TABLE OF CONTENTS Algebraic quantum theory usually deals with Hilbert spaces. This Chapter addresses quantum models involving Hilbert manifolds and Hilbert bundles, but we restrict our consideration to particular Hilbert bundles over smooth finite-dimensional manifolds. For instance, this is the case of time-dependent quantum systems and quantum models depending on classical parameters.
Since a Hilbert space E is a real Banach space and the unitary group U( E) is a Banach-Lie group, we in fact deal with Banach manifolds and bundles. Their differential geometry is similar to differential geometry of finite-dimensional manifolds and bundles in main. In particular, the inverse mapping theorem and the Frobenius theorem hold. For instance, this is not the case of Fr chet manifolds. We refer the reader to [262] for the wider class of infinite-dimensional smooth manifold and principal bundles modelled on so called convenient locally convex topological vector spaces.
We start with the notion of a real Banach manifold [273; 422]. Banach manifolds are defined similarly to finite-dimensional smooth manifolds, but they are modelled on Banach spaces, not necessarily finite-dimensional. Passing later to complex Hilbert manifolds, we also refer the reader to Section 2.6 for analogy to finite-dimensional Hermitian and K hler manifolds.
Let us recall some particular properties of (infinite-dimensional) real Banach spaces (see Section 10.5A for a general case of topological vector spaces). Let us note that a finite-dimensional Banach space is always provided with an Euclidean norm.
Given Banach spaces E and H, every continuous...
