  In condensed matter physics, Kramers-Wannier duality implies that the state
disordered by quantum fluctuations or temperature actually corresponds with an
ordered state formed from the topological excitations of the 'original' ordered
state. At first sight it might appear to be impossible to observe this dual
order using means associated with the original order. Although true for Ising
models, we consider in this paper the well known vortex duality, in particular
in the quantum interpretation in 2+1D where it is associated with the quantum
phase transition from a superfluid to a Bose Mott insulator. Here, the
disordered Mott insulating state is at the same time a dual superconductor
corresponding with a Bose condensate of vortices. We present a simple formalism
making it possible to compute the velocity propagator associated with the
superfluid in terms of the degrees of freedom of the dual theory. The Mott
insulator is characterized by a doublet of massive modes and we demonstrate
that one of these modes is nothing else than the longitudinal photon (gauged
second sound) of the dual superconductor. For increasing momenta, the system
rediscovers the original order, and the effect on the velocity correlator is
that the longitudinal photon looses its pole strength. The quantum critical
regime as probed by the velocity correlator is most interesting. We demonstrate
that at infinite wavelength the continua of critical modes associated with
second sound and the dual longitudinal photon are indistinguishable. However,
at finite momenta they behave differently, tracking the weight reshuffling
found in the quasiparticle spectrum of the disorder state closely.
