The Lattice Formulation

In lattice field theory, spacetime is discretized on a hypercubic lattice with spacing $a$. The partition function becomes:

$$Z = \int \mathcal{D}\phi \, e^{-S[\phi]}$$

where the action is now a sum over lattice sites:

$$S = a^4 \sum_x \left[ \frac{1}{2} \sum_\mu (\partial_\mu \phi)^2 + \frac{m^2}{2} \phi^2 + \frac{\lambda}{4!} \phi^4 \right]$$

Discretized Derivatives

The lattice derivative is defined as:

$$\partial_\mu \phi(x) = \frac{\phi(x + a\hat{\mu}) - \phi(x)}{a}$$

Monte Carlo Integration

We sample field configurations using the Metropolis algorithm:

import numpy as np

def metropolis_step(phi, S, beta, delta=0.5):
    """Single Metropolis update sweep."""
    N = phi.shape[0]
    for i in range(N):
        for j in range(N):
            phi_old = phi[i, j]
            phi_new = phi_old + np.random.uniform(-delta, delta)
            
            dS = S(phi_new, i, j) - S(phi_old, i, j)
            
            if dS < 0 or np.random.random() < np.exp(-beta * dS):
                phi[i, j] = phi_new
    return phi

Continuum Limit

Physical observables must be extrapolated to $a \to 0$:

$$\langle O \rangle_{\text{phys}} = \lim_{a \to 0} \langle O \rangle_a$$