The directed random walk representation of the Dirac propagator is presented using different algebraic spinor spaces. In particular, Majorana, Even, Dirac, and Chevalley–Crumeyrolle spinor spaces are used. For these cases, the spin connections were calculated and, from them, the gauge potentials. Moreover, for the Chevalley–Crumeyrolle algebraic spinor space, the use of the Dirac propagator as the inverse of the generator of a Markov process in the McKane–Parisi–Sourlas theorem is suggested.