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 γ i j {\displaystyle \gamma _{ij}} and given by
g μ ν d x μ d x ν = ( − N 2 + β k β k ) d t 2 + 2 β k d x k d t + γ i j d x i d x j . {\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 γ i j {\displaystyle \gamma _{ij}} is the field, and its conjugate momenta are denoted as π i j {\displaystyle \pi ^{ij}} . The Hamiltonian is a constraint (characteristic of most relativistic systems)
H = 1 2 γ G i j k l π i j π k l − γ
( 3 ) R = 0 , {\displaystyle {\mathcal {H}}={\frac {1}{2{\sqrt {\gamma }}}}G_{ijkl}\pi ^{ij}\pi ^{kl}-{\sqrt {\gamma }}\,{}^{(3)}\!R=0,}
where γ = det ( γ i j ) {\displaystyle \gamma =\det(\gamma _{ij})} , and G i j k l = ( γ i k γ j l + γ i l γ j k − γ i j γ k l ) {\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
tr ( ( g − 1 d g ) 2 ) − ( tr ( g − 1 d g ) ) 2 . {\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 the operator
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