Direct Approach to Quantum Tunneling

The decay rates of quasistable states in quantum field theories are usually calculated using instanton methods. Standard derivations of these methods rely in a crucial way upon deformations and analytic continuations of the physical potential and on the saddle-point approximation. While the resulting procedure can be checked against other semiclassical approaches in some one-dimensional cases, it is challenging to trace the role of the relevant physical scales, and any intuitive handle on the precision of the approximations involved is at best obscure. In this Letter, we use a physical definition of the tunneling probability to derive a formula for the decay rate in both quantum mechanics and quantum field theory directly from the Minkowski path integral, without reference to unphysical deformations of the potential. There are numerous benefits to this approach, from nonperturbative applications to precision calculations and aesthetic simplicity.

Figure 1

An example 1D potential exhibiting quantum tunneling from the region FV to the region $R$.

Figure 2

The probability $P_{\mathrm{FV}}(T)$ of funding a wave function in FV at time $t$ for the toy 1D potential in Fig. 1. This curve is computed by numerically integrating the Schrödinger equation beginning with a Gaussian wave packet localized near $a$ at $t=0$. The false-vacuum probability falls exponentially, $P_{\mathrm{FV}}\sim e^{-\mathrm{\Gamma }T}$, for times intermediate between the false-vacuum sloshing time $T_{\text{slosh}}$ and the time when nonlinearities set in and the flux starts flowing back into the FV region.