Compressed QFT refresher

Start with the result you need, not with the first page of a course. This pathway is for a returning reader who can still recognize QFT but cannot yet reconstruct one calculation with its assumptions and checks visible.

Make this decision now. If you can derive your chosen result from stated inputs, explain where each assumption enters, and pass two independent checks, go directly to the specialist choice at the end. If one line fails, review only the dependency responsible for that line and then resume the same derivation. If you need a sequential first graduate course instead, use QFT I.

Choose one derivation, not a subject

“Review QFT” is not a finishable task. A useful target is one formula, identity, or controlled approximation that you can reconstruct on a blank page. It should name five things:

result:
starting action, state, or observable:
kinematics and regime:
approximation order and allowed inputs:
two checks and the specialist question it unlocks:

Make the target narrow enough that success is unambiguous.

Too broad	A usable target
Review Ward identities	Derive the contact terms in the complex-scalar two-point Ward identity and check that their integral vanishes
Remember scattering theory	Derive the tree-level $\phi\phi\to\phi\phi$ cross section from $-\lambda\phi^4/4!$, including the final-state factor
Relearn renormalization	Show how a one-loop logarithm cancels the running of a coupling through the calculated order

The calculation need not be advanced. It needs to be complete enough that a sign, normalization, symmetry, dimension, or limit could prove it wrong.

Filled target template

result:
  sigma(phi phi -> phi phi) = lambda^2/(32 pi s) at tree level

starting action, state, or observable:
  one massive real scalar in the Minkowski vacuum,
  L_int = -lambda phi^4/4!, covariantly normalized one-particle states

kinematics and regime:
  elastic 2 -> 2 center-of-mass scattering; full solid angle;
  two identical final particles; energies below any omitted threshold

approximation order and allowed inputs:
  first order in lambda; free external residues Z = 1

two checks and specialist question:
  [sigma] = mass^(-2); exchanging the final particles changes nothing
  next question: how does this become a loop-corrected, infrared-safe observable?

This target is precise enough to decide where a reconstruction fails. It does not require reviewing gauge theory, renormalization, or EFT before the tree calculation begins.

Test only the dependencies the derivation uses

Attempt the target first. Use this table only after you know which step is missing.

Dependency	Quick reconstruction test	Open only if the test fails
Objects and correlators	Name the field, state, observable, and ordering prescription; explain whether the target is a correlator, response, amplitude, or rate. For a Gaussian theory, recover the two-point function as the inverse quadratic kernel with its boundary condition.	Quantum fields, states, and observables and Functional integrals and correlators
Symmetry	Localize the relevant transformation and retain current, contact, explicit-breaking, boundary, and possible Jacobian terms. State the identity the complete calculation must obey.	Symmetry, currents, and Ward identities
Perturbation and LSZ	Derive the propagators and vertices from the declared action; distinguish connected, amputated, and on-shell objects; include external residues, flux, phase space, sums, averages, and identical-particle factors.	Perturbative expansion and Feynman rules and LSZ reduction and tree amplitudes
Loops and RG	Classify UV, IR, and threshold regions; state the regulator and renormalization condition; verify that explicit logarithms and implicit running cancel through the retained order.	Loops and regularization and Renormalization and the renormalization group
EFT and infrared structure	Use this row only if the target removes a heavy mode, crosses separated scales, or includes unresolved massless radiation. Match and power count the first omitted term, or define the measurement and test its soft and collinear limits.	Effective field theory and matching and Infrared-safe observables and synthesis

A mathematical operation can fail even when the QFT concept is familiar. If the obstacle is a Fourier transform, distribution, tensor contraction, variation, Lorentz representation, or operator manipulation, use the smallest matching readiness review and return to the same line.

The omissions are intentional:

• A correlator or response calculation may not use LSZ.
• A tree calculation does not require a loop or RG review.
• A theory with no removed heavy scale does not need an EFT step merely because EFT is important elsewhere.
• A massive exclusive tree process does not need an infrared-safety analysis before its leading result is reconstructed.
• A numerical target adds statistical and reproducibility checks; it does not replace the analytic assumptions behind the quantity being computed.

