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Visualising quantum effective action calculations in zero dimensions

Millington, Peter; Saffin, Paul M



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.


Millington, P., & Saffin, P. M. (2019). Visualising quantum effective action calculations in zero dimensions. Journal of Physics A: Mathematical and Theoretical, 52(40),

Journal Article Type Article
Acceptance Date Aug 1, 2019
Online Publication Date Sep 12, 2019
Publication Date Oct 4, 2019
Deposit Date Sep 17, 2019
Publicly Available Date Dec 11, 2019
Journal Journal of Physics A: Mathematical and Theoretical
Print ISSN 1751-8113
Electronic ISSN 1751-8121
Publisher IOP Publishing
Peer Reviewed Peer Reviewed
Volume 52
Issue 40
Article Number 405401
Keywords Modelling and Simulation; Statistics and Probability; Mathematical Physics; General Physics and Astronomy; Statistical and Nonlinear Physics
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Additional Information Journal title: Journal of Physics A: Mathematical and Theoretical; Article type: paper; Article title: Visualising quantum effective action calculations in zero dimensions; Copyright information: © 2019 IOP Publishing Ltd; Date received: 2019-05-28; Date accepted: 2019-08-01; Online publication date: 2019-09-12


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