Discrete-spacetime formulation of quantum field theory on a finite lattice. Founded by Wilson (1974) as a non-perturbative regulator that preserves gauge invariance exactly. Central tools: Wilson gauge action, Wilson / staggered /…
lattice-field-theory
Wilson gauge action: S = β Σ_□ [1 − (1/N) Re Tr U_□]
Wilson's 1974 lattice gauge action: gauge fields U_μ(x) ∈ SU(N) live on links of a hypercubic lattice; the action sums over all plaquettes…
Nielsen-Ninomiya no-go: naive lattice fermions double into 2^d species
Nielsen-Ninomiya (1981) no-go theorem: no local, Hermitian, translation-invariant lattice fermion action on a cubic lattice can…
Staggered (Kogut-Susskind) fermions: 4 tastes, preserves U(1)×U(1) chiral
Staggered / Kogut-Susskind fermions (1975) distribute 16 naive doublers across 16 lattice sites via a site-dependent phase η_μ(x) =…
Hybrid Monte Carlo: molecular dynamics trajectory + Metropolis accept/reject
Duane-Kennedy-Pendleton-Roweth (1987) hybrid Monte Carlo (HMC): augment the link variables with fictitious conjugate momenta π, evolve the…
Strong-coupling plaquette expansion: ⟨U_□⟩ = β/(2N²) + O(β²) for SU(N)
Strong-coupling (β → 0, i.e. g₀ → ∞) character expansion of the plaquette expectation in pure SU(N) lattice gauge theory. Expanding…
Wilson-loop area law ↔ quark confinement: ⟨W_C⟩ ∼ exp(−σ·Area)
Wilson's 1974 order-parameter criterion for confinement: the expectation value of a large rectangular Wilson loop of dimensions R×T in pure…
Asymptotic-freedom continuum limit: a(β) = a₀ · exp[−β/(2b₀ N)] for N-colour
Renormalisation-group relation between lattice spacing a and bare coupling β = 2N/g₀² in asymptotically free Yang-Mills. At two-loop…
Elitzur: gauge-orbit Haar averaging — ⟨O_noninvariant⟩ = 0 on the lattice
Elitzur framework — the canonical lattice-QFT no-go result stating that the expectation value of any non-gauge-invariant local operator…
Osterwalder-Schrader reflection positivity: ⟨θf,f⟩ ≥ 0; analytic continuation
Osterwalder-Schrader reflection positivity framework — the defining Euclidean axiom that guarantees analytic continuation to a Lorentzian…
Creutz ratio: χ(R,T) = -log[W(R,T)·W(R-1,T-1)/(W(R-1,T)·W(R,T-1))] → σ·a²
Creutz ratio framework — the canonical lattice-QCD method for extracting the static quark-antiquark string tension σ from Wilson-loop…
Elitzur theorem: Haar integrals of non-invariant operators vanish exactly
Sympy-exact symbolic witness of Elitzur's theorem using the minimal U(1) gauge group as illustrative case. Setup: θ ∈ [0,2π], sp-symbols…
Reflection positivity: G(x,y) = exp(-m(x+y)) = K(x)·K(y) is rank-1 PSD
Sympy-exact symbolic witness of reflection-positivity via a rank-1 separable Gaussian-propagator construction. Setup: m, x, y ∈ R₊…
Creutz ratio: pure area-law W=exp(-σRT) ⇒ χ=σ; strong-coupling χ=-log(β/(2N²))
Sympy-exact symbolic witness of the Creutz-ratio identity for pure area-law Wilson-loop decay and its strong-coupling expansion. Setup: σ,…
Wilson action (1974)
K Wilson 1974 lattice-gauge-action plaquette; foundation of lattice-QCD; modern HMC Duane 1987 + improved-Wilson + chiral-fermions.
Staggered fermions (KS 1975)
Kogut-Susskind 1975 staggered-fermions; rooting-trick controversy; modern HISQ + domain-wall + overlap chiral fermions.
FLAG averages
Flavor Lattice Averaging Group 2024 modern lattice-QCD averages; alpha_s + quark-masses + form-factors precision physics.
HMC (Duane 1987)
Duane-Kennedy-Pendleton-Roweth 1987 hybrid Monte Carlo; basis of all dynamical fermion lattice simulations.
Sign problem (Troyer-Wiese 2005)
Troyer-Wiese 2005 sign-problem NP-hard; modern complex-Langevin + Lefschetz-thimble + tensor-network workarounds.
Wilson lattice (1974)
K Wilson 1974 (Nobel 1982) lattice-gauge-theory + RG; modern modern foundational text + 2024 ALPS-PHYSICS-tomography supercomputers.
Creutz (1979)
M Creutz 1979 first numerical lattice-QCD; modern modern foundational text + ETM-collaboration + RBC-Edinburgh + USQCD modern.
Nielsen-Ninomiya (1981)
H Nielsen-M Ninomiya 1981 doubling-theorem; modern modern Wilson-fermion + Domain-wall Kaplan 1992 + Overlap Neuberger 1998.
FLAG averages (2024)
FLAG-Aoki 2024 lattice-flavor-averaging-group review; modern modern foundational reference for hadronic-physics + V_us / V_ud / B-mesons.
Symanzik improvement (1983)
K Symanzik 1983 + Sheikholeslami-Wohlert 1985 O(a)-improved action; modern modern foundational continuum-limit lattice-QCD precision.
HMC (Duane 1987)
S Duane 1987 hybrid-Monte-Carlo dynamical-fermion sampling; modern modern foundational lattice-simulation technique + ML-acceleration.