Hall and Reginatto show that Quantum Mechanics can be derived from Classical Mechanics (Hamilton-Jacobi formalism)
by adding the Heisenberg uncertainty relation, in form of an equality, as postulate.

Abstract. An exact uncertainty principle, formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.