Quantization of the Universe as a Dirac Particle: Geometrical   Unification of Gravitation, Dark Energy and Multiverses
  A Friedmann--Robertson--Walker Universe is studied with a dark energy component represented by a quintessence field. The Lagrangian for this system, hereafter called the Friedmann--Robertson--Walker--quintessence (FRWq) system, is presented. It is shown that the classical Lagrangian reproduces the usual two (second order) dynamical equations for the radius of the Universe and for the quintessence scalar field as well as a (first order) constraint equation.   It is noted that the quintessence field (dark energy) is related with the tlaplon field (appearing on dynamical theories of spacetime torsion). Our approach naturally unifies gravity and dark energy, as it is obtained that the Lagrangian and the equations of motion are those of a relativistic particle moving on a two dimensional conformally flat spacetime, whereas the conformal metric factor is related to the dark energy scalar field potential. We proceed to quantize the system in three different schemes. First, we assume the Universe as a spinless particle (as it is common in literature), obtaining a quantum theory for a Universe described by the Klein--Gordon equation. Second, we push the quantization scheme further assuming the Universe as a Dirac particle, and therefore constructing its corresponding Dirac and the Majorana theories. With the different theories we calculate the expected values for the scale factor of the Universe. They depend on the type of quantization scheme used. The differences between the Dirac and Majorana schemes are highlighted. The implications of the different quantization procedures are discussed. Finally the possible consequences for a multiverse theory of the Dirac and Majorana quantized Universe are briefly considered. 
Sergio A. Hojman, Felipe A. Asenjo and Carlos A. Rubio
gr-qc