Quantum Chromodynamics 2026, 11 - Lattice QCD Calculations
QCD can be calculated on a lattice through a path-integral or functional-integral representation of QM and QFT. A particle propagating from xi at time ti to a point xf(tf) with trajectory x(t) and velocity v(t) of the particle connecting the initial and final points, determining the corresponding action S defined classically through its Lagrangian. This gives the basis for a path integral method of computation (7.1.1.). The operators of the path integral method have determinable expectation values through G(t1, t2) = ⟨0 | T(q(t1)q(t2))|0⟩ in the case of single-body QM systems. The time evolution of a general state can be expressed through a complete set of energy eigenstates. By way of Wick-rotation, the singularities can be removed and the resulting integrals become computable (7.1.2.). On a lattice, the infinite number of field variables needs to be reduced to a finite number, through discretizing space and time. This can be achieved through hypercubic equally spaced lattices in space and time, utilizing in the easiest mode, some unified lattice spacing constant (Eq. 7.24). The resulting Lagrangian is gauge invariant w.r.t. local phase rotations of the fields in colour space. The colour orientation in this case acts like parallel charge transport (7.1.3.).
Using the Euclidean formulation of the gluonic action, and keeping in mind that it must be gauge invariant, a simple nontrivial gauge invariant expression on the lattice can be formulated, using the link variables. If the lattice spacing is very small, a single link can be written as Uμ(x) ≈ 1 + iaAμ(x) pointing in the μ-direction (Eq. 7.30). Otherwise, one must divide the integral over the path between x and x + aeμ. In this case, the leading order derivative is very similar:
(Eq. 7.32). This expression is gauge-invariant. When applying a SU(3) gauge transformation to the point x, it resolves to
(Eq. 7.36). By application to a lattice plaquette, one eventually recovers the Wilson action
(Eq. 7.49). (7.1.4)
In the expression for the path integral and the observables calculated in the path-integral formalism, perform the integration over the group elements for every link on the lattice. SU(2) link variables are unitary 2 × 2 matrices. For integration in SU(2), integrate over every link on the lattice (7.1.5). Other field variables can be used in discretized spacetime. In 1D, the FT of a function f defined on the lattice is periodic (Eq. 7.65), which restricts the momentum integration to the first Brillouin zone. The emerging preliminaries give rise to a discretization structure for field equations of motion. As part of this, the Laplacian of the Klein-Gordon equation is discretized, setting the metric tensor to δμν, and applying a 3-point formula (p. 457). The equations of motion for a spin-1/2 field does not function via the Laplacian, as it does not feature in the Dirac equation. Instead, discretize via the γ-matrices. However, the momenta vary between -π/a and π/a, where one would expect continuity. This is the fermion doubling problem. It is usually addressed by removing the superfluous solutions. The Wilson method adds an additional term to cancel the terms at the corner of the Brillouin zone (p. 459).