Exact integrability conditions for cotangent vector fields

In Quantum Hydro-Dynamics the following problem is relevant: let (ρ,Λ)∈W1,2(Rd,Ld,R+)×L2(Rd,Ld,Rd) be a finite energy hydrodynamics state, i.e. Λ = 0 when ρ= 0 and E=∫Rd12|∇ρ|2+12Λ2Ld<∞.The question is under which conditions there exists a wave function ψ∈ W1 , 2(Rd, Ld, C) such that ρ=|ψ|,J=ρΛ=ℑ(ψ¯∇ψ).The second equation gives for ψ=ρw smooth, | w| = 1 , that iΛ=ρw¯∇w. Interpreting ρLd as a measure in the metric space Rd, this question can be stated in generality as follows: given metric measure space (X, d, μ) and a cotangent vector field v∈ L2(T∗X) , is there a function w∈ W1 , 2(X, μ, S1) such that dw=iwv.Under some assumptions on the metric measure space (X, d, μ) (conditions which are verified on Riemann manifolds with the measure μ= ρVol or more generally on non-branching MCP (K, N)), we show that the necessary and sufficient conditions for the existence of w is that (in the case of differentiable manifold) ∫v(γ(t))·γ˙(t)dt∈2πZfor π-a.e. γ, where π 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 ψ=ρw is in W1 , 2(X, μ, C) whenever ρ∈W1,2(X,μ,R+).