Perturbative Expansion and Feynman Rules

A perturbative rule set is obtained by a chain of controlled translations: split free from interacting evolution, expand the ordered interaction, reduce free fields by Wick contraction, quotient labeled contractions into decorated graphs, Fourier transform the action, and finally check the result by an independent functional derivation or a gauge identity. This chapter supports three entry routes—Dyson and Wick derivation, direct action-to-rule translation, and gauge-fixed diagrammatics—and brings them to the same exit: given a declared action and state prescription, you can construct every propagator and vertex, attach signs and symmetry factors, isolate the connected contribution, and name the assumptions under which the result is meaningful.

The chapter treats scalar, Dirac, derivative, and generic covariantly gauge-fixed rules. It does not provide model-specific Standard Model catalogs, renormalized loop predictions, or a proof that the interaction picture exists as an exact global unitary in an infinite-volume interacting theory. Those topics are treated elsewhere.

From the action to a checked perturbative contribution

For a real scalar example,

$$\mathcal L =\frac12\partial_\mu\phi\partial^\mu\phi -\frac12m^2\phi^2 -\frac{\lambda}{4!}\phi^4,$$

the chapter’s chain is

$$\begin{array}{c} i\partial_tU_I=H_IU_I \\[2pt] \Downarrow\ \text{ordered expansion} \\[2pt] \mathrm T\exp\!\left(i\int\mathcal L_{\mathrm{int}}\right) \\[2pt] \Downarrow\ \text{free Wick contractions} \\[2pt] \text{decorated graphs with }1/S_G \\[2pt] \Downarrow\ \text{Fourier transform} \\[2pt] \dfrac{i}{p^2-m^2+i0},\quad -i\lambda,\quad (2\pi)^d\delta^{(d)}\!\left(\sum p\right). \end{array}$$

Dividing by $Z[0]$ removes source-independent vacuum bubbles, while $W[J]=-i\log(Z[J]/Z[0])$ generates correlators connected to all external insertions. For fermions, graded permutations and closed loops add signs. For derivative interactions, each derivative acts on its own field before momentum conservation is used. For gauge fields, the quadratic kernel is inverted only after gauge fixing, and ghost contributions enter complete Slavnov–Taylor checks. These complementary derivations and cautions are standardly developed across Weinberg 1995, §§ 3.5, 6.1–6.3, and 9.3–9.4, pp. 143–144, 261–285, and 389–398, Schwartz 2014, §§ 7 and 14.3, pp. 78–100 and 261–264, and Srednicki 2007, §§ 9, 45, and 71–72, pp. 71–86, 282–291, and 420–426.

Choose a route

Reader goal	Suggested route	What you can verify at the end
Derive diagrammatics from time evolution	The Interaction Picture and Dyson Series → Wick Expansion for Interacting Fields → Diagrammatics and Symmetry Factors	Reproduce the $1/n!$ ordered expansion, Wick multiplicities, and $1/S_G$ for a scalar graph
Translate a known action efficiently	Momentum-Space Feynman Rules → Fermion Signs and Closed Loops or Derivative Interactions and Contact Terms	Produce propagators, deltas, loop measures, momentum numerators, and statistics signs from the written action
Normalize and cross-check correlators	Connected, Disconnected, and Vacuum Diagrams → Functional Derivation of Perturbation Theory	Show why vacuum bubbles cancel, why $-i\log Z$ is connected, and why source differentiation reproduces Wick graphs
Use an already fixed gauge	Close the direct action-to-rule route, then enter Gauge-Fixed Perturbation Rules, Ghost Diagrams, and Identity Checks	Derive gauge and ghost ingredients and state the complete-amplitude Ward or Slavnov–Taylor check

The first route is the best first graduate pass. The second and third are focused routes, not replacements for their hard dependencies. Gauge fixing, BRST theory, and model-specific gauge actions are imported rather than rederived here.

Before you begin

Use the questions below to identify any background worth revisiting before you begin.

