Generalization of saddle-point methods for the nonperturbative approximation of integrals.

What is it about?

The concept of the saddle-point approximation is well-studied for the approximation of integrals that have integrands which are sharply peaked. Such kind of integrals where this classical saddle-point method works are abundant in physics, e.g. Quantum Field Theory or Statistical Mechanics. However, when strong fluctuations are strong, this method gets inaccurate and generalizations of the concept are needed. A new method inspired by the Faddeev-Popov trick is shown in this paper, allowing to manipulate the integral into several parts with its own generations of "effective actions". These will, dependent on their depth of recursive Faddeev-Popov procedure, capture the effects of strong fluctuations. At some point these can be handled with ordinary methods. The advantage is that very high precision can be achieved, even with very strong fluctuations.

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Photo by Steve Johnson on Unsplash

Photo by Steve Johnson on Unsplash

Why is it important?

It may give more insights into physical systems described by path integrals which have strong fluctuations like Quantum systems or systems at a phase transition.

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