Functional Techniques in Classical Mechanics

I usually do not go to conferences which have in the title the standard fashionable words ”Quantum Gravity” (QG). I made an exception this time because my dear friend Giampiero had the good taste of creating two sessions: one dedicated to QG and another to ”Foundations of Quantization”(FQ). In fact I belong to that minority which thinks that we should not only try to attack the second of the two horns of the problem of Quantum Gravity, that is Gravity, but also the first one that is the Quantum. Gravity is the queen of geometrical theories, and I feel we should better understand QM from a more geometrical point of view before putting the two theories together. Attempts in this direction already exist like for example the method called ”Geometric Quantization”(see ref. [1] for a review). We feel anyhow that method should be made less cumbersome and understood from a more physical point of view. With this goal in mind we proved [2] that, by formulating classical mechanics (CM) via functional methods [3], the standard geometric quantization rules become equivalent to freezing to zero some Grassmannian partners of time. This may throw some light on the geometrical aspects of quantization and work is in progress on it [2]. In this paper we will limit ourselves to reviewing the classical mechanics part of this project that means the geometrical structures which enter the functional approach to CM. 2. FUNCTIONAL APPROACH TO CLASSICAL MECHANICS