Try this	Ready when	Repair or entry
Explain why $H_I(t_1)H_I(t_2)$ cannot generally be put in an ordinary exponential	You name noncommutativity and time ordering	Review Fock Space, Vacuum, and Particle Number, then enter the Dyson page
Write the free scalar ordered two-point function including its boundary value	You give $i/(p^2-m^2+i0)$ in momentum space	Review Scalar Propagators, Ordered Correlators, and Sources
Count the complete pairings of four centered Gaussian scalar fields	You obtain three pairings and distinguish the free theorem from an interacting correlator	Review Wick’s Theorem and Free Gaussian Factorization
Fourier transform $\partial_\mu\phi$	You obtain $-ip_\mu\widetilde\phi(p)$ and keep track of which field is differentiated	Enter the momentum-rule page; use the Fourier page linked there if needed
Explain why a non-Abelian gauge quadratic form cannot be inverted before gauge fixing	You identify gauge null directions and do not call the gauge propagator physical	Repair with The Faddeev–Popov Construction and Ghosts, Auxiliary Fields, and Gauge-Parameter Dependence

Exact chapter guide

1. The Interaction Picture and Dyson Series. Derives ordered evolution from $i\partial_tU_I=H_IU_I$, explains the simplex factorial, and states switching, regulator, and Haag-theorem limits. It requires free Fock-space evolution and hands the ordered products to Wick expansion.
2. Wick Expansion for Interacting Fields. Applies the free theorem inside the normalized Dyson numerator and denominator, with a complete first-order $\phi^4$ two-point count. It requires the Dyson page and the free Wick theorem, then hands labeled contractions to graph combinatorics.
3. Diagrammatics and Symmetry Factors. Defines decorated graphs as quotients of labeled contractions and checks $S_G=2,8,2$ in representative scalar examples. It requires interacting Wick expansion and hands weighted graphs to momentum space.
4. Momentum-Space Feynman Rules. Inverts quadratic kernels, derives vertices and momentum deltas, and proves the loop count $L=I-V+1$. It is the core action-to-rule derivation used by the later pages.
5. Fermion Signs and Closed Loops. Separates external graded permutations, open-line matrix order, and the minus per closed fermion loop. It requires the momentum-space rule derivation and the Dirac propagator, and prepares fermionic tree and loop amplitudes.
6. Derivative Interactions and Contact Terms. Derives momentum numerators with identical-field factors, checks integration by parts, and retains contact terms from differentiated time ordering. It requires the momentum-space rule derivation and prepares derivative EFT and gauge vertices without performing operator-basis reduction.
7. Connected, Disconnected, and Vacuum Diagrams. Proves vacuum exponentiation, normalization, and the partition of full correlators into connected blocks. It requires graph combinatorics and prepares the connected amplitude selected by LSZ.
8. Functional Derivation of Perturbation Theory. Replaces fields by $(1/i)\delta/\delta J$, reproduces Wick pairings from the free Gaussian, and checks the same vacuum and connected decomposition independently. It requires momentum rules and the generating functional.
9. Gauge-Fixed Perturbation Rules, Ghost Diagrams, and Identity Checks. Imports a fixed Faddeev–Popov action, derives gauge and ghost rules, and distinguishes gauge-dependent ingredients from complete Ward or Slavnov–Taylor checks. It requires the momentum-space rule derivation and the gauge-fixing pages in the symmetry volume.

Convention and sign bridge

The chapter inherits the site’s Lorentzian $e^{iS}$, mostly-minus metric, Fourier transform, Hermitian gauge generators, and Feynman $+i0$ conventions. The recurrent consequences are:

Datum	Convention used here	Check
Fourier derivative	$\partial_\mu\mapsto-ip_\mu$	total derivative gives $-i\sum p_\mu=0$ at an integrated vertex
scalar propagator	$i/(p^2-m^2+i0)$	$(\Box+m^2)D_F=-i\delta^{(d)}$
scalar interaction	$\mathcal L_{\mathrm{int}}=-\lambda\phi^4/4!$	vertex is $-i\lambda$
momentum orientation	all momenta incoming	every vertex carries $\delta^{(d)}(\sum p)$
fermion contraction	$\psi$ paired with $\bar\psi$ in declared order	external exchange and closed-loop signs follow graded permutations
gauge fixing	local $R_\xi$-type action stated before inversion	physical on-shell result, not each graph, is $\xi$ independent

Sources that define the propagator without its numerator $i$, use $e^{+ipx}$ for the inverse transform, or take anti-Hermitian gauge generators move signs among these entries. Translate the entire action–propagator–vertex chain and check a pole or Ward identity; do not repair one sign in isolation.

The scalar thread

One scalar calculation runs through the chapter. Starting from $-\lambda\phi^4/4!$:

• the Dyson series supplies $(-i\lambda/4!)^V/V!$;
• Wick expansion of the first-order two-point function finds $12$ connected tadpole contractions and $3$ vacuum contractions;
• graph quotienting turns those counts into $1/2$ and $1/8$;
• Fourier transformation supplies $-i\lambda$, scalar propagators, and one loop measure;
• normalization removes the $1/8$ vacuum term;
• source differentiation independently reproduces the surviving $1/2$ tadpole coefficient.

