Orientation, conventions, and study contract

Quantum field theory becomes manageable when every calculation answers four questions: What is the physical object? Which assumptions define it? Which conventions affect its formula? What check would reveal a mistake? This page helps you choose an honest entry into Core QFT, set up those habits once, and continue without repeating material you can already use.

Core QFT is a conceptual roadmap and refresher. If you want a sequential first course with exercises and worked solutions, take QFT I.

Choose your entry point

Start with the route that matches the work you can do now—not the title of a course you once took.

• First pass. Begin here, follow the modules in order, and complete both the fermion and spin-one branches before perturbation theory.
• Focused review. Open the first module whose central result you cannot reproduce with its assumptions visible. Follow a prerequisite link only when that missing skill blocks the calculation.
• Research refresher. Begin from the observable or method you need, then move backward until every input is explicit. The compressed refresher turns that process into a short route.

The Core route eventually uses five capabilities: linear and tensor reasoning; Fourier transforms, distributions, and Green functions; variational classical-field reasoning; Lorentz, causal, and spin reasoning; and quantum states and operators. The opening classical-action lesson leans most directly on linear/tensor and variational reasoning; later lessons draw on the others. The readiness page offers short work-based checks and focused reviews. A check describes one piece of work, not your identity or potential, and there is no total score.

Capability	Small piece of work	Check or review
Linear and tensor reasoning	Explain how a vector, covector, and linear map change under a basis change; identify an invariant.	Mathematics diagnostic
Fourier, distribution, and Green-function reasoning	Transform a derivative and state which boundary, support, or pole condition selects a Green function.	Mathematics diagnostic
Variational classical-field reasoning	Vary an action, retain its surface term, and state which boundary data justify the field equation.	Classical fields and relativity diagnostic
Lorentz, causal, and spin reasoning	Classify an interval and distinguish a field transformation law from a one-particle representation.	Classical fields and relativity diagnostic
Quantum state and operator reasoning	Distinguish an algebra, state, observable, spectrum, and time evolution in a familiar system.	Quantum mechanics diagnostic

Begin with more than a Lagrangian

A Lagrangian density is important data, but by itself it does not specify a quantum field theory or a prediction. Before calculating, write down enough of the following to make the question unambiguous:

Question	Data to make explicit
What system?	Fields or observables, spacetime, dimension, geometry, and relevant symmetries
What physical situation?	State, boundary or initial conditions, sources, scales, and kinematic regime
What approximation?	Expansion parameter, perturbative order, regulator, effective-theory cutoff, and neglected terms
What is measured?	Correlator, spectrum, response, amplitude, cross section, decay rate, or other operational quantity
What makes the result trustworthy?	Dimensions, symmetry identities, normalization, limiting cases, regulator independence, or comparison with known data

For a fuller account of why states and observables are part of the definition, read What Is a Quantum Field Theory?. The next Core module deliberately begins with a classical action; later modules add the quantum state, observables, regularization, and limiting procedures needed for a prediction.

Track conventions that can change the formula

Core pages inherit Conventions and normalizations. The most frequently used defaults are the mostly-minus metric $\eta_{\mu\nu}=\operatorname{diag}(+,-,-,-)$, natural units $\hbar=c=k_{\mathrm B}=1$, and the Fourier pair

$$\widetilde f(p)=\int \mathrm d^d x\,e^{+ip\cdot x}f(x), \qquad f(x)=\int\frac{\mathrm d^d p}{(2\pi)^d}\, e^{-ip\cdot x}\widetilde f(p),$$

so that $\partial_\mu\mapsto -ip_\mu$. The reference page remains the source of truth; this summary only prepares the first calculation.

Do not copy a large convention table into every set of notes. Instead, add a short calculation header containing only choices that are consequential here:

object and regime:
state or boundary condition:
local normalization or sign choice:
regulator, scheme, and approximation order:
one convention-independent check:

When a paper or textbook uses different signs or normalizations, translate the formula and then check an invariant quantity: a pole location, commutator, Ward identity, rate, or round trip back to the original convention. Different notation is not different physics, but a failed invariant check may expose a real mismatch.

Study by reconstructing the argument

For each module, keep a compact four-line note:

result I need:
assumptions and local choices:
one calculation or identity I can reproduce:
next question this result unlocks:

Reading a derivation creates familiarity. Reproducing its decisive step shows whether the definitions and normalizations are under control. Testing an invariant or limiting case asks a different question: whether the result is internally consistent. Use all three, but do not confuse them.

If you get stuck, name the smallest missing operation—varying an action, transforming a derivative, handling a distribution, classifying a Lorentz representation, or separating a state from an operator—and use the matching focused review. Return to the same line of the calculation as soon as that operation is secure.

Match the evidence to the claim

A theorem, an analytic calculation, a numerical run, and a literature synthesis can support different kinds of conclusions. For any result, ask: What was established? Under which assumptions? By what evidence?

• For a theorem, identify the hypotheses and conclusion.
• For a calculation, reproduce the central step and test a sign, dimension, symmetry, normalization, or limit.
• For an approximation, name the expansion parameter, regime, truncation, and expected failure boundary.
• For a numerical or research claim, inspect uncertainty and provenance, then check that the output actually supports the stated conclusion.

This distinction matters later: a successful computation can verify an implementation without establishing that its model describes the physical regime, while a cited theorem may not apply once one of its hypotheses is changed.

Check your orientation

Before continuing, try these without looking back.

1. With the Fourier convention above, derive the replacement $\partial_\mu\mapsto -ip_\mu$ by integrating by parts. State the boundary or decay condition you used.
2. Someone writes $\mathcal L=\tfrac12(\partial\phi)^2-V(\phi)$ and asks for a scattering prediction. Name four kinds of information still missing.
3. A source uses the opposite metric signature. Name one quantity you would use to verify that your translated result describes the same physics.

Answer guide

1. Insert the transform, move $\partial_\mu$ from $f(x)$ to $e^{+ip\cdot x}$, and discard the boundary term under the stated decay, periodicity, or test-function condition. The derivative of the exponential gives $+ip_\mu$, so integration by parts supplies the minus sign.
2. Possible answers include the spacetime and dimension; the field content and parameters; a state or asymptotic prescription; boundary conditions; a regulator and renormalization prescription; the expansion order; and the observable with its external-state normalization and kinematics.
3. Suitable checks include a physical pole or mass shell, the causal support of a propagator, a normalized commutator, a Ward identity, or a measurable rate. State the translation completely before comparing.

If the reasoning is clear, continue to Classical fields, actions, and local dynamics. If the first task is the obstacle, use the Fourier, distributions, and Green functions review. If the second task feels underspecified rather than merely unfamiliar, that is the right instinct: identifying missing physical data is part of doing QFT.

References

• Anthony Duncan, The Conceptual Framework of Quantum Field Theory, Oxford University Press, 2012, doi:10.1093/acprof:oso/9780199573264.001.0001.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.