Quantum theory from classical mechanics near equilibrium

We consider classical theories described by Hamiltonians H(p, q) that have a non-degenerate minimum at the point where generalized momenta p and generalized coordinates q vanish. We assume that the sum of squares of generalized momenta and generalized coordinates is an integral of motion. In this situation, in the neighborhood of the point \(p=0, q=0\), the quadratic part of a Hamiltonian plays a dominant role. We suppose that a classical observer can observe only physical quantities corresponding to quadratic Hamiltonians and show that in this case, he should conclude that the laws of quantum theory govern his world.