Quantum mechanics of a spin particle in a curved spacetime with absolute time
The authors extend the geometric formulation of nonrelativistic charged scalar particles given by two of them [\textit{A. Jadczyk} and \textit{M. Modugno}, Proc. XXI Internat. Conf. Differ. Geom. Methods in Theoretical Physics, Tianjin 1992, 543-556, World Scientific, Singapore (1992; per bibl.)] to particles with spin. The spacetime is a four-dimensional manifold with a Galileian structure and Newtonian connection. It is described here as a fibre bundle \(E\) over the one-dimensional time manifold, the fibres being equipped with a Riemannian metric. The connection is metric compatible and given on the first jet bundle \(J^1E\). It is determined by the metric on the fibres and a cosymplectic (or Poisson) form \(\Omega\) on the spacetime \(E\). For a scalar particle a quantum bundle \(Q\) is then a Hermitian line bundle over spacetime (rather than over phase space as in geometric quantization) with a connection whose curvature is the cosymplectic form on spacetime. For particles with spin the quantum bundle is replaced by a tensor product of \(Q\) and a spinor bundle (of complex dimension 2). Analogues of the Schrödinger and Schrödinger-Pauli equations are then derived. Since these quantum bundles are over spacetime, a polarization is not needed in this approach, but the class of quantizable functions is nevertheless restricted, essentially to quadratic functions in the velocity. While most constructions involve an observer field explicitly, it is verified that the main physical content of the equations is invariant under changes of the observer. For the flat Galilein spacetime, the standard nonrelativistic quantum mechanics is recovered.
scientific article; zbMATH DE number 1087268
scientific article