Problems With Complex Actions

TL;DR

This paper discusses the challenges of defining Euclidean functional integrals for actions with complex terms, emphasizing the need for analytic continuation into complex fields and illustrating the failure of traditional methods with an explicit example.

Contribution

It demonstrates the failure of standard quantization procedures for complex actions and highlights the importance of analytic continuation through complex critical points.

Findings

Standard real-field critical points often do not exist for complex actions

Analytic continuation into complex fields is necessary for proper quantization

Perturbative methods can fail when dealing with complex actions

Abstract

We consider Euclidean functional integrals involving actions which are not exclusively real. This situation arises, for example, when there are $t$-odd terms in the the Minkowski action. Writing the action in terms of only real fields (which is always possible), such terms appear as explicitly imaginary terms in the Euclidean action. The usual quanization procedure which involves finding the critical points of the action and then quantizing the spectrum of fluctuations about these critical points fails. In the case of complex actions, there do not exist, in general, any critical points of the action on the space of real fields, the critical points are in general complex. The proper definition of the function integral then requires the analytic continuation of the functional integration into the space of complex fields so as to pass through the complex critical points according to the method of steepest descent. We show a simple example where this procedure can be carried out explicitly. The procedure of finding the critical points of the real part of the action and quantizing the corresponding fluctuations, treating the (exponential of the) complex part of the action as a bounded integrable function is shown to fail in our explicit example, at least perturbatively.