On half-form quantization of Lagrangian manifolds and quantum mechanics in phase space
Let \((Z, \omega)\) be a symplectic space, \(Lag\) the manifold of \(n\)-planes of \(Z\) on which \(\omega\) vanishes identically, \(Lag_{\infty}\) the universal covering of \(Lag\) and \(Lag_q\) the coverings of all orders \(q=1,2,\ldots\) of \(Lag\). In some previous papers the author studied the Maslov index on \(Lag_{\infty}\) \(m:Lag_{\infty}\times Lag_{\infty}\rightarrow {\mathbb{Z}}\) and the Maslov index on \(Lag_q\) \(m_q:Lag_q\times Lag_q \rightarrow {\mathbb{Z}}_q\). Let \(V\) be a Lagrangian manifold immersed in \(Z\). Choosing once for all an element \({\hat L}_0\in Lag_4\) the author defines a map \(m_0:\breve V\ni \breve z \rightarrow m_0(\breve z)\in {\mathbb{Z}}_4\) where \(\breve V\) is the universal covering of \(V\). This construction yields the definitions of two new objects: 1) metaplectic half-forms on \(\breve V\) 2) Lagrangian catalogues (LC in short) which are collections of functions \(\psi\) defined on \(\breve V\) and which are generalizations of the wave functions of quantum mechanics. Also are defined LC on \(V\) by the condition: \(\gamma \psi=\psi\) for every \(\gamma \in \pi_1(V)\). If this later condition is satisfied for every \(\psi\), the Lagrangian manifold \(V\) is called quantized. The author obtains characterizations of quantization and the local structure of LC. Also it is proved that the quantization is preserved by Hamiltonian flows. In the last section the notion of projection of a LC on ''configuration space'' \({\mathbb{R}}^n\) is defined and it is proved that the projection is an asymptotic solution of the Schrödinger equation with rapidly oscillating initial data when the Hamiltonian is quadratic.
scientific article; zbMATH DE number 1028935