The formal loop space approach to classical and quantum Hamiltonian systems

In this thesis we focus on the study of the formal loop space and its applications to classical and quantum Hamiltonian systems. In particular in the first part of this work we generalize the basic tools of finite dimensional differential geometry to the formal loop space defining in this environment the notions of function, Poisson bracket between functions, coordinate transformation, multivector and Poisson cohomology. The Second part is spent on the exposition and the proof of two fundamental theorems on Poisson geometry of the formal loop space: the Dubrovin and Getzler Theorems. These results allow to simplify the form of Poisson brackets of a particular type (called hydrodynamic) by means of an appropriate change of coordinates on the formal loop space. In particular the Getzler theorem can be viewed as a generalization of the Weinstein theorem in finite dimensional Poisson geometry.