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 and given by
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 is the field, and we denote its conjugate momenta as . The Hamiltonian is a constraint (characteristic of most relativistic systems)
where and 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
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
One can apply the operator to a general wave functional of the metric where:
which would give a set of constraints amongst the coefficients . This means the amplitudes for 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 as an independent field so that the wave function is .