Holography from quantum cosmology

The Weyl-Wigner-Groenewold-Moyal formalism of deformation quantization is applied to the closed Friedmann-LemaEtre-Robertson-Walker cosmological model. We show that the phase space average for the surface of the apparent horizon is quantized in units of the Planck's surface, and that the total entropy of the Universe is also quantized. Taking into account these two concepts, it is shown that 't Hooft conjecture on the cosmological holographic principle in radiation and dust dominated quantum universes is satisfied as a manifestation of quantization. This suggests that the entire Universe (not only inside the apparent horizon) can be seen as a two-dimensional information structure encoded on the apparent horizon.