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Abstract

The effective potential V(n,a) for the Euclidean p(ø)₂ quantum field theory is defined to be the Fenchel transform (convex conjugate) of the pressure in an external field, and is shown to be finite. The parameter h is Planck’s constant divided by 2π. The classical limit (h↓0) of the effective potential is shown to be the convex hull of the classical potential P(a) + 1/2m²a². For values of a for which the classical potential is equal to its convex hull and has a nonvanishing second derivative, the usual one-particle irreducible loop expansion for the effective potential is shown to be asymptotic as (h↓0), using a uniformly convergent (as h↓0) high temperature cluster expansion and irreducibility properties of the Legendre transform. For the same values of a, V is shown to be analytic in a for sufficiently small h. Finally an example is given for a double well classical potential where the one-particle irreducible loop expansion fails to be asymptotic, and the true asymptotics are obtained.