A discrete scheme for the Dirac and Klein–Gordon equations

Free fermionic propagators on a lattice

“…In this section we will obtain some properties of this function. First, we will show the equivalence between (33) and the discrete propagator given in [8]. By using the asymptotic formula (9) we have that lim…”

“…for an alternating change of sign [8,7]. By using an asymptotic expression for the Hermite zeros we find that C Nµ becomes proportional to the standard measure of a Riemann integral in each variable (the difference between two consecutive lattice points):…”

“…In this subsection we only present the main results of our discretization scheme; proofs and further applications can be found in [7,8,9,10,11,12]. Let us consider the non-equispaced four dimensional lattice constructed with the set of nodal points x µ j (µ denotes the Lorentz index and j = 1, 2, .…”

“…In this context, a new approach to the problem of discretization of fields is presented in [7]. This approach is based on a non-equispaced lattice, performed through the zeros of Hermite polynomials, where both discrete derivatives and Fourier transform have support [8,9]. The technique yields a projection of the quantum algebras on a finite linear space yielding matrix representations for the partial derivatives that produce discrete hermitian actions with nonlocal kinetic terms.…”

A Nonlocal Discretization of Fields

“…The classical discrete propagators approach to their continuum forms when the number of nodes tends to infinity and a relationship between this technique and that of path-integrals can be established. Such a technique can be seen as a projection of the quantum algebras on a finite linear space yielding matrix representations for the partial derivatives that produce hermitian actions with nonlocal kinetic terms [9,11]. This scheme yields Dirac operators that are free from some drawbacks exhibited in the standard lattice method as the fermion doubling, the chiral symmetry breaking for the massless case or the yielding of different dispersion relations for bosons and fermions.…”