A Nonlocal Discretization of Fields

Abstract

A nonlocal method to obtain discrete classical fields is presented. This technique relies on well-behaved matrix representations of the derivatives constructed on a non-equispaced lattice. The drawbacks of lattice theory like the fermion doubling or the breaking of chiral symmetry for the massless case, are absent in this method.

Free fermionic propagators on a lattice

“…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, .…”

“…Thus, the discrete version of a dynamical differential variable operating on such functions is essentially the matrix obtained under the replacement ∂ µ → SD µ S −1 . The error arising from this procedure can be estimated in special cases [12] and can be related to the complement of ψ(x) with respect to U [7]. Thus, if ψ(x) ∈ U and Ψ denotes the N × 1 vector of components…”

“…The similarity transformation given by S changes Ψ into itself except for an alternating change of sign along each direction when N µ → ∞ [7], therefore we may use D µ instead SD µ S −1 as a discrete representation of ∂ µ . Let g(x µ ) be a given function and g and g ′ the vectors of components g(x µ j ) and ∂g(x µ j )/∂x µ , respectively.…”

“…The use of nonlocal operators can be found among others (see for example [3]- [6]) in spite that the locality of the Dirac operator is in general a desired property. 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].…”