A corrected Maslov index for complex saddle trajectories

TL;DR

This paper addresses sign errors in saddle point approximations in quantum mechanics by proposing a practical method to correct phase calculations involving complex saddle trajectories, improving accuracy in phase correction.

Contribution

It introduces a new approach linking complex saddle points to real trajectories to avoid sign errors caused by determinant zeros in phase calculations.

Findings

Deformed complex time contours can fix sign errors.

A practical method links saddles to real trajectories.

The approach improves phase correction accuracy.

Abstract

Saddle point approximations, extremely important in a wide variety of physical contexts, require the analytical continuation of canonically conjugate quantities to complex variables in quantum mechanics. An important component of this approximation's implementation is arriving at the phase correction attributable to caustics, which involves determinantal prefactors. The common prescription of using the inverse of half a certain determinant's total accumulated phase sometimes leads to sign errors. The root of this problem is traced to the zeros of the determinants at complex times crossing the real time axis. Deformed complex time contours around the zeros can repair the sign errors that sometimes occur, but a much more practical way is given that links saddles back to associated real trajectories and avoids the necessity of locating the complex time zeros of the determinants.