3 Tunnelling

The action (2) is crucial in our semiclassical quantization program, since it can be used to quantize the system via a path integral approach. Here we are going to consider the tunnelling process from the $R \equiv 0$ solution to the bounce solution, within the saddle-point approximation. This gives the possibility to estimate the following approximated amplitude

$$A_{\mathrm{s.p.}} \propto \exp \left( - S^{(\mathrm{e})}_{\mathrm{eff.}} \right) ,$$ (3)

where $S^{(\mathrm{e})}_{\mathrm{eff.}}$ is the Euclidean effective action obtained by analytically continuing the action (2) to the Euclidean sector. In order to simplify some expressions, we introduce the following adimensional quantities:

$$x = k R \quad , \quad t = k \tau \quad , \quad \alpha = \fra... ...} \quad , \quad \beta = \frac{\Lambda_{-} - \Lambda_{+}}{k^2}.$$

Moreover it is a well known result that the adimensional version of the equation of motion (1) can be cast in the following form

$$(x ')^2 + V(x) = 0 ,$$

where the prime now denotes the derivative with respect the adimensional time $t$. The potential $V (x)$ is given by

$$V(x) = 1 - \frac{x^2}{x _{0} ^{2}} ,$$ (4)

where $x_0=2/\sqrt{(1+2\alpha+\beta^2)}$ is the adimensional turning radius, provided that the argument of the square root is positive. If this condition holds, there is a bounce trajectory, otherwise we have only the $x=0$ solution. Starting from (2), (3) and (4) it is possible to evaluate the Euclidean action on the tunnelling trajectory, i.e on the segment $[0,x _{0}]$. The final result can be expressed as

$$S^{(\mathrm{e})}_{\mathrm{eff}}=\frac{x_0 ^{(N-3)/2}}{2(N-2)k^{(N-2)}} \left[ (\beta-\epsilon) J(N,C_{\sigma}) \right],$$ (5)

where $J (N,p)$ is a function of the dimension of spacetime, $N$, and of

$$C_{\sigma} = 1 - \left(\frac{\beta - \sigma}{2}\right)^2 x_0^2; \\$$

with

$$\sigma _{\pm} = \pm 1 .$$

A detailed description of the functional form of $J (N,p)$ is beyond the scope of this contribution and can be found elsewhere [7]. We just remark one important feature of it, namely that $J (N,p)$ is defined only when $p < 1$, a condition that, according to the form of $C _{\sigma}$, is always satisfied in our case.