Random Walk Representation of the Lattice Fermionic Propagators and the Quark Model

A representation of the continuum fermionic propagator as a sum of directed random walks on a lattice is presented. Also a random walk representation for the lattice fermionic propagators is developed for the case of the naive, the Wilson, and the Kogut-Susskind fermions. For the naive fermions the phenomenon of fermion doubling appears as having $2^{D}$ distinct spin factors being associated with a single path in $D$-dimensions. In the case of the Wilson and the Kogut-Susskind fermions, in the naive continuum limit, the path integral representation coincides with the path integral representation for the continuum fermionic propagator. Using this representation the Green's functions of lattice QCD involving quark operators are written as a sum over the paths of valence quark, the gauge fields and the sea quarks being integrated out. Possible advantages of such a representation are illustrated by showing how one can use numerical simulations to obtain a heuristic insight into the relationship between QCD and the constituent quark model.