The thread is a combinatorial and convention check, not a finite prediction: $D_F(0)$ is ultraviolet singular and renormalization lies downstream. It also ceases to represent fermion and gauge theories where graded ordering, constraints, ghosts, or additional tensor structure matter.

Chapter synthesis

The central statements have different logical status:

• the Dyson formula is a derivation from the interaction-picture evolution equation within its regulated or asymptotic scope;
• Wick reduction is an operator theorem for free interaction-picture fields, applied term by term;
• the symmetry factor is a combinatorial quotient of labeled contractions;
• momentum rules are Fourier transforms and kernel inversions with a boundary prescription;
• the loop minus is a graded-permutation consequence;
• vacuum cancellation and connected generation are exponential and logarithmic identities; and
• gauge independence is a property of properly defined physical quantities constrained by Ward or Slavnov–Taylor identities, not of individual diagrams.

Together they answer the chapter’s governing question: the action becomes a perturbative contribution only after state and boundary data, combinatorics, statistics, normalization, and validation are all supplied. A diagram alone is not that contribution.

Review the chapter

Use the questions below to identify topics worth revisiting.

Mode	Prompt	A satisfactory response	Repair
Retrieval	State the order-$n$ Dyson term	Includes $(-i)^n/n!$, $n$ time integrals, and $\mathrm T$	Dyson page
Derivation check	Explain the $1/2$ tadpole factor in $\phi^4$ theory	Counts $4\times3$ contractions against $4!$	Wick and symmetry-factor pages
Representation change	Derive $-i\lambda$ and the vertex delta from $-\lambda\phi^4/4!$	Uses $e^{iS}$, cancels $4!$, and integrates the local vertex	Momentum-rule page
Sign diagnosis	Locate all signs in a closed Yukawa fermion loop	Separates vertices, propagators, one loop minus, and the trace	Fermion-sign page
Contact diagnosis	Apply $(\Box+m^2)$ to $D_F$	Obtains $-i\delta^{(d)}$ and does not set the result to zero	Derivative/contact page
Connectedness	Distinguish a vacuum bubble from a product of two externally connected components	Cancels only the first in $Z[J]/Z[0]$ and removes the second by taking $\log Z$	Connected-diagram page
Independent check	Reproduce the scalar four-point pairings from $Z_0[J]$	Finds all three once with the correct powers of $i$	Functional page
Gauge check	Replace an external gauge polarization by its momentum	Tests the complete amplitude and states the non-Abelian Slavnov–Taylor qualification	Gauge-rule page
End-to-end transfer	Starting from $\mathcal L_{\mathrm{int}}=-g\chi\phi^2/2$, organize the order-$g^2$ connected $\chi$ two-point contribution	Obtains vertex $-ig$, residual graph factor $1/2$, two internal $\phi$ propagators, one regulated loop measure, and states whether external legs and the overall delta have been retained	Dyson → Wick → symmetry-factor → momentum-rule pages

For the end-to-end transfer, the amputated one-loop kernel is

$$i\Pi_\chi(p) =\frac12(-ig)^2 \int\frac{\mathrm d^d\ell}{(2\pi)^d} \frac{i}{\ell^2-m_\phi^2+i0} \frac{i}{(\ell+p)^2-m_\phi^2+i0}.$$

The two vertices cancel the Dyson $2!$ after their positions are assigned, while exchange of the two identical internal $\phi$ lines leaves $S_G=2$. The connected correlator additionally carries the two external $\chi$ propagators and $(2\pi)^d\delta^{(d)}(p+p')$; the amputated kernel above carries neither. This distinction is the chapter’s shortest complete normalization check.

Where to continue

• Convert connected correlators into scattering observables: Asymptotic States, LSZ, and Scattering Observables adds stable isolated poles, residues, state normalization, and phase space.
• Assemble complete trees and gauge checks: Tree Amplitudes and Gauge Consistency applies this chapter’s rule set to contact and exchange amplitudes.
• Renormalize loop expansions and organize EFT operators: Renormalization and Effective Field Theory treats counterterms, schemes, running, field redefinitions, and operator mixing.
• Use model-specific gauge rules: Gauge Theories and the Standard Model supplies declared groups, representations, symmetry breaking, and process catalogs.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge: Cambridge University Press, 1995. First edition; 2005 paperback, 2012 printing consulted. DOI.