Checkpoints

A checkpoint is for one short decision: what should you study next? Choose the nearest phase boundary below, produce the requested work on paper or in a notebook, and use the first decisive failure signal to select an action. Aim for a focused attempt rather than a survey of everything you remember.

This page is a reader-facing study aid, not an exam. It creates no formal assessment, score, stored result, credential, or claim of completion. Keep the capabilities separate: strength on a propagator calculation does not cancel a missing boundary condition, Ward check, or uncertainty statement. You may use your normal study notes. If you inspect a worked solution, retry with changed data so that the next attempt tests reasoning rather than recall.

1. Preparation — variations, inverses, and states

Use this checkpoint before beginning the Core sequence or whenever a later derivation repeatedly fails at its mathematical setup.

Observable tasks.

• Vary a scalar action on a region with boundary. Display the bulk Euler–Lagrange term and the complete surface term, then state boundary data that make the variational problem well posed. The Robin-wall exercise is a concrete version.
• Starting from a Fourier convention, recover the distribution associated with a declared pole prescription and explain which boundary or time-ordering information it selects. Use the pole-prescription exercise as the model.
• Insert a complete set of energy eigenstates into a two-point function. Identify the spectral weights, time phases, and state being used, as in the spectral-correlator exercise.

Review before continuing if any one of these occurs.

• A surface term is discarded without a boundary condition or falloff statement.
• An “inverse” is written without a pole prescription, support condition, or other selecting data.
• An operator, a state, and an expectation value are treated as the same kind of object, or the spectral weights are assigned without matrix elements.

Continue. Move to local actions and admissible variations and then to algebras, representations, and states when each artifact states its assumptions and passes a direct substitution or dimension check.

Review one dependency. Choose only the first broken task: boundary variation, Fourier distributions, or quantum spectral reasoning. Follow its linked exercise, then close the solution before retrying.

Retry with changed data. Change one Robin coefficient, reverse the pole boundary value, or replace the two-level spectrum by three nondegenerate levels. Reproduce the same reasoning with the new input.

2. Free fields and correlators

Use this checkpoint after the canonical scalar and functional-integral lessons, before perturbative interactions.

Observable tasks.

• Derive the scalar mode coefficient from the equal-time commutator and show that the normal-ordered Hamiltonian has positive one-particle energies. Check your normalization against the equal-time algebra.
• Obtain the same Feynman two-point function from canonical time ordering and from the regulated Gaussian source functional. Match the pole prescription, residue, and overall factor using the independent canonical check.
• Differentiate $Z[J]$ and $W[J]=-i\log Z[J]$ far enough to explain why the free four-point function is nonzero while its connected part vanishes. Compare with full and connected correlators.
• Transfer the inverse and sign check to a free Dirac field by completing the operator-and-Grassmann propagator exercise.

Review before continuing if any one of these occurs.

• The field expansion and oscillator commutator do not reproduce $[\phi,\pi]=i\delta$, or a negative-energy particle is introduced to repair a sign.
• The canonical and Gaussian propagators disagree in their $i0$, residue, or factor of $i$.
• Derivatives of $Z$ and $\log Z$ are treated as interchangeable, or Grassmann derivatives are reordered without their sign.

Continue. Begin the perturbative split when the canonical and functional calculations agree on the same regulated object, the $Z$ versus $W$ distinction is explicit, and the fermionic transfer has a sign check.

Review one dependency. Return to the scalar normalization, Gaussian cross-check, connected-generating functional, or Dirac propagator linked to the first mismatch. Do not repeat all four topics.

Retry with changed data. Change the scalar mass, exchange the oscillator normalization convention, or reverse the order of two Grassmann sources and predict every induced change before calculating.

3. Interactions and scattering

Use this checkpoint after perturbative rules and LSZ, before loop corrections.

Observable tasks.

• Derive the $-i\lambda$ quartic vertex from $-\lambda\phi^4/4!$ and independently derive the one-vertex tadpole symmetry factor. The contact-factor exercise and tadpole exercise test different counting questions.
• Start from a connected correlator with an isolated stable-particle pole and take the joint on-shell residue. State what changes when the object is only amputated or remains off shell, using the LSZ residue construction.
• For tree-level $\phi\phi\to\phi\phi$, go from $\mathcal M=-\lambda$ to the total cross section. State the flux, phase-space measure, incoming averages, identical-final factor, and mass dimension, as required in the rate construction.

Review before continuing if any one of these occurs.

• A factorial is inserted by analogy rather than traced to the action, Wick contractions, a diagram automorphism, or final-state phase space.
• External propagators are deleted without taking a pole residue, or the momentum-conservation delta function is included inside $\mathcal M$.
• A finite amplitude is called a cross section without flux, phase space, and state sums, or the final rate has the wrong mass dimension.

Continue. Move to singular-region classification when the correlator, amputated function, S-matrix amplitude, and observable remain distinct throughout one complete calculation.

