Functional methods in classical and quantum mechanics Prof. Giuseppe Furlan, Full Professor Prof. Ennio Gozzi, Associate Professor Dr. Danilo Mauro, Postdoctoral fellow In the last 30 years the path-integral method has become one of the most useful tool both in quantum mechanics and in quantum field theory. Recently we have proved that a path-integral formulation can be given even to classical mechanics.This is nothing else that the functional counterpart of the operatorial approach to classical mechanics pioneered in the 30's by Koopman and von Neumann (KvN). Over the last year we have explored few issues. The first one is the role played in the KvN formalism by the gauge fields. In particular we have studied the classical analog of the minimal coupling and the relative gauge phases present in a classical version of the Aharonov-Bohm type of experiments. A second issue we have investigated is the following one: we have extended the KvN Hilbert space to include the differential forms associated to the classical phase-space. This extended Hilbert space presents some very peculiar features. In particular we proved that it cannot be endowed with a positive definite scalar product unless we abandon the request of unitarity for the time evolution of some systems. These systems seems to be related to the cahotic ones and we have given physical reasons why this must be so. This analysis, together with previous work, has thrown new light on concepts like ergodicity, Lyauponov exponents and similar. It would be interesting to expand this formalism to classical fluids and see if the tools of differential forms can provide new insights on issues such as turbolence and similar. Another topic presently under investigation is the issue of quantization of classical systems once they are formulated via the KvN formalism (or the equivalent path-integral version ). It seems to emerge that the the most natural route to quantization is via the so called geometric quantization technique. In our framework we achieve this type of quantization by freezing to zero two Grassmannian partners of time which appear naturally when we introduce the concept of differential forms in the KvN approach as we did above. This research seems to throw an intriguing light on such seemingly unrelated issues as quantum mechanics and time. INTERNATIONAL COLLABORATIONS: E.Deotto (MIT, USA); M.Reuter (Mainz University, Germany); A.A.Abrikosov (jr) (ITEP, Moscow); I.Fiziev (Sofia University, Bulgaria).

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