Lattice QCD Discretization and the Dirac Operator Spectrum — E8 Intelligence Research

FINDING: Lattice QCD discretizes spacetime into a finite 4D grid to compute quantum chromodynamics non-perturbatively, with the Dirac operator's spectrum encoding fermion dynamics. | MATH: Lattice spacing \\(a\\); gauge links \\(U_\\mu(x) = e^{i a g A_\\mu(x)}\\); Wilson action \\(S_W = \\beta \\sum_{\\text{plaquettes}} \\left(1 - \\frac{1}{3} \\text{Re Tr } U_{\\mu\\nu}\\right)\\), \\(\\beta = 6/g^2\\); Dirac operator \\(D = \\gamma_\\mu \\nabla_\\mu + m\\) (naive), Wilson term \\(-\\frac{r a}{2} \\nabla^2\\) to remove doublers; domain-wall fermions add a fifth dimension \\(L_s\\) with mass \\(m_f = m_0 - m_{\\text{res}}\\); staggered fermions reduce spinor doubling via \\(4^d\\) rooting. | CONNECTION: The lattice is a discrete complex plane scaffold — a hypercubic lattice with \\(Z_4\\) rotational symmetry (crystallographic point group \\(O_h\\) in 3D, extended to 4D). The Dirac operator's spectrum on a finite lattice approximates a circle/ellipse in the complex plane (for free fermions, eigenvalues lie on a circle of radiu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com