\(\theta\)-structures in quantum theory in view of geometric quantization
The authors present a geometric quantization scheme from which topological effects \((\theta\)-structures) such as the Aharonov-Bohm effect should emerge naturally. Presupposing a classical phase space of the form \(M=T^*Q\), they restrict themselves to quantization with respect to the Schrödinger polarization and proceed in three steps. (i) Consider the universal covering manifold \(\tilde Q\) as a \(\pi_ 1(Q)\)- principal fibre bundle over configuration space \(Q\). (ii) Choose an irreducible unitary representation of \(\pi_ 1(Q)\) on \(C^ n\) and construct the associated Hermitian vector bundle with \(n\)-dimensional fibre over \(Q\). (iii) Construct the quantum mechanical Hilbert space \({\mathcal H}\) as the set of square-integrable sections in this associated bundle. Whenever \(\pi_ 1(Q)\) admits nonequivalent irreducible unitary representations, there is an ambiguity in this construction. This is the origin of \(\theta\)-structures. Moreover, the authors show that the elements of \({\mathcal H}\) can be reinterpreted as functions \(\tilde Q\to C^ n\), whose transformation behavior under \(\pi_ 1(Q)\) can be characterized in terms of a natural flat connection. The paper is almost exclusively concerned with the construction of the Hilbert space. The question of how to assign self-adjoint operators to phase space functions is only briefly touched upon.
scientific article; zbMATH DE number 167641