Quantum structures in Einstein general relativity. (English) Zbl 0977.83009

Vitolo, Raffaele

Lett. Math. Phys. 51, No. 2, 119-133 (2000).

The author presents a generalization of the geometric covariant formulation of the Galilei classical mechanics and quantum mechanics on a curved spacetime. This generalization is related to the geometric quantization. He defines an Einstein general relativistic quantum structure in analogy to the known Galilean. This new quantum structure is a Hermitian complex line bundle over spacetime endowed with a universal connection having curvature proportional to the cosymplectic form. The classification and conditions for existence of such structures are shown. They are similar to those of Kostant Souriau. A novel feature is the inclusion of the topology of the spacetime. As examples the author discusses: Minkowski spacetime, Schwarzschild spacetime, Kerr-Newman spacetime, Dirac monopole, the Aharonov-Bohm effect.

Reviewer: A.Frydryszak (Wrocław)

MSC:

83C15	Exact solutions to problems in general relativity and gravitational theory
53C05	Connections (general theory)
81S10	Geometry and quantization, symplectic methods
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
55N05	Čech types
58A20	Jets in global analysis
14F40	de Rham cohomology and algebraic geometry
70H40	Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics

Keywords:

Einstein general relativity; particle mechanics; jets nonlinear connections cosymplectic forms geometric quantization; Schwarzschild spacetime; Kerr-Newman spacetime; Dirac monopole