Classical-Quantum dualities

In quantum field theory, the generating functional is the functional Fourier transform of e^iS, with S the classical action. Therefore, applying the inverse transform, a classical object, e^iS, can be expressed in terms of a path integral over quantum objects. This suggests that it may be possible to formulate a theory where classical mechanics is seen as the ''exact'' result and quantum mechanics is recovered in a ''quantum limit'', dual to the usual classical limit. The aim of this thesis is to explore this idea using a Hilbert space formulation for classical mechanics. In particular, we analyze the Koopman-von Neumann operatorial formulation of classical mechanics and derive from it a path integral representation. Then, we review the Wigner-Weyl (WW) formalism, usually applied in the formulation of quantum mechanics in phase space, and use it to reformulate classical mechanics in the quantum Hilbert space. Using the WW formalism, we derive two possible realizations of the quantum limit. First, we show that the role of Planck's constant h can be reversed because the limit h->0 can be interpreted as a limit in which the classical algebraic structure reduces to the quantum non-commutative structure. This suggests to interpret, at a dynamical level, the limit h->0 as a classical-quantum interface and to to consider the two theories, classical and quantum, on equal footing. Then, we show that applying the WW formalism to the classical Liouville equation, which governs the dynamics of classical statistical ensembles, it is possible to consider a ''local quantum approximation'' where the classical dynamics reduces to the quantum one if the states are sufficiently localized. This gives an alternative quantization procedure, where quantum dynamics is derived using a limit process.