Wheeler–DeWitt equation

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)

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

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

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

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 ${\displaystyle \Psi [\gamma ,\omega ]}$.

The Wheeler–DeWitt equation[1] is a functional differential equation. It is ill-defined in the general case, but very important in theoretical physics, especially in quantum gravity. It is a functional differential equation on the space of three dimensional spatial metrics. The Wheeler–DeWitt equation has the form of an operator acting on a wave functional; the functional reduces to a function in cosmology. Contrary to the general case, the Wheeler–DeWitt equation is well defined in minisuperspaces like the configuration space of cosmological theories. An example of such a wave function is the Hartle–Hawking state. Bryce DeWitt first published this equation in 1967 under the name "Einstein–Schrödinger equation"; it was later renamed the "Wheeler–DeWitt equation".[3]

• Herbert W. Hamber and Ruth M. Williams (2011). "Discrete Wheeler-DeWitt Equation". Physical Review D. 84 (10): 104033. arXiv:1109.2530. Bibcode:2011PhRvD..84j4033H. doi:10.1103/PhysRevD.84.104033. S2CID 4812404.
• Herbert W. Hamber, Reiko Toriumi and Ruth M. Williams (2012). "Wheeler-DeWitt Equation in 2+1 Dimensions". Physical Review D. 86 (8): 084010. arXiv:1207.3759. Bibcode:2012PhRvD..86h4010H. doi:10.1103/PhysRevD.86.084010. S2CID 119229306.