Three concrete refresher missions

Mission 1: recover a charged-scalar Ward identity

For a complex scalar with global transformation $\delta\phi=i\phi$ and $\delta\phi^\dagger=-i\phi^\dagger$, reconstruct

$$\partial_\mu \left\langle \mathrm T\{j^\mu(x)\phi(y)\phi^\dagger(z)\} \right\rangle = \left[ \delta^{(d)}(x-z)-\delta^{(d)}(x-y) \right] \left\langle \mathrm T\{\phi(y)\phi^\dagger(z)\} \right\rangle .$$

A complete reconstruction identifies the state and time ordering, derives the current by localizing the global transformation, fixes both contact signs, and states why boundary flux and a regulated Jacobian matter. Integrating over $x$ makes the two contacts cancel, as required for a neutral two-point function. The operator and functional derivations are compared in Weinberg 1995, §10.4, pp. 442–448.

Use the objects/correlators and symmetry rows. Do not add LSZ unless the goal is to turn the insertion identity into an on-shell amplitude statement.

Mission 2: turn a quartic interaction into a rate

Starting from

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

recover the contact vertex, connected four-point function, on-shell amplitude, and total elastic cross section for two identical massive real scalars:

$$\mathcal M_{\mathrm{tree}}=-\lambda, \qquad \sigma_{\mathrm{tree}}=\frac{\lambda^2}{32\pi s}.$$

A complete result shows where the $4!$ cancels, where external propagators are removed, and why the full final-state solid angle requires $1/2!$. Use the perturbation/LSZ row. The miniature derivation below supplies the comparison.

Mission 3: make one-loop scale independence visible

For a finite dimensionless quantity

$$F(Q) = g^2(\mu) +g^4(\mu) \left[ A\ln\frac{Q^2}{\mu^2}+C \right] +O(g^6), \qquad \beta(g)=b g^3+O(g^5),$$

show that fixed-bare scale independence requires $A=b$, integrate the one-loop flow, and check that choosing $\mu=Q$ reproduces the fixed-order logarithm after re-expansion. If the target compares schemes, also perform the finite round trip $g'=g+c g^3$, which requires $C'=C-2c$ at this order. The fixed-bare construction is developed in Collins 1984, §§7.1–7.3, pp. 169–185.

Use the loops/RG row. Add EFT only if the actual target also crosses a heavy threshold or evolves operators in a lower-energy theory.

Miniature reconstruction: quartic interaction to cross section

Take one real scalar of mass $m>0$ with

$$\mathcal L = \frac12(\partial\phi)^2 -\frac12m^2\phi^2 -\frac{\lambda}{4!}\phi^4.$$

The interaction exponential contributes $-i\lambda/4!$ at first order. In the connected four-point function, the four labeled external fields can attach to the four fields at the vertex in $4!$ ways. The factors cancel, leaving the vertex $-i\lambda$ and

$$\begin{aligned} \widetilde G^{(4)}_{c,\mathrm{tree}} ={}& (2\pi)^4\delta^{(4)} \left(\sum_{r=1}^{4}q_r\right)(-i\lambda) \\ &\times \prod_{r=1}^{4} \frac{i}{q_r^2-m^2+i0}. \end{aligned}$$

This is still a connected correlator with external poles. For canonically normalized free external fields, $Z=1$. Multiplying by the four inverse poles, taking the joint on-shell limit, and stripping the single overall momentum-conservation delta function gives

$$i\mathcal M_{\mathrm{tree}}=-i\lambda, \qquad \mathcal M_{\mathrm{tree}}=-\lambda.$$

The reduction requires isolated stable one-particle poles and asymptotic states; it is not a rule for arbitrary backgrounds or unstable external particles. Those assumptions are part of the original LSZ result Lehmann, Symanzik, and Zimmermann 1955, pp. 205–225.

For elastic equal-mass scattering in the center-of-mass frame,

$$\frac{\mathrm d\sigma}{\mathrm d\Omega} = \frac{1}{64\pi^2s} \frac{|\mathbf p_f|}{|\mathbf p_i|} \frac{1}{S_f} |\mathcal M|^2.$$

Here $|\mathbf p_f|=|\mathbf p_i|$. Integrating over the full labeled solid angle counts the two identical final particles twice, so $S_f=2!$. Therefore

$$\frac{\mathrm d\sigma_{\mathrm{tree}}}{\mathrm d\Omega} = \frac{\lambda^2}{128\pi^2s}, \qquad \sigma_{\mathrm{tree}} = 4\pi\frac{\lambda^2}{128\pi^2s} = \frac{\lambda^2}{32\pi s}.$$

The covariant normalization, flux, and phase-space factors used here are derived in Schwartz 2014, §5.1, pp. 57–63.

