Lattice Gauge Theory: Discretizing Spacetime for Non-Perturbative QCD Calculations — E8 Intelligence Research

Abstract

FINDING: Lattice gauge theory provides a discrete, computationally tractable formulation of quantum field theories (e.g., QCD) by discretizing spacetime onto a hypercubic lattice, enabling non-perturbative calculations of particle masses and gauge field dynamics. | MATH: Wilson action \( S_G = \beta \sum_{\text{plaquettes}} \left(1 - \frac{1}{N}\text{Re}\,\text{Tr}\,U_{\mu\nu}\right) \), with \(\beta = 2N/g^2\) for SU(N); link variables \(U_\mu(x) = e^{i a g A_\mu(x)}\); lattice spacing \(a\) sets UV cutoff; heavy-quark mass extraction via meson mass ratios \(m_H/m_S\) from Fermilab method, using one-loop lattice perturbation theory. | CONNECTION: The hypercubic lattice is a **crystallographic lattice** (Bravais lattice, \(Z^4\) in Euclidean spacetime) — its symmetry group is the hyperoctahedral group \(B_4\), a finite reflection group (Weyl group of \(D_4\) root system). The plaquette structure encodes **gauge curvature** as a holonomy around elementary squares, mirroring the discrete Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com