Quantum structures in Einstein general relativity

We introduce the notion of a “quantum structure” on an Einstein general rela- tivistic classical spacetime M . It consists of a line bundle over M equipped with a connection fulfilling certain conditions. We give a necessary and sufficient condition for the existence of quantum structures, and classify them. The existence and classifi- cation results are analogous to those of geometric quantisation (Kostant and Souriau), but they involve the topology of spacetime, rather than the topology of the configura- tion space. We provide physically relevant examples, such as the Dirac monopole, the Aharonov–Bohm effect and the Kerr–Newman spacetime.