Canonical formalism and quantization of world-line in a curved background metric

Abstract

After discussing three various approaches to the canonical formalism for a world-line in a curved background metric we develop a new covariant formalism. It is based on the Hamiltonian which, for τ=s, is equal to the proper mass and it generates translations in proper times. The resulting Poisson-bracket relations are equivalent to the geodesic equation. In the quantized theory the classical 4-velocity\(u^\mu \)is replaced by the operator\(\gamma ^\mu \) (Dirac matrices); the resulting Heisenberg equations are quantum analogue of the Papapetrou's equation for a spinning particle in a gravitational field. Our theory also predicts in a natural way the existence of an infinite bare mass term in the lagrangian for the Dirac (or Klein-Gordon) equation and thus provides a deeper understanding of the «renormalization» procedure.