The loop expansion for the effective potential in the \(P(\Phi)_ 2\) quantum field theory
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.
one particle irreducible Feynman diagrams
single scalar boson field with polynomial selfinteraction
scientific article; zbMATH DE number 3969418