Quantum Mechanics in General Relativity

Abstract

Having started with the general formulation of the quantum theory of the real scalar field (QFT) in the general Riemannian space--time $ V_{1,3} $, the general--covariant quasinonrelativistic quantum mechanics of a point-like spinless particle in $ V_{1,3} $ is constructed. To this end, for any normal geodesic 1+3--foliation of $ V_{1,3} $, a space $\Phi^-$ of asymptotic in $c^{-1}$ solutions of the field equation is specified, which can be mapped to a space $\Psi$ of solutions of a Schr\"odinger equation with an (asymptotically) Hermitean hamiltonian and the Born probabilistic interpretation of the vectors of $\Psi$. The basic operators of the momentum and the spatial position of the particle acting in $\Psi$ generated by the corresponding observables of QFT include relativistic corrections, and therefore differ generally from those which follow for the geodesic motion in $ V_{1,3} $ from the canonical postulates of quantization. In particular, the operators of coordinates do not commute as well as the operators of the conjugate momenta, except the cases of Cartesian coordinates in the Minkowski space--time or of the exact nonrelativistic limit $(c^{-1} = 0)$. This approach provides QFT in the general $ V_{1,3} $ in the Fock representation with a particle interpretation based on the Born interpretation of wave functions.