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

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 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.