  We derive the Josephson relation for a dilute Bose gas in the framework of an
auxiliary-field resummation of the theory in terms of the normal- and
anomalous-density condensates. The mean-field phase diagram of this theory
features two critical temperatures, T_c and $T^*, associated with the presence
in the system of the Bose-Einstein condensate (BEC) and superfluid state,
respectively. In this context, the Josephson relation shows that the superfluid
density is related to a second order parameter, the square of the
anomalous-density condensate. This is in contrast with the corresponding result
in the Bose gas theory without an anomalous condensate, which predicts that the
superfluid density is proportional to the BEC condensate density. Our findings
are consistent with the prediction that in the temperature range between T_c
and T^* a fraction of the system is in the superfluid state in the absence of
the BEC condensate. This situation is similar to the case of dilute Fermi
gases, where the superfluid density is proportional to the square of the gap
parameter. The Josephson relation relies on the existence of zero energy and
momentum excitations showing the intimate relationship between superfluidity
and the Goldstone theorem.
