  This work rectifies the hydrodynamic equations commonly used to describe the
superfluid velocity field in such a way that vortex dynamics are also taken
into account. In the field of quantum turbulence, it is of fundamental
importance to know the correct form of the equations which play similar roles
to the Navier-Stokes equation in classical turbulence. Here, such equations are
obtained by carefully taking into account the frequently overlooked multivalued
nature of the $U(1)$ phase field. Such an approach provides exact analytical
explanations to some numerically observed features involving the dynamics of
quantum vortices in Bose-Einstein condensates, such as the universal $t^{1/2}$
behavior of reconnecting vortex lines. It also expands these results beyond the
Gross-Pitaevskii theory so that some features can be generalized to other
systems such as superfluid $^{4}$He, dipolar condensates, and mixtures of
different superfluid systems.

Quantum Gases