Stochastic dynamics from O(2N) fractional Laplacian vector model and O(N) vector model free energy in finite temperature

We explore O(2N) vector model with fractional Laplacian, $\sqrt{-\nabla^2}$ in $d$-dimension and its Hamiltonain dynamics which is described by a Schrodinger type equation. This equation is a kind of current conservation equation, where one can define a current $j(ϕ^a)$ of a probability $P(ϕ^a)$, wh