Lattice and nonperturbative methods

Replace continuum points by sites separated by spacing $a$ and derivatives by finite differences. For a free scalar with a nearest-neighbor kinetic term, each momentum component enters through $2\sin(ap_\mu/2)/a$. Expanding the sine at small $ap_\mu$ recovers the continuum momentum with corrections of order $a^2$. This is a statement about a specified discretization; improved actions can cancel leading errors, and interacting observables can have additional logarithmic factors.

Gauge theories place group-valued parallel transporters on links rather than simply replacing every gauge potential by an unconstrained site vector. Products around elementary plaquettes give gauge-invariant local terms in the Wilson action. Its conventional parameter $\beta_{\rm lat}=2N/g_0^2$ for SU(N) is a bare lattice parameter, not the renormalization-group beta function. To approach a continuum theory one tunes bare parameters while holding chosen physical quantities fixed as $a$ decreases.

Insert energy eigenstates between an operator and its adjoint in a zero-temperature Euclidean correlator. After subtracting any vacuum expectation value and setting the vacuum energy to zero, the connected correlator is a sum of decaying exponentials. If the operator overlaps the lightest state in the selected channel, sufficiently long Euclidean time suppresses heavier states relative to it. The decay rate therefore exposes an energy gap without identifying a literal particle trajectory through sampled fields.

Taking the logarithm of neighboring correlator values defines an effective mass. A plateau can indicate ground-state dominance, but it must be assessed with correlated statistical errors and changes of the fitting window or operator basis. Finite temporal extent introduces backward-propagating contributions, often requiring a hyperbolic-cosine fit. At long times the signal can decay faster than its uncertainty, so simply moving to later times is not a cost-free cure for excited-state contamination.

When the Euclidean weight is nonnegative, importance sampling can estimate expectation values without enumerating every field configuration. Consecutive samples are correlated, so the effective sample count can be much smaller than the number stored. Thermalization, autocorrelation, topology freezing, finite spatial volume, and finite lattice spacing all require checks. Statistical precision improves with more suitable samples, but it does not remove a systematic error caused by the wrong volume or an unextrapolated regulator.

At nonzero baryon chemical potential and in other settings, the weight can become complex or change sign. Sampling its absolute value and reweighting leaves a small average phase that can fall exponentially with volume, producing the sign problem. There is no general efficient classical remedy for arbitrary theories. Tensor networks, quantum simulation, contour methods, and specialized expansions address particular regimes with their own limits. None should be advertised as a universal replacement for continuum extrapolation and a transparent error budget.