Review one dependency. Revisit either contraction counting, the LSZ pole logic, or the rate formula—not the entire interaction sequence. Redo the specific linked calculation from its action or state normalization.

Retry with changed data. Replace the identical final scalars by distinguishable species, or rescale the interpolating field while keeping the physical state fixed. Predict which factors change and which observable does not.

4. Loops, scales, and gauge consistency

Use this checkpoint after regularization, renormalization, and the first gauge theory lessons.

Observable tasks.

• Evaluate one logarithmically divergent integral with a cutoff and in dimensional regularization. Match the logarithm’s coefficient, identify regulator-dependent finite terms, and keep a physical threshold cut separate from the ultraviolet divergence. The dimensional-expansion exercise supplies a compact test.
• For a one-loop quantity with explicit $\log(Q^2/\mu^2)$, differentiate at fixed bare data and recover the required cancellation with the running coupling. Then perform one finite scheme change in both directions. Use explicit–implicit scale cancellation and the scheme-change round trip.
• Build a gauge-consistency sheet for one calculation: count the Proca and Maxwell physical polarizations, state the gauge fixing, contract a complete on-shell amplitude or conserved-source response with the gauge momentum, and explain whether ghosts interact. The polarization-to-Ward bridge and gauge-theory checklist supply the checks.

Review before continuing if any one of these occurs.

• Ultraviolet, infrared, threshold, and regulator singularities are given one common explanation, or a regulator is removed before cancellation.
• The regulator, renormalization scheme, $\mu$, and physical scale $Q$ are treated as interchangeable.
• Gauge fixing is used as a physical polarization condition, a covariant polarization sum is applied before the Ward test, or a non-Abelian loop calculation omits interacting ghosts.
• A Ward or Slavnov–Taylor contraction is demanded of one incomplete diagram when only the complete stated object obeys it.

Continue. Move to matching across scales when the regulated answer has the right dimensions and analytic structure, its residual scale dependence begins beyond the retained order, and its physical gauge check is independent of auxiliary choices.

Review one dependency. Choose singular-region analysis, RG cancellation, or gauge redundancy according to the first failure signal. Keep the other two results separate rather than using them as compensating evidence.

Retry with changed data. Change the regulator, choose a different $Q/\mu$, or compare an Abelian conserved-current exchange with a non-Abelian amplitude. State the expected invariant result before recomputing.

5. EFT and observable synthesis

Use this checkpoint when moving from renormalized amplitudes to a controlled low-energy or infrared-safe prediction.

Observable tasks.

• Integrate out the heavy scalar in the worked model, match the amplitude through a declared power of $q^2/M^2$, and estimate the first omitted term. Use complete the square and test the truncation as independent algebraic and approximation checks.
• Define a family of measurement functions $F_n$ for one bin. Verify its soft and collinear limits, then write the NLO real–virtual subtraction formula with the counterterm evaluated at the correct mapped multiplicity. The thrust limit check and finite-residual exercise provide concrete data.
• Assemble a fixed-order plus resummed prediction without double counting. Re-expand it to the retained order, distinguish $\mu_R$, $\mu_F$, and sector scales, and separate perturbative, parametric, long-distance, and numerical uncertainties. Check additive matching and the hadronic scale exercise.

Review before continuing if any one of these occurs.

• Matching compares unlike external states, schemes, gauges, or infrared prescriptions, or the claimed accuracy contains operators and loops of inconsistent order.
• $F_{n+1}$ fails to approach $F_n$ in a soft or collinear limit, or a local subtraction term is removed from the real contribution without being integrated and added at lower multiplicity.
• Resummed and fixed-order terms are simply added, incoming-hadron factorization is omitted where needed, or scale variation is reported as a probability without an uncertainty model.

Continue. Continue only when each task has produced its own artifact. For a combined example, choose an outcome-driven continuation such as scattering phenomenology or renormalization and EFT for research when one named observable has a declared domain, normalization, perturbative content, matching prescription, integration method, and separated uncertainty statement.

Review one dependency. Return only to power counting and matching, measurement-function limits and subtraction, or RG resummation and uncertainties—whichever first broke the prediction.

Retry with changed data. Increase $q^2/M^2$, move the event-shape bin toward or away from an endpoint, or replace the lepton beam by a hadron beam. Reassess the expansion, logarithms, factorization inputs, and uncertainty statement rather than copying the previous conclusion.

A copyable checkpoint note

Make one scratch note per observable task. Do not total the notes or turn them into a phase score.

Checkpoint:
Task (one capability only):
Changed or supplied data:
Assumptions and conventions:
Artifact produced:
Independent check:
Decisive failure signal, if any:
Next action: Continue / Review one dependency / Retry with changed data
Dependency to review or datum to change:

The note is useful only if it changes the next study action. Keep it private, discard it, or revise it as you prefer; it does not certify anything. If several tasks fail, begin with the earliest dependency in the chain and retry the later tasks only after that input has changed.