In this paper, we study the physical meaning of the wavefunction of the universe. With the continuity equation derived from the Wheeler-DeWitt (WDW) equation in the minisuperspace model, we show that the quantity $rho(a)=|psi(a)|^2$ for the universe
is inversely proportional to the Hubble parameter of the universe. Thus, $rho(a)$ represents the probability density of the universe staying in the state $a$ during its evolution, which we call the dynamical interpretation of the wavefunction of the universe. We demonstrate that the dynamical interpretation can predict the evolution laws of the universe in the classical limit as those given by the Friedmann equation. Furthermore, we show that the value of the operator ordering factor $p$ in the WDW equation can be determined to be $p=-2$.
An interesting idea is that the universe could be spontaneously created from nothing, but no rigorous proof has been given. In this paper, we present such a proof based on the analytic solutions of the Wheeler-DeWitt equation (WDWE). Explicit solutio
ns of the WDWE for the special operator ordering factor p=-2 (or 4) show that, once a small true vacuum bubble is created by quantum fluctuations of the metastable false vacuum, it can expand exponentially no matter whether the bubble is closed, flat or open. The exponential expansion will end when the bubble becomes large and thus the early universe appears. With the de Broglie-Bohm quantum trajectory theory, we show explicitly that it is the quantum potential that plays the role of the cosmological constant and provides the power for the exponential expansion of the true vacuum bubble. So it is clear that the birth of the early universe completely depends on the quantum nature of the theory.
Overlap between Hartree-Fock-Bogoliubov(HFB) vacua is very important in the beyond mean-field calculations. However, in the HFB transformation, the $U,V$ matrices are sometimes singular due to the exact emptiness ($v_i=0$) or full occupation ($u_i=0$
) of some single-particle orbits. This singularity may cause some problem in evaluating the overlap between HFB vacua through Pfaffian. We found that this problem can be well avoided by setting those zero occupation numbers to some tiny values (e.g., $u_i,v_i=10^{-8}$). This treatment does not change the HFB vacuum state because $u_i^2,v_i^2=10^{-16}$ are numerically zero relative to 1. Therefore, for arbitrary HFB transformation, we say that the $U,V$ matrices can always be nonsingular. From this standpoint, we present a new convenient Pfaffian formula for the overlap between arbitrary HFB vacua, which is especially suitable for symmetry restoration. Testing calculations have been performed for this new formula. It turns out that our method is reliable and accurate in evaluating the overlap between arbitrary HFB vacua.
