Loops, Knots, Gauge Theories and Quantum Gravity 2026, 28 - Lattice Techniques

A cubic lattice in 3D euclidean spacetime with lattice sites labeled by a triplet of integers n and the unit vectors pointing along lattice orientation (note +/-) is constructed so that each link corresponds to two possible arrangements (n, μ) = (n + μ, -μ). Associate an Ising spin σ = ±1 to each link. Define a transformation to flip all spins with one lattice site and an action of S(σ) = -β&sum_{n, μ, ν}σ(n, μ)σ(n + μ, ν)σ(n + μ + ν, -μ)σ(n + μ, -ν) where β is the coupling constant, so that S is invariant under the gauge transformation at n (Eq. 6.1). The order parameter for this model is the magnetization, which, according to Wegner, could be written as the gauge invariant W_γ = Π_l∈γσ(l) (Eq. 6.2). At small values for β, its mean is proportioned as the e^-area(γ), and at large values, it decreases as e^-length(γ) (6.2.1). Occasionally it's more immediate to write a discretized version of the discretized covariant theory on a Euclidean spacetime lattice which more immediately retains the symmetry. The transfer matrix method for for a single particle in a potential, for example is close to the Euclidean path integral formulation of QM and statsitical mechanics. The vacuume energy is identified with the free energy of the statistical mechanical system (6.2.2). Applied to the Z(2) theory, the action can be split into two types of terms: One iterating over the time-like links and one iterating over the spatial plaquettes (Eq. 6.25). The gauge is fixed so the spins associated to the temporal links take the value +1. The spatial spins associated with the spatial surface, the partition function can be expressed as the actions, summed over the spatial spin configurations (Eq. 6.28), which can be rewritten in terms of "flip operators" T (6.2.3). The loop representation arises from these operators as they do in the TM theories (6.2.4).