# Initial wave-function of the universe is arbitrary

## TL;DR

This paper shows that in quantum cosmology, the initial wave-function of the universe can be arbitrary, highlighting the fundamental ambiguity and complexity of initial conditions in quantum gravity models.

## Contribution

It demonstrates that the Hamiltonian in a quantum gravity-scalar system can be uniquely defined and that the initial wave-function is arbitrary, emphasizing the initial condition problem.

## Key findings

- Hamiltonian is uniquely defined without ordering ambiguity.
- The Hamiltonian degenerates to zero at the initial state.
- The initial wave-function can be any vector in the Hilbert space.

## Abstract

We consider quantization of the gravity-scalar field system in the minisuperspace approximation. It turns out that in the gauge fixed deparametrized theory where the scale factor plays the role of time, the Hamiltonian can be uniquely defined without any ordering ambiguity as the square root of a self-adjoint operator. Moreover, the Hamiltonian degenerates to zero and the Schr\"{o}dinger equation becomes well behaved as the scale factor vanishes. Therefore, there is no technical or physical obstruction for the initial wave-function of the universe to be an arbitrary vector in the Hilbert space, which demonstrates the severeness of the initial condition problem in quantum cosmology.