}.$ One can apply the operator to a general wave functional of the metric ${\widehat {\mathcal {H}}}\Psi [\gamma ]=0$ where:

$\Psi [\gamma ]=a+\int \psi (x)\gamma (x)dx^{3}+\int \int \psi (x,y)\gamma (x)\gamma (y)dx^{3}dy^{3}+…$ which would give a set of constraints amongst the coefficients $\psi (x,y,…)$ . This means the amplitudes for $N$ 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 $\omega (g)$ as an independent field so that the wave function is $\Psi [\gamma ,\omega ]$ .

## 2 thoughts on “Wheeler–DeWitt equation (motivation and background)”

1. shinichi Post author

# Wheeler–DeWitt equation

Wikipedia

https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation

The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity.

In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called ‘problem of time‘. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar “group” (which is the diffeomorphism group on-shell).

## Motivation and background

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

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

2. shinichi Post author

(sk)

理論物理と応用数学で用いられるホイーラー・ドウィット方程式は、宇宙全体の波動関数が量子重力理論の中で満たすべき方程式で、その着想は、量子重力理論を構築するための指針となったが、いくつかの重要な問題点を指摘されている：

• この方程式は時間を変数に含んでいない
• この方程式には数学的な欠陥が認められ、意味のない無数の解が得られてしまう