Bounds for a Bose condensate in dimensions v ? 3

A stronger version of the Bogoliubov inequality is used to derive an upper bound for the anomalous average ¦x)>|s of an interacting nonrelativislic Bose fielda(x) at a finite temperature. This bound is ¦a(x)²|s <pR, whereR satisfies 1 -R = (RT/2T _c ^v/2, withv the dimensionality, andT _c the critical temperature in the absence of interactions. The formation of nonzero averages is closely related to the Bose-Einstein condensation and ¦|·² is often believed to coincide with the mean densitypa of the condensate. We have found nonrigorous arguments supporting the inequality po ? ¦|·², which parallels the result of Griffiths in the case of spin systems.