Quantum theory from classical mechanics near equilibrium

Abstract 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 $$p=0, q=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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.