Geometry in quantum theory II: Infinite-dimensional fiber bundles

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. This is the space the pure states of a C*-algebra A associated to the same irreducible representation of A in a Hilbert space E. Therefore, it plays a prominent role in many quantum models. For instance, it has been suggested to consider a loop in the projective Hilbert space, instead of a parameter space, in order to describe Berry's phase.

Sections of a Hilbert bundle over a smooth finite-dimensional manifold X make up a particular locally trivial continuous field of Hilbert spaces. Conversely, one can think of any locally trivial continuous field of Hilbert spaces or C*-algebras as being the module of sections of a topological fibre bundle. Given a Hilbert space E, let B be some C*-algebra of bounded operators in E. The following fact reflects the non-equivalence of Schrodinger and Heisenberg quantum pictures. There is the obstruction to the existence of associated (topological) Hilbert and C*-algebra bundles E->X and B->X with the typical fibres E and B, respectively. Firstly, transition functions of E define those of B, but the latter need not be continuous, unless B is the algebra of compact operators in E. Secondly, transition functions of B need not give rise to transition functions of E. This obstruction is characterized by the Dixmier--Douady class of B in the third Cech cohomology of X.

There is a problem of the definition of a connection on C*-algebra bundles which comes from the fact that a C*-algebra need not admit non-zero bounded derivations. An unbounded derivation  of a C*-algebra A obeying certain conditions is an infinitesimal generator of a strongly (but not uniformly) continuous one-parameter group of automorphisms of A. Therefore, one may introduce a connection on a C*-algebra bundle in terms of parallel transport curves and operators, but not their infinitesimal generators. Moreover, a representation of A does not imply necessarily a unitary representation of its strongly (not uniformly) continuous one-parameter group of automorphisms. In contrast, 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.

Instantwise geometric quantization of time-dependent mechanics is phrased in terms of Hilbert bundles over the time axis R. Holonomy operators in a Hilbert bundle with a structure finite-dimensional Lie group are well known to describe the non-Abelian geometric phase phenomena. At present, holonomy operators in Hilbert bundles attract special attention in connection with quantum computation and control theory.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric and Algebaic Topological Methods in Quantum Mechanics (WS, 2005)
G.Sardanashvily, G.Giachetta, What is a geometry in quantum theory, arXiv: hep-th/0401080
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2009)