Wheeler-DeWitt Universe Wave Function in the presence of stiff matter

We study the Wheeler-DeWitt (WDW) equation close to the Big-Bang. We argue that an interaction dominated fluid (speed of sound equal to the speed of light), if present, would dominate during such an early phase. Such a fluid with $p=ρ\propto1/a^{6}$ generates a term in the potential of the wave function of the WDW equation proportional to $-1/a^{2}.\$This very peculiar quantum potential, which embodies a spontaneous breaking of dilatation invariance, has some very remarkable consequences for the wave function of the Universe: $Ψ(a)$ vanishes at the Big-Bang: $Ψ(0)=0$; the wave function $Ψ(a)$ is always real; a superselection rule assures that the system is confined to $a\geq0$ without the need of imposing any additional artificial barrier for unphysical negative $a$. These results do not depend on the operator-ordering problem of the WDW equation.

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