Exercises
The Readiness and Core lessons already contain worked checks, exercises, and hidden answer guides. This page helps you find and use that practice without copying it into a second exercise bank.
What this page is. It is a reader-facing index to unscored practice inside the lessons. It is not a formal scored exercise bank, does not restrict access to any page, and does not award or certify mastery. Each prompt, hint when present, and solution stays in the lesson where its assumptions and conventions are explained.
Turn one exercise into three passes
Section titled “Turn one exercise into three passes”The useful unit of practice is not “read prompt, read solution.” Work the same idea three times.
- Attempt it closed. Before opening any help, write the requested object, the assumptions you think are needed, and at least one check the answer should pass. Leave a visible record of where the reasoning stops.
- Reveal help selectively. If hints are available, use them in order and stop as soon as you can continue. If the lesson offers only a solution or answer guide, read just the first useful line or equation, close it, and resume. Compare conventions, signs, dimensions, limits, and evidence—not just the final expression.
- Redo it with changed data. Change one consequential feature: a sign convention, boundary condition, spacetime dimension, mass hierarchy, kinematic limit, covariance, or numerical tolerance. Explain which steps survive and which must change.
A clean first attempt followed by a changed-data retry is stronger evidence of understanding than a polished transcription. When the retry fails, return to the smallest lesson section that supplied the missing move.
Choose the kind of work you need
Section titled “Choose the kind of work you need”Start from the failure you can see in your current calculation.
| Kind of work | Choose it when you need to | Good first stops |
|---|---|---|
| Conceptual | distinguish fields, states, particles, observables, assumptions, or domains | orientation check and fields, states, and observables |
| Derivation | reconstruct an equation from an action, symmetry, algebra, or source functional | classical actions, free scalar, and symmetry and Ward identities |
| Computation | control integrals, combinatorics, scales, expansions, or numerical error | loops and regularization, renormalization and RG, and numerical reproducibility |
| Evidence | decide whether a calculation, comparison, or uncertainty supports the claim | tests that can fail, correlated uncertainty, and infrared synthesis |
Many good exercises use more than one mode. A loop integral is computational while it is being evaluated, derivational when its regulator dependence is explained, and evidential when its residual scale dependence is used to bound a claim.
Check preparation before repairing it
Section titled “Check preparation before repairing it”Use a diagnostic section when you do not yet know whether the obstacle is background knowledge or the new QFT idea. These checks are untimed and separate: strength in one area should not hide a specific gap in another.
- Mathematics: choose linear and tensor methods, Fourier transforms, distributions, and Green functions, or complex and asymptotic methods. If one move blocks you, use the corresponding exercises in linear and tensor methods, Fourier and Green functions, or complex and asymptotic methods.
- Classical fields and spacetime: try variational and classical-field reasoning separately from Lorentz, causal, and spin reasoning. Repair only the blocked side with variational classical fields or relativity, Lorentz symmetry, and spin.
- Quantum mechanics: use states, spectra, and measurement and commutators, symmetry, and pictures as distinct checks. The quantum states and operators exercises rebuild the spectral and picture-changing moves used by the Core.
- Probability and statistics: keep probability and independence, physical ensembles, and correlated-sample uncertainty separate. Use the statistics repair exercises when one of those distinctions fails.
- Computation and evidence: begin with precision and convergence, then ask whether your tests can fail and whether another person could reproduce the calculation from a clean description. The numerical reproducibility exercises provide a focused retry.
Once a changed-data retry succeeds, return to the QFT calculation immediately. Preparation is a loop back into the subject, not a separate course to finish.
Foundations
Section titled “Foundations”Use these lessons to make the object of the calculation and its assumptions explicit.
- Orientation and conventions: check your orientation tests whether you can translate conventions and identify what a Lagrangian does not specify by itself.
- Classical fields, actions, and local dynamics practices source terms, surface variations, and the connection between symmetry and field equations.
- Quantum fields, states, and observables asks you to classify fields, states, correlators, particles, and observables without treating them as interchangeable.
A useful changed-data retry replaces a boundary condition, rescales a field, or asks which claims remain invariant after a convention change.
