Summary: In Quantum Hydro-Dynamics the following problem is relevant: let \((\sqrt{\rho },\Lambda ) \in{W^{1,2}}({\mathbb{R}}^d,{\mathcal{L}}^d,{\mathbb{R}}^+) \times L^2({\mathbb{R}}^d,\mathcal L^d,{\mathbb{R}}^d)\) be a finite energy hydrodynamics state, i.e. \( \Lambda = 0\) when \(\rho = 0\) and \[ E = \int_{{\mathbb{R}}^d} \frac{1}{2} \big | \nabla \sqrt{\rho } \big |^2 + \frac{1}{2} \Lambda^2 {\mathcal{L}}^d < \infty. \] The question is under which conditions there exists a wave function \(\psi \in{W^{1,2}}({\mathbb{R}}^d,{\mathcal{L}}^d,{\mathbb{C}})\) such that \[ \sqrt{\rho } = |\psi |, \quad J = \sqrt{\rho } \Lambda = \operatorname{Im} \big ( {\bar{\psi }} \nabla \psi ). \] The second equation gives for \(\psi = \sqrt{\rho } w\) smooth, \(|w| = 1\), that \(i \Lambda = \sqrt{\rho } {\bar{w}} \nabla w\). Interpreting \(\rho{\mathcal{L}}^d\) as a measure in the metric space \({\mathbb{R}}^d\), this question can be stated in generality as follows: given metric measure space \((X,d,\mu )\) and a cotangent vector field \(v \in L^2(T^* X)\), is there a function \(w \in{W^{1,2}}(X,\mu ,{\mathbb{S}}^1)\) such that \[ dw = i w v. \] Under some assumptions on the metric measure space \((X,d,\mu )\) (conditions which are verified on Riemann manifolds with the measure \(\mu = \rho \text{Vol}\) or more generally on non-branching \(\text{MCP}(K,N)\)), we show that the necessary and sufficient conditions for the existence of \(w\) is that (in the case of differentiable manifold) \[ \int v(\gamma (t)) \cdot{\dot{\gamma }} (t)\, dt \in 2\pi{\mathbb{Z}} \] for \(\pi \)-a.e. \( \gamma \), where \(\pi\) is a test plan supported on closed curves. This condition generalizes the condition that the vorticity is quantized. We also give a representation of every possible solution. In particular, we deduce that the wave function \(\psi = \sqrt{\rho } w\) is in \(W^{1,2}(X,\mu ,{\mathbb{C}})\) whenever \(\sqrt{\rho } \in W^{1,2}(X,\mu ,{\mathbb{R}}^+)\).