(a)eiS(a)/\Psi(a) \sim e^{\pm i S(a)/\hbar}
with S(a)S(a) the classical action.
2. Bohmian Quantum Potential
To characterize the tunneling dynamics, we define the quantum potential Q(a)Q(a) in the Bohmian sense:
Q(a)=22m1R(a)d2R(a)da2Q(a) = -\frac{\hbar^2}{2m} \frac{1}{R(a)} \frac{d^2 R(a)}{d a^2}
where (a)=R(a)eiS(a)/\Psi(a) = R(a) e^{i S(a)/\hbar} and m=1m = 1 (in Planck units) is an effective mass for the configuration variable.
The effective quantum Hamilton--Jacobi equation becomes:
(dSda)2+Veff(a)+Q(a)=0\left( \frac{dS}{da} \right)^2 + V_{\text{eff}}(a) + Q(a) = 0
This equation governs the non-classical evolution through classically forbidden regions, i.e., for Veff(a)+Q(a)>0V_{\text{eff}}(a) + Q(a) > 0.
3. Effective Potential and Tunneling Barrier
The effective potential governing the birth of the universe is given by:
