The Physics of Quantum Creation from Nothing
In Quantum Cosmology, the evolution of the spatial metric scale factor a is governed by the Wheeler-DeWitt equation -\hbar^2 \frac{d^2\Psi}{da^2} + V(a)\Psi(a) = 0. The universe starts at zero spatial volume (a = 0) and undergoes quantum tunneling through a potential barrier V(a) = \kappa a^2 - \frac{\Lambda}{3}a^4 + m^2 a^2 into classical Lorentzian de Sitter inflationary expansion at the nucleation turning point a_0.
Vilenkin Proposal: Demands outgoing wave boundary conditions at infinity, resulting in nucleation probability P \propto e^{-2 S_E}.
Hartle-Hawking Proposal: The "No-Boundary" wavefunction calculates path integrals over smooth Euclidean geometries, leading to probability P \propto e^{+2 S_E}.