From the equations of motion to the canonical commutation relations

The problem of whether or not the equations of motion of a quantum system determine the commutation relations was posed by E. P. Wigner in 1950. A similar problem (known as “The Inverse Problem in the Calculus of Variations”) was posed in a classical setting as back as in 1887 by H. Helmoltz and has received great attention also in recent times. The aim of this paper is to discuss how these two apparently unrelated problems can actually be discussed in a somewhat unified framework. After reviewing briefly the Inverse Problem and the existence of alternative structures for classical systems, we discuss the geometric structures that are intrinsically present in Quantum Mechanics, starting from finite-level systems and then moving to a more general setting by using the Weyl-Wigner approach, showing how this approach can accommodate in an almost natural way the existence of alternative structures in Quantum Mechanics as well.