The Non–linear and Quadratic Quantization Programs

Abstract

AbstractAfter 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 \(*\)–Lie algebra of homogeneous quadratic Boson fields denoted \(\hbox {heis}_{2,d}(\mathbb {C})\). Then we recall the main notions related to the complex symplectic Lie algebra \(sp(2d, \mathbb {C})\) and how it is possible to define a natural involution \(\diamond \) on it, thus obtaining a \(*\)–Lie algebra \( sp_{\diamond }(2d, \mathbb {C})\). We prove that, with this involution, there is a \(*\)–isomorphism between \(\hbox {heis}_{2,d, cls}(\mathbb {C})\) and \( sp_{\diamond }(2d,\mathbb {C})\). We show that that the real Lie sub–algebra \(sp_{\diamond ,skew}(2d,\mathbb {C})\) of \(sp_{\diamond }(2d,\mathbb {C})\) consisting of its skew–adjoint elements is isomorphic, as a real \(*\)–Lie algebra to \(sp_{-}(2d,\mathbb {R})\), but the involution allowing this isomorphism is not the restriction to \(sp(2d,\mathbb {R})\) of the \(\diamond \)–involution. We recall the expressions of the vacuum characteristic functions of quadratic Weyl operators, i.e. exponentials of elements of \(sp_{\diamond ,skew}(2d,\mathbb {C})\). 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 \(sp_{\diamond }(2d,\mathbb {C})\), and we give a necessary and sufficient condition for diagonalizability.KeywordsQuantum decomposition of a classical random fieldNon–linear quantizationSymplectic Lie algebraQuadratic quantization