  Classical mechanics (CM), like quantum mechanics (QM), can have an
operatorial formulation. This was pioneered by Koopman and von Neumann (KvN) in
the 30's. They basically formalized, via the introduction of a classical
Hilbert space, earlier work of Liouville who had shown that the classical time
evolution can take place via an operator, nowadays known as the Liouville
operator. In this paper we study how to perform the coupling of a point
particle to a gauge field in the KvN version of CM. So we basically implement
at the classical operatorial level the analog of the minimal coupling of QM. We
show that, differently than in QM, not only the momenta but also other
variables have to be coupled to the gauge field. We also analyze in details how
the gauge invariance manifests itself in the Hilbert space of KvN and indicate
the differences with QM. As an application of the KvN method we study the
Landau problem proving that there are many more degeneracies at the classical
operatorial level than at the quantum one. As a second example we go through
the Aharonov-Bohm phenomenon showing that, at the quantum level, this
phenomenon manifests its effects on the spectrum of the quantum Hamiltonian
while at the classical level there is no effect whatsoever on the spectrum of
the Liouville operator.
