Probabilistic Representation and Fall-Off of Bound States of Relativistic Schr\"odinger Operators with Spin 1/2

A Feynman-Kac type formula of relativistic Schr"odinger operators with unbounded vector potential and spin 1/2 is given in terms of a three-component process consisting of Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic field with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound states is established separately for growing and decaying potentials by using martingale methods.