Zero BEC State Amplitude, and BEC Unnecessary to Define Phase \phi

We define the "BEC state" to be the many-body wavefunction where all particles are in the same one-body state. Using an argument analogous to Anderson's Orthogonality Catastrophe, we argue that for interacting particles the amplitude of the BEC state within the many-body wavefunction goes to zero in the thermodynamic limit. This does not mean that there is no condensate. However, we argue that, if the excitations satisfy the Landau criterion, then the absence of a finite amplitude for the BEC state, or the absence of a condensate, do not prevent the definition of the phase function \phi, from which the superfluid velocity follows.