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The conceptual spine

See how the central ideas of QFT fit together

Core QFT is a guided roadmap through fields, quantization, correlators, interactions, scattering, renormalization, gauge theory, effective field theory, and infrared-safe observables. Use it to understand the dependency map, refresh a calculation, or connect detailed readings across QFT.org.

Want a taught course?QFT I is the sequential first course; this roadmap is a cross-site conceptual bridge and refresher.Open QFT I
Want a flexible roadmap?Follow the phases below, or enter at the first result you cannot yet reproduce.Check preparation

Five phases

From an action to a controlled observable

After functional integrals, study fermions alongside the vector-field → Ward-identity branch. The default roadmap completes both before perturbation theory; their distinct inputs converge again in QED and Yang–Mills. Once loops and RG are secure, gauge theory and EFT are compatible branches; the order shown keeps one prediction narrative.

  1. 01

    Foundations

    Fix the physical question, conventions, action, boundary data, and the distinction between a field, a state, and an observable.

    1. Orientation, conventions, and study contractChoose an entry point and lock the conventions used downstream.
    2. Classical fields, actions, and local dynamicsDerive local field equations while retaining boundary and symmetry assumptions.
    3. Quantum fields, states, and observablesSeparate fields, states, observables, particles, and formulations.
  2. 02

    Free fields and correlators

    Quantize the scalar field, connect canonical and functional descriptions, then learn the distinct structures carried by fermions and spin-one fields.

    1. Canonical quantization and the free scalarFix scalar normalization, spectrum, and the causal propagator.
    2. Functional integrals and correlatorsRecover Gaussian correlators from sources and connect them to canonical quantization.
    3. Fermions, spin, and anticommutationControl Dirac spinors, anticommutation, positive-energy states, and signs.
    4. Vector fields and gauge redundancySeparate constraints, gauge redundancy, gauge fixing, and physical polarizations.
  3. 03

    Symmetry, interactions, and scattering

    Use symmetry and Ward identities to control perturbation theory, then connect correlators to normalized amplitudes and rates.

    1. Symmetry, currents, and Ward identitiesTurn symmetries into current constraints with contact and breaking terms visible.
    2. Perturbative expansion and Feynman rulesDerive Feynman rules, combinatorics, and statistics from the action.
    3. LSZ and tree amplitudesConvert correlators into normalized amplitudes, rates, and consistency checks.
  4. 04

    Loops, scales, and gauge dynamics

    Regulate loop integrals, define finite quantities, follow their scale dependence, and apply the consistency checks of QED and Yang–Mills theory.

    1. Loops and regularizationDiagnose singular regions and compare regulators before renormalizing.
    2. Renormalization and the renormalization groupDefine finite inputs, follow scale dependence, and distinguish scheme from physics.
    3. QED and Yang–Mills theoryCombine matter, gauge fields, color, ghosts, and Ward or Slavnov–Taylor checks.
  5. 05

    Effective theories and observables

    Use effective field theory, matching, infrared safety, factorization, and uncertainty to say where a prediction is reliable and where it breaks down.

    1. Effective field theory and matchingMatch a controlled low-energy theory, power count omissions, and estimate breakdown.
    2. Infrared-safe observables and synthesisDefine what is measured, combine real and virtual radiation, and state uncertainties and limits.

Use the roadmap actively

Read less, reproduce more

  • Before a module: state the result you are trying to obtain and the assumptions it needs.
  • During the detailed reading: track conventions, normalizations, boundary conditions, and approximation order.
  • Before continuing: reproduce the module’s central calculation or identity and test one limit or invariant.
  • When blocked: use the named readiness route, repair one skill, and return to the same calculation.

Where the core leads

Choose the next question, not a completion badge

The final synthesis asks whether you can define an observable, assemble its calculation, check its symmetries and scale dependence, estimate its uncertainty, and state its domain. That is a useful basis for choosing a specialist direction; it is not a claim that all QFT is perturbative or described by an ordinary S-matrix.

Compare specialist pathways