Feynman diagrams in QED

Feynman diagrams in QED are graphical representations of terms in the perturbative expansion of scattering amplitudes for quantum electrodynamics. Each diagram corresponds, via a precise dictionary of Feynman rules, to a specific mathematical contribution to the S-matrix element, organized by powers of the fine-structure constant

α \approx 1/137

. They apply to electromagnetic interactions of charged leptons and photons at energies where perturbation theory converges. Introduced by Richard Feynman around 1948–1949, formalized in parallel by Dyson.

A Feynman diagram is a topological graph whose internal lines (propagators) and vertices encode a term in the Dyson series for the time-ordered scattering operator. In QED, the only vertex couples two fermion lines to one photon line with strength $- i e γ^{μ}$ , so every physical process is built from electrons/positrons (solid lines with arrows) emitting and absorbing photons (wavy lines). The diagrams matter because they convert the formidable bookkeeping of perturbative field theory into an intuitive, almost mechanical algorithm — one draws all topologically distinct graphs of a given order, translates each into an integral, and sums them. The surprising fact is that internal lines need not satisfy $p^{2} = m^{2}$ : virtual particles are off-shell, mere labels for propagators, yet the result agrees with experiment to better than one part in a trillion (the electron $g - 2$ ).

The Core Insight

The interaction picture lets us write the S-matrix as a time-ordered exponential of the interaction Hamiltonian. Wick's theorem expands the time-ordered product of field operators into normal-ordered terms times contractions (propagators). Each surviving term in this combinatorial explosion has a unique pictorial signature: external legs are uncontracted fields hitting initial/final states, internal lines are contractions, and vertices are insertions of the interaction. A Feynman diagram is nothing more than a mnemonic for one such Wick contraction pattern — and the Feynman rules are the once-and-for-all dictionary that bypasses re-deriving Wick's theorem each time.

Formal Statement

The QED Lagrangian and S-Matrix

Quantum electrodynamics is defined by the Lagrangian density

L_{QED} = \overset{ˉ}{ψ} (i γ^{μ} D_{μ} - m) ψ - \frac{1}{4} F_{μν} F^{μν} - \frac{1}{2 ξ} (\partial_{μ} A^{μ})^{2}

where $ψ$ is the Dirac spinor field of mass $m$ and charge $- e$ , $D_{μ} = \partial_{μ} + i e A_{μ}$ is the gauge-covariant derivative, $A_{μ}$ is the photon field, $F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}$ , $γ^{μ}$ are the Dirac matrices, and $ξ$ is the gauge-fixing parameter (Feynman gauge $ξ = 1$ , Landau gauge $ξ = 0$ ).

⟳Start from the free Dirac and Maxwell Lagrangians; impose local

U (1)

invariance by promoting

\partial_{μ} \to D_{μ}

; the gauge-fixing term is added to make the photon propagator invertible.

The scattering operator is the Dyson series

S = T exp (- i \int d^{4} x H_{int} (x)) = n = 0 \sum \infty \frac{( - i ) ^{n}}{n !} \int d^{4} x_{1} \dots d^{4} x_{n} T {H_{int} (x_{1}) \dots H_{int} (x_{n})}

with the QED interaction $H_{int} (x) = e \overset{ˉ}{ψ} (x) γ^{μ} ψ (x) A_{μ} (x)$ .

⟳Integrate the Schwinger–Tomonaga / interaction-picture evolution equation

i \partial_{t} U = H_{int} U

iteratively; time-ordering arises from the nested integration limits via the Dyson trick symmetrizing the integrand.

The Three QED Propagators and the Vertex

The momentum-space Feynman rules in covariant gauge are:

Fermion propagator:

S_{F} (p) = \frac{i ( \neq p + m )}{p ^{2} - m ^{2} + i