I am trying to study several quantum cosmology models. The standard procedure for quantization consists typically in several steps:

  1. People write the theory as an action, or Hamiltonian $H(p^i,q^j)$ with canonical conjugated variables.
  2. Then, the conjugated variables are promoted to operators $\hat{q}^i$ and $\hat{p}^j$ and the Hilbert space of the theory is given by the square integrable functions over the $q^i$'s with scalar product $\left<\psi(q^i), \phi(q^i)\right>=\int dq^i\sqrt{-\det{g}} \ \psi(q^i)\phi^*(q^i)$, where $\sqrt{-\det{g}}$ defines an "appropriate measure". This measure is usually defined so that the Hamiltonian can be written as $\hat{H}=\square$ (plus some extra terms if there is for example a cosmological constant).

Apparently this way of ordering the operators in the Hamiltonian and defining the scalar product is well motivated by Hawking and Page, but I don't see what advantage it can have. In principle there are other possible ways of writing the Hamiltonian and other possible scalar products one could define.

Any hint will be more than appreciated.


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