On the anomalous density of a dilute homogeneous Bose gas

Measurement of numerical values of the anomalous density, $\sigma$, which plays important role in Bose – Einstein condensation, and, especially, determination of its sign, has been a long standing problem. We develop Hartree – Fock – Bogoliubov theory taking account arbitrary phase of the condensate wave function. We show that, the sign of $\sigma$ directly related to the phase, and, hence is not observable. Despite this, its absolute value can be extracted from measurements of the sound velocity and condensed fraction. We present theoretical prediction for $\mathrm{\mid }\sigma \mathrm{\mid }$ for a BEC in a uniform box.