Solutions and feedback

A solution is most useful after it changes what you can do unaided. Preserve your own attempt, compare the reasoning at the first consequential difference, and then close the solution before trying a related problem. This page gives a practical method for doing that with the solved work throughout Readiness and Core QFT.

There is no site-wide score or formal assessment attached to these solutions. They are public study tools. A correct final expression is useful evidence about one task, not a credential or a claim that every related capability has been established.

Before you open the solution

Save four pieces of your attempt:

1. Your reading of the question. Restate the requested object, assumptions, domain, and accuracy in one or two sentences.
2. Your conventions. Record the metric, Fourier sign, normalization, regulator, basis, or statistical definition that can affect the result.
3. Your last justified line. Stop where a specific step becomes unclear; do not replace the gap with “then it follows.”
4. Two checks. Try dimensions and one physical or mathematical limit before looking for help.

An incomplete derivation with these notes is much easier to diagnose than an unsupported answer. If you cannot restate the task, return to its surrounding lesson before consulting the calculation.

Use hints progressively

When a lesson provides a collapsed solution rather than separate hints, make your own hint ladder. Reveal only as much as you need:

1. Principle. Which symmetry, variational rule, representation, expansion, or probability identity governs the step?
2. Structure. Which decomposition, contour, diagram, basis, or limiting regime makes that principle usable?
3. Intermediate relation. What equation should connect the last secure line to the desired result?
4. Full comparison. Only now compare the complete derivation and checks.

For example, if a Fourier-space Green function has the wrong sign, first ask which transform convention is in use. Next derive the symbol of the differential operator. Only then compare the pole prescription or inverse transform. This sequence distinguishes a convention mismatch from a failed inverse.

Useful starting points include the solved work in mathematical preparation, free scalar fields, Ward identities, and renormalization and running.

Compare reasoning, not just answers

Locate the first difference that can affect the conclusion. Later differences may be consequences of that one.

Difference	Question to ask	Useful next action
Convention	Would both results agree after translating signs, Fourier factors, state normalization, or index placement?	Write the translation explicitly and recompute one invariant check.
Algebra or combinatorics	Is a derivative, contraction, symmetry factor, Jacobian, or degeneracy missing?	Recalculate the smallest affected subexpression and test dimensions or a simple case.
Physical assumption	Did one derivation change the state, boundary condition, gauge treatment, scale hierarchy, or measured object?	State both domains; do not average incompatible answers.
Approximation	Are different perturbative orders, regulators, expansions, or truncations being compared?	Re-expand both results to the same order and compare the first omitted term.
Evidence	Does a numerical run establish convergence, uncertainty, or only execution?	Add a benchmark, refinement test, independent method, or changed-data reproduction.
Scope	Is a local result being treated as a universal claim?	Rewrite the conclusion with its hypotheses and failure boundary.

Equivalent calculations may look very different. Compare invariant quantities, pole locations, conserved charges, normalized rates, or basis-independent predictions before deciding that notation signals disagreement.

Worked feedback example: one missing symmetry factor

Consider massless $2\to2$ scattering in a real scalar theory with

$$\mathcal L_{\mathrm{int}}=-\frac{\lambda}{4!}\phi^4.$$

The tree amplitude is $\mathcal M=-\lambda$. Suppose an attempt reports

$$\frac{\mathrm d\sigma}{\mathrm d\Omega} =\frac{\lambda^2}{64\pi^2s}, \qquad \sigma=\frac{\lambda^2}{16\pi s}.$$

The dimensions and angular integration are consistent, so “check the algebra” would be poor feedback. The earliest consequential difference is that the two final particles are identical. When the full final-state phase space is integrated, it carries $1/2!$:

$$\frac{\mathrm d\sigma}{\mathrm d\Omega} =\frac{1}{2!}\frac{\lvert\mathcal M\rvert^2}{64\pi^2s} =\frac{\lambda^2}{128\pi^2s}, \qquad \sigma=\frac{\lambda^2}{32\pi s}.$$

The same physics can be represented by integrating only a nonduplicated half of the angular domain and omitting the explicit $1/2!$. Mixing the half-domain and full-domain conventions would introduce another factor of two.

Good feedback therefore has four parts:

Observation: the amplitude and dimensions are correct, but the reported full-sphere rate counts each identical final state twice.
Cause: the final-state phase-space symmetry factor is missing.
Correction: include $1/2!$, or integrate a nonduplicated angular domain, but not both.
Transfer test: repeat the calculation for distinguishable final scalars; the factor is then absent.

This example is developed with the LSZ normalization and phase-space conventions in LSZ reduction and tree amplitudes.

What a complete solution should expose

A useful solution does more than display the official route. As applicable, it should make visible:

• the assumptions, conventions, and allowed identities;
• the requested chain of reasoning rather than only the final expression;
• dimensions, signs, normalization, symmetry, analytic structure, limiting cases, or numerical tests;
• the approximation order and first omitted contribution;
• alternative valid methods and the translation between their conventions; and
• what the result establishes—and what it does not.

If a long proof or computation is intentionally omitted, the solution should still state the hypotheses, decisive idea, and exact source for the missing step. A citation is not a substitute for explaining how the cited result applies.

Give feedback that creates a next action

Use a compact note:

task:
first secure result:
first consequential difference:
classification:
  preparation / convention / algebra / physics / approximation / evidence / scope
why it changes the conclusion:
smallest repair:
changed-data retry:
check that could falsify the repair:

The classification matters because the remedies differ. A missing Fourier identity calls for focused preparation; a sign convention calls for translation; an unjustified truncation calls for an error test; and an undefined observable calls for reframing the problem.

Feedback should be specific without pretending there is only one valid presentation. “Wrong sign” is less useful than “with the stated Fourier convention, each derivative contributes $-ip_\mu$; translating that step changes the pole denominator.” “Needs more evidence” is less useful than “repeat at two finer resolutions and fit the predicted leading cutoff power.”

Check a solution independently

For a consequential result, use at least one check that does not repeat the same derivation:

• derive it in another representation, gauge, basis, or regularization scheme;
• test a free, Abelian, static, threshold, ultraviolet, or infrared limit;
• compare a Ward identity, conserved quantity, spectral sign, or normalization;
• reproduce a numerical fixture in a clean environment and vary resolution or random seed;
• compare two lawful sources that play different roles, such as an original result and a later review; or
• ask whether changed data or boundary conditions produce the predicted response.

Agreement is strongest when the checks fail in different ways. Two implementations sharing the same formula or copied input are not fully independent.

When the task or solution may be wrong

Do not force your work to agree. Isolate the smallest disputed statement, record the route and heading, show the competing calculations in the same conventions, and identify the consequence. Check the surrounding assumptions and cited source before proposing a change.

The site’s editorial and review process explains how scientific claims are checked. Use Contributing to report a reproducible issue or propose a correction. Do not include private data, copyrighted answer keys, or credentials in a public report.

Continue

• Browse solved exercises when you want a task matched to the topic you are studying.
• Use a phase checkpoint when the question is whether to continue or review one dependency.
• Choose a synthesis project when you are ready to combine several ideas into one result.
• Return to Practice to choose by purpose.