Gravitation, geometry, and nonrelativistic quantum theory

Abstract

In Cartan's description, classical particles freely falling in a Newtonian gravitational field follow geodesics of a curved spacetime. We cast this geodesic motion into generalized Hamiltonian form and quantize it by Dirac's constraint method in a coordinate-independent way. The Dirac constraint takes a simplified form in special noninertial frames (nonrotating, rigid, Galilean, and Gaussian). Transformation theory of the state function allows us to compare descriptions of a given quantum state by two different observers and to illustrate how the principle of equivalence works for quantum systems. In particular, we show that quantum states of a particle moving in a homogeneous gravitational field and of the gravitational harmonic oscillator can be reduced to the study of plane waves in an appropriate frame.