Solution to the fermion doubling problem for supersymmetric theories on the transverse lattice

Abstract

Species doubling is a problem that infects most numerical methods that use a spatial lattice. An understanding of species doubling can be found in the Nielsen-Ninomiya theorem which gives a set of conditions that require species doubling. The transverse lattice approach to solving field theories, which has at least one spatial lattice, fails one of the conditions of the Nielsen-Ninomiya theorem nevertheless one still finds species doubling for the standard Lagrangian formulation of the transverse lattice. We will show that the Supersymmetric Discrete Light Cone Quantization (SDLCQ) formulation of the transverse lattice does not have species doubling.

N=(1,1) super-Yang–Mills on a (2+1)-dimensional transverse lattice with one exact supersymmetry

“…which is free from the fermion species doubling problem [22]. Furthermore, one can check that this Q − commutes with P + obtained from L; [Q − , P + ] = 0.…”

“…( 6) is satisfied. Before discussing the physical constraint in more detail, let us point out the fact that this naive Lagrangian formulation is not free from the fermion species doubling problem, while our SDLCQ formulation that we will introduce in the next section actually is [22]. Nonetheless, the constraint equation would still be valid since the constraint equation (4,6) was derived from δL δA − n − ∂ + δL δ∂ + A − n = 0 in which we do not have any problematic terms responsible for the doubling problem, i.e.…”

“…The transverse lattice formulation of N = 1 SYM theory in 3+1 dimension presented in the previous section has several undesirable features. First and foremost the naive Lagrangian suffers from the fermion species doubling problem [22]. Second, the supersymmetric structure of the theory is completely hidden.…”

“…Instead we seem to have supersymmetric Hamiltonian of 8 th order in fields. However, since this SDLCQ Hamiltonian is free from the doubling problem [22] and since the supercharge Q − α , where α = 1, 2, is of 5 th order and it is this Q − α that we make use of for our calculations, we think that this SDLCQ formulation is still more advantageous than the naive DLCQ formulation. There can, of course, be many discrete formulations that correspond to the same continuum theory and it is therefore desirable to search for a better one.…”

“…In fact it is well known that the transverse lattice suffers from the doubling problem [21]. However, the authors have found that SDLCQ formulation of a transverse lattice model is automatically free of the doubling problem [22].…”