A cosmological viewpoint on the correspondence between deformed phase-space and canonical quantization

TL;DR

This paper demonstrates that in cosmology, deforming phase-space yields the same physical results as canonical quantization, provided diffeomorphism invariance is preserved, using a Probabilistic Evolutionary Process for comparison.

Contribution

It establishes the equivalence between phase-space deformation and canonical quantization in cosmology, ensuring invariance preservation and using specific models for validation.

Findings

Solutions of Wheeler-DeWitt equation match those of phase-space deformation with sign differences.

Deformation method preserves diffeomorphism invariance.

Equivalence holds under specific conditions for phase-space deformation.

Abstract

We employ the familiar canonical quantization procedure in a given cosmological setting to argue that it is equivalent to and results in the same physical picture if one considers the deformation of the phase-space instead. To show this we use a Probabilistic Evolutionary Process (PEP) to make the solutions of these different approaches comparable. Specific model theories are used to show that the independent solutions of the resulting Wheeler-DeWitt equation are equivalent to solutions of the deformation method with different signs for the deformation parameter. We also argued that since the Wheeler-DeWitt equation is a direct consequence of diffeomorphism invariance, this equivalence is only true provided that the deformation of phase-space does not break such an invariance.