# Visualising quantum effective action calculations in zero dimensions

## TL;DR

This paper explores the 2PI effective action in a zero-dimensional quantum field theory, providing explicit analytic and numerical methods, graphical insights, and clarifying the role of saddle points and convexity.

## Contribution

It offers a complete analytic and numerical treatment of the 2PI effective action in zero dimensions, including graphical representations and analysis of convexity and saddle points.

## Key findings

- Complete analytic approximations are achievable in zero dimensions.
- Graphical representations help visualize the effective action and its components.
- The Maxwell construction is explained in the context of multiple saddle points.

## Abstract

We present an explicit treatment of the two-particle-irreducible (2PI) effective action for a zero-dimensional quantum field theory. The advantage of this simple playground is that we are required to deal only with functions rather than functionals, making complete analytic approximations accessible and full numerical evaluation of the exact result possible. Moreover, it permits us to plot intuitive graphical representations of the behaviour of the effective action, as well as the objects out of which it is built. We illustrate the subtleties of the behaviour of the sources and their convex-conjugate variables, and their relation to the various saddle points of the path integral. With this understood, we describe the convexity of the 2PI effective action and provide a comprehensive explanation of how the Maxwell construction arises in the case of multiple, classically stable saddle points, finding results that are consistent with previous studies of the one-particle-irreducible (1PI) effective action.