Complex time and the Gaussian approximation

It is shown that, in general, full agreement between the semiclassical approximation of the path integral and the WKB method of ordinary quantum mechanics is achieved only by the use of complex-time classical solutions. This technique is applied to compute the shift in energy levels due to tunneling and to calculate the lifetimes of virtual states. It is emphasized that the exponential time decay of the kernel is only approximately valid for finite time and that for $T\to \infty$ a power-law behavior applies. The Gaussian approximation of the kernel corroborates these statements.