Our treatment is based on a straightforward application of the quantum-field theory of atoms. Although the key result is obtained within the Hartree-Fock-Bogoliubov-Popov (HFB-Popov) approximation we believe it will be found to be valid in a wider context. The essential aspects of this result are as follows. First, current BEC systems are produced in atom traps, and so they are intrinsically inhomogeneous. Second, the condensed and noncondensed (thermal) fractions of the atomic gas are distributed differently over the trap, and they make separate and distinct contributions to the local coherence function.
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