---
title: "Quantum field theory, coarse-graining and critical phenomena"
description: "Eight connected chapters and five interactive Labs, from coupled oscillators to free-field correlations, parameter matching and critical scaling."
canonical_url: "https://sandboxphysics.com/learn/field-theory/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/learn/field-theory/"
---

FR01 · A course with experiments

# Quantum field theory, coarse-graining and critical phenomena

A coarse photograph loses fine detail but keeps the shape of a mountain. A field theory can also lose short-distance detail and keep long-distance predictions. The interesting part is working out which parameters must change along the way.

Start with a chain of springs. Quantize its normal modes, calculate correlations, then put the Euclidean field on a finite lattice. By the last chapter you can check a low-energy match, challenge a data collapse and read a small-epsilon flow diagram.

Ideal analytic reference; dimensionless reference units. $\omega_{\mathrm{lat}}^2=0.25+4\sin^2(k/2),\quad \omega_{\mathrm{cont}}^2=0.25+k^2$The curves agree for long wavelengths; the difference grows near the lattice cutoff.

L1 begins with forces and waves. L2 uses quantum mechanics; L3 introduces matrices and statistical errors. L4 needs perturbation theory. Each chapter names the tools it needs and explains the next new step. You can read the derivations and examples before opening a Lab.

[L1 · 01 How a chain of oscillators becomes a field Derive a field equation from coupled springs and see how a lattice changes short waves.](https://sandboxphysics.com/learn/field-theory/from-oscillators-to-fields/) [L1 · 02 Symmetry tells us what flows Use energy flow and a local wave equation to connect Noether’s theorem with causal propagation.](https://sandboxphysics.com/learn/field-theory/symmetry-and-causality/) [L2 · 03 Quantizing the normal modes Build the free scalar vacuum from harmonic oscillators and calculate a mode’s zero-point variance.](https://sandboxphysics.com/learn/field-theory/quantizing-a-field/) [L2 · 04 What a correlation function predicts Work from a Gaussian generating function through Wick contractions to a scalar scattering amplitude.](https://sandboxphysics.com/learn/field-theory/correlators-to-scattering/) [L3 · 05 Putting a Euclidean field on a lattice Define the lattice action, derive its covariance and separate spacing, volume and sampling errors.](https://sandboxphysics.com/learn/field-theory/euclidean-lattice-and-errors/) [L3 · 06 Changing resolution while keeping predictions Integrate out alternate sites, match low momenta and test a prediction that was not fitted.](https://sandboxphysics.com/learn/field-theory/keeping-low-energy-physics/) [L3 · 07 Testing a convincing data collapse Derive a finite-size scaling curve and expose a false critical point with held-out sizes.](https://sandboxphysics.com/learn/field-theory/finite-size-scaling/) [L4 · 08 Reading a Wilson–Fisher flow diagram Reproduce a one-loop flow near four dimensions, compare critical exponents and locate the limits of universality.](https://sandboxphysics.com/learn/field-theory/wilson-fisher-and-universality/)

## Three places to test your predictions

[Vacuum correlations →Change mass, spacing and boundary conditions; compare sampled fields with the exact finite-lattice covariance.](https://sandboxphysics.com/labs/vacuum-correlations/) [Coarse-graining and parameter flow →Remove short-distance variables, preserve low momenta and test a held-out prediction.](https://sandboxphysics.com/labs/renormalization-flow/) [Finite-size scaling →Fit small sizes, then reveal large sizes to see whether an attractive collapse survives.](https://sandboxphysics.com/labs/finite-size-scaling/)

The lattice experiments use finite one-dimensional Euclidean Gaussian scalar models. Their samples represent that probability distribution. The final chapter separately studies a one-loop interacting flow near four dimensions. Model results, analytic references and source-reading scope are labeled where they appear.