A Lagrangian path integral approach to the qubit

Abstract

A Lagrangian description of the qubit based on Schwinger’s picture of Quantum Mechanics that allows for a Feynman-like computation of its probability amplitudes is presented. The Lagrangian is a function on the groupoid that describes the qubit and at the same time determines a self-adjoint element on its associated algebra. Feynman’s paths are replaced by histories on the groupoid which form a groupoid again, and a simple method to compute the sum over all histories is discussed. The unitarity of the theory described in this way imposes quantization conditions on the parameters determining the Lagrangian, and some particular instances are solved completely.