The Non–linear and Quadratic Quantization Programs

After a short outline of the notion of canonical quantum decomposition of a classical random field and of its connection with the program of non–linear quantization, we concentrate our attention on quadratic quantization. We introduce the (formula presented) –Lie algebra of homogeneous quadratic Boson fields denoted heis (formula presented). Then we recall the main notions related to the complex symplectic Lie algebra sp(2 d, C) and how it is possible to define a natural involution (formula presented) on it, thus obtaining a (formula presented) –Lie algebra (formula presented). We prove that, with this involution, there is a (formula presented) –isomorphism between heis (formula presented) consisting of its skew–adjoint elements is isomorphic, as a real (formula presented) –Lie algebra to (formula presented), but the involution allowing this isomorphism is not the restriction to sp(2 d, R) of the (formula presented) –involution. We recall the expressions of the vacuum characteristic functions of quadratic Weyl operators, i.e. exponentials of elements of (formula presented)(2 d, C). We describe the Lie groups associated with the symplectic algebra. Finally we discuss the problems of diagonalizability and vacuum factorizability for quadratic fields, i.e. elements of (formula presented), and we give a necessary and sufficient condition for diagonalizability.