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Wednesday, 3 October 2012

Infinite-dimensional differential geometry

In quantum models, one deals with infinite-dimensional smooth Banach and Hilbert manifolds and (locally trivial) Hilbert and C*-algebra bundles. The definition of smooth Banach (and Hilbert) manifolds follows that of finite-dimensional smooth manifolds in general, but infinite-dimensional Banach manifolds are not locally compact, and they need not be paracompact. In particular, a Banach manifold admits the differentiable partition of unity if and only if its model space does. It is essential that Hilbert manifolds satisfy the inverse function theorem and, therefore, locally trivial Hilbert bundles are defined. However, they need not be bundles with a structure group.

Infinite-dimensional Kahler manifolds provide an important example of Hilbert manifolds. In particular, the projective Hilbert space of complex rays in a Hilbert space E is such a Kahler manifold.

Note that differential geometry of Banach and Hilbert bundles over a finite-dimensional smooth manifold X can be formulated in terms of differential geometry of modules over a ring of smooth functions on XIn particular, connections on a Hilbert bundle over a smooth manifold can be defined both as particular first order differential operators on the module of its sections. This is the case of time-dependent quantum mechanics where X=R.


G.Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005) #

G.Sardanashvily, Differential Geometry of Module and Rings. Application to Quantum Theory. (LAP, 2012)  #

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