The loop expansion for the effective potential in the \(P(\Phi)_ 2\) quantum field theory
type: Item
instance of: scholarly article
title: The loop expansion for the effective potential in the \(P(\Phi)_ 2\) quantum field theory
zbMATH Open document ID: 0601.46068
DOI: 10.1007/BF01209294
author: Gordon Slade
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: In quantum field theory the effective potential is usually calculated approximately by expanding it in a power series in \(\hslash\) and keeping the first few terms. The coefficient of \(\hslash^ n\) in this expansion is given by the sum of all n loop one-particle irreducible (1-PI) Feynman diagrams. In the paper under review this expansion is studied from a mathematical point of view in the \(P(\Phi)_ 2\) quantum field theory. The \(P(\Phi)_ 2\) quantum field theory is the theory of a single scalar boson field with polynomial selfinteraction in \(1+1\) space-time dimensions, and is mathematically very well-understood. It is shown that in the one-phase region, i.e., for values of the classical field a for which the classical potential \(U_ 0(a)=P(a)+m^ 2a^ 2\) is equal to its convex hull and has nonvanishing second derivative, the 1-PI loop expansion is asymptotic as \(\hslash \downarrow 0\). An example is given of a double-well classical potential for which the 1-PI loop expansion is not asymptotic for values of the classical field lying between the two minima of the classical potential. The true asymptotics here involve connected rather than 1-PI diagrams, and at the one-loop level give the linear interpolation of the one-loop approximation from the one-phase region. In a later work [Commun. Math. Phys. 104, 573-580 (1986); see also \textit{L. O'Raifeartaigh, A. W. Wipf} and \textit{H. Yoneyama}, Nucl. Phys. B 271, 653-680 (1986)] it is shown that in the one-phase region and for small \(\hslash\) the effective potential, defined as a Legendre transform, is equal to the vacuum energy density under the constraint of fixed average field, in the infinite volume limit.
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 81T08
zbMATH DE Number: 3969418
zbMATH Keywords: one particle irreducible Feynman diagrams
zbMATH Keywords: effective potential
zbMATH Keywords: \(P(\Phi )_ 2\) quantum field theory
zbMATH Keywords: single scalar boson field with polynomial selfinteraction
zbMATH Keywords: loop expansion
zbMATH Keywords: Legendre transform
zbMATH Keywords: vacuum energy density
MaRDI profile type: Publication
cites work: Measures on Schwartz distribution space and applications to \(P(\varphi)_2\) field theories
cites work: Laplace's method for Gaussian integrals with an application to statistical mechanics
cites work: Q3910965
cites work: Convex Analysis
cites work: Q3872705
full work available at URL: https://doi.org/10.1007/bf01209294
OpenAlex ID: W1999443604
description: scientific article; zbMATH DE number 3969418