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Functional Methods in Classical and Quantum mechanics

In 1931, Koopmann and von Neuman proposed a Hilbert space and operatorial formulation of classical mechanics. Since 1988, we have written various papers which give a path-integral formulation of this version of classical mechanics and we have compared it with the path-integral version for quantum mechanics proposed long ago by Feynman. Let us call the first one the CPI (for Classical Path-Integral) and the second one the QPI (for Quantum Path-Integral). We have shown that the weights in these two path-integrals are the same and what changes is only the arguments entering those weights. Moreover, the QPI can be derived from the CPI via a dimensional reduction involving time and two of its anticommuting partners. This transition from CPI to QPI is nothing other than a quantization method, and corresponds to a method known in the literature as "geometric quantization". Our approach has simplified things and thrown some light on the actual geometry of "geometric quantization". We have also found that in the space made of time and its two anticommuting partners, a metric can be introduced. One family of these metrics gives us the CPI, while another one gives us the QPI. We are now using renormalization-type methods to make the opposite transition, i.e. from the QPI to the CPI.

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