Four checks close the reconstruction:

• In four dimensions, $[\lambda]=0$, so $[\mathcal M]=0$ and $[\sigma]=-2$.
• The amplitude is unchanged when identical external scalars are exchanged.
• A local contact interaction has no exchange pole.
• Integrating over one permutation-ordered hemisphere with no $1/2!$ gives the same total rate; using both prescriptions would divide by two twice.

If this was the result you needed, stop. Loop corrections, running, EFT, and infrared analysis become relevant only when the next physical question requires them.

Skip and re-enter without restarting

Use this procedure in plain terms:

1. Try the derivation without notes.
2. Mark the first line you cannot justify.
3. Open the one Core lesson or readiness review that teaches that operation.
4. Redo the failed line with a changed momentum assignment, parameter, or example so that recognition alone is not enough.
5. Resume the original derivation at the next line.

Skip a Core lesson when you can already produce the particular result your target uses and pass its checks. Do not require yourself to reproduce every result on that lesson. Conversely, remembering a final formula is not enough when you cannot explain the assumption or normalization that makes it true.

Examples make the re-entry rule concrete:

• If you derive $\mathcal M=-\lambda$ but obtain a cross section twice as large, return to flux, phase space, and the identical-final-state factor in LSZ and tree amplitudes. Do not restart with Gaussian integrals.
• If the explicit logarithm and running coupling add instead of cancel, return to the fixed-bare derivative in renormalization and the RG. Keep the already-checked loop coefficient.
• If you cannot say whether a pole belongs to a correlator, an asymptotic particle, or a measured rate, return first to fields, states, and observables before applying LSZ.

If the target itself changes, write a new target template and retest the dependencies. That is the only reason to restart the selection.

Self-checks

1. Diagnose the factor-of-two error

For the quartic-scalar mission, you obtain $\mathcal M=-\lambda$ and $\sigma=\lambda^2/(16\pi s)$ after integrating over the full solid angle. Which dependency failed, and what is the smallest repair?

Solution

The amplitude is correct, so the interaction normalization, Wick combinatorics, and LSZ amputation need not be repeated. The rate is twice the correct result because the full labeled phase space counts the two identical final scalars twice. Insert $S_f=2!$ in the rate formula, or integrate over one permutation-ordered half of phase space without that factor. The smallest repair is the flux-and-phase-space section of LSZ and tree amplitudes.

2. Check the RG sign

For Mission 3, compute $\mu\,dF/d\mu$ through $O(g^4)$. What must $A$ be, and which terms can be ignored at this order?

Solution

Running the tree term gives

$$\beta(g)\frac{\partial g^2}{\partial g} = 2g\beta(g) = 2b g^4+O(g^6).$$

The explicit logarithm gives

$$\mu\frac{\partial}{\partial\mu} \left( A g^4\ln\frac{Q^2}{\mu^2} \right) = -2A g^4+O(g^6).$$

Differentiating $g^4$ inside the one-loop term starts at $O(g^6)$ and is beyond the stated accuracy. Hence

$$\mu\frac{dF}{d\mu} = 2(b-A)g^4+O(g^6),$$

so $A=b$. If the two retained contributions have the same sign, either the logarithm was differentiated incorrectly or the beta-function convention was changed without translating the formula.

References

• John C. Collins, Renormalization, Cambridge University Press, 1984, doi:10.1017/CBO9780511622656.
• Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann, “On the Formulation of Quantized Field Theories,” Il Nuovo Cimento 1 (1955), 205–225, doi:10.1007/BF02731765.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.

Choose one exact specialist continuation

Choose by the next result you will produce, not by the broadest subject name.

Next result	Continue with
An amplitude, phase-space integral, factorized cross section, or precision observable	Scattering calculations for phenomenology
A QED, Yang–Mills, QCD, Standard Model, hadron, or nuclear calculation	QFT for particle and nuclear physics
A matched operator basis, running Wilson coefficient, multiscale EFT, or controlled EFT error	Renormalization and EFT for working researchers
A many-body Green function, collective phase, response, critical point, or topological observable	QFT for quantum matter
A curved-spacetime field, cosmological correlator, horizon effect, or gravity EFT result	QFT for gravity and cosmology
A theorem-level statement with its hypotheses and framework distinctions explicit	Mathematical foundations for physicists
A physical construction, approximation, or observable connected to a formal framework	Physical foundations for mathematicians
A reproduced lattice, bootstrap, transport, or inference result with independent numerical checks	Computational field theory onboarding

If no row names the result you want, return to Learning pathways and sharpen the target. Otherwise, open exactly one specialist pathway and carry the reconstructed derivation with you.