Covariant Schrödinger operator. (English) Zbl 1057.81050
Summary: We analyse the Schrödinger operator for a quantum scalar particle in a curved spacetime which is fibred over absolute time and is equipped with given spacelike metric, gravitational field and electromagnetic field. We approach the Schrödinger operator in three independent ways: in terms of covariant differentials induced by the quantum connection, via a quantum Lagrangian and directly by the only requirement of general covariance. In particular, in the flat case, our Schrödinger operator coincides with the standard one.
MSC:
81T20 | Quantum field theory on curved space or space-time backgrounds |
81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
58A20 | Jets in global analysis |
58J90 | Applications of PDEs on manifolds |
81V17 | Gravitational interaction in quantum theory |