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In this paper we develop a gapless theory of BEC which can be applied to bothtrapped and homogeneous gases at zero and finite temperature. The many-bodyHamiltonian for the system is written in a form which is approximatelyquadratic with higher order cubic and quartic terms. The quadratic part isdiagonalized exactly by transforming to a quasiparticle basis, while thenon-quadratic terms are dealt with using first and second order perturbationtheory. The conventional treatment of these terms, based on factorizationapproximations, is shown to be inconsistent.  Infra-red divergences can appear in individual terms of the perturbationexpansion, but we show analytically that the total contribution beyondquadratic order is finite. The resulting excitation spectrum is gapless and theenergy shifts are small for a dilute gas away from the critical region,justifying the use of perturbation theory. Ultra-violet divergences can appearif a contact potential is used to describe particle interactions. We show thatthe use of this potential as an approximation to the two-body T-matrix leadsnaturally to a high-energy renormalization.  The theory developed in this paper is therefore well-defined at both low andhigh energy and provides a systematic description of Bose-Einstein condensationin dilute gases. It can therefore be used to calculate the energies and decayrates of the excitations of the system at temperatures approaching the phasetransition.Comment: 39 pages of Revtex. 1 figur

A gapless theory of Bose-Einstein condensation in dilute gases at finite temperature