Free fields and correlators
Section titled “Free fields and correlators”This phase is most useful when normalizations, propagators, statistics, or physical degrees of freedom are unclear.
| Lesson practice | What to carry into the next pass |
|---|---|
| Canonical free scalar | oscillator normalization, one-particle norm, energy, and a convention translation that leaves physics unchanged |
| Functional integrals and correlators | Gaussian inversion, source differentiation, pole selection, and an independent canonical check |
| Fermions and spin | spinor completeness, anticommutation, antiparticle interpretation, and Grassmann signs |
| Vector fields and gauge redundancy | constraint counts, physical polarizations, conserved-source contractions, and the massless-limit boundary |
For a paired check, derive a propagator or equal-time algebra by one route and verify it by another. The fermion propagator exercise makes that comparison explicit.
Symmetry, interactions, and scattering
Section titled “Symmetry, interactions, and scattering”These lessons connect identities and combinatorics to normalized amplitudes and rates.
- Symmetry, currents, and Ward identities develops current signs, contact terms, boundary flux, and the distinction between breaking and an anomaly.
- Perturbative expansion and Feynman rules practices contraction counting, vertex factors, and independent loop momenta. The quartic contact-factor exercise is a compact first test.
- LSZ and tree amplitudes connects pole residues and correlators to amplitudes, flux, phase space, polarization checks, and the boundary of ordinary LSZ.
Redo one item after rescaling the interpolating field or changing a polarization representative. The amplitude or rate should change only when the physical input changes.
Loops, RG, gauge theory, EFT, and infrared observables
Section titled “Loops, RG, gauge theory, EFT, and infrared observables”Use this phase when the calculation depends on a regulator, auxiliary scale, gauge description, hierarchy, or measurement definition.
| Lesson practice | Best diagnostic question |
|---|---|
| Loops and regularization | Can you recover the divergence, logarithm, dimension, and power count with two independent checks? |
| Renormalization and RG | Does explicit scale dependence cancel parameter running through the retained order? |
| QED and Yang–Mills theory | Do covariant-derivative signs, ghost structure, color algebra, and Ward or Slavnov–Taylor checks agree? |
| Effective field theory and matching | Is the operator basis complete enough, and is the first omitted power small in the stated regime? |
| Infrared-safe observables and synthesis | Is the measurement insensitive to unresolved limits, and do subtraction, factorization, evolution, and matching fit one accuracy claim? |
Good short retries include the dimensionally regulated integral, explicit–implicit scale cancellation, EFT truncation test, and additive matching check. Change the scale ratio or retained order and state which error estimate must change with it.
Build a 30-, 60-, or 90-minute session
Section titled “Build a 30-, 60-, or 90-minute session”These are agendas, not calibrated completion times or difficulty labels. The clock tells you when to stop, record the unfinished step, and resume later; an exercise may occupy more than one session.
30 minutes: one clean retry
Section titled “30 minutes: one clean retry”- 5 minutes: choose one diagnostic or lesson exercise and state its output, assumptions, and decisive check.
- 15 minutes: attempt it without opening help.
- 5 minutes: reveal the smallest useful part of the answer guide or solution and correct the first consequential error.
- 5 minutes: change one datum or convention and redo the affected step.
This format works especially well for the orientation check or one focused readiness repair.
60 minutes: compare two formulations
Section titled “60 minutes: compare two formulations”- 10 minutes: choose one normalization or propagator item from the canonical scalar exercises.
- 20 minutes: complete a closed attempt and record its normalization and pole prescription.
- 20 minutes: use one functional-integral exercise to recover the corresponding information from a Gaussian or source derivative.
- 10 minutes: compare the two routes, then change the oscillator normalization or source convention and verify that the physical correlator is unchanged.
90 minutes: follow a consistency chain
Section titled “90 minutes: follow a consistency chain”- 20 minutes: extract a pole and logarithm with the dimensionally regulated integral.
- 20 minutes: test how that logarithm participates in scale cancellation.
- 20 minutes: examine a controlled expansion with the EFT truncation test.
- 20 minutes: connect resummed and fixed-order information with the additive matching check.
- 10 minutes: write one paragraph separating regulator dependence, renormalization-scale dependence, EFT power corrections, and matching accuracy. Mark any step that still relies on an opened solution.
Continue with the Core roadmap when you want the conceptual sequence, return to Readiness when a prerequisite move is blocking the work, or choose a specialist pathway when you can name the calculation you want to carry farther.