Self-Consistent Excitations in an Inhomogeneous Bose Condensate

Abstract

The many-body description of an inhomogeneous Bose system interacting through two-body forces is considered. The functional formulation of Bosonic field theory is utilized to derive exact relations between the field variables. In addition to sources generating field correlation functions, a source is introduced to generate density correlations. A Legendre transformation is used to introduce the order parameter and the particle density as independent variables. This simplifies the physical interpretation of the relations between correlators. The external potential prevents the use of the momentum representation, and the configuration space single particle propagator and the density–density correlation function are derived as functionals of each other. These relations constitute a formal proof that, even in the confined case, the two propagators contain the same elementary excitations when the symmetry is broken. In the homogeneous case this result is long known, but its validity in the trapped system provides a basis for approximations in its condensed phase. The problem in terms of coupled correlation functions only is formulated; in this manner the terms deriving from symmetry breaking are explicitly displayed in the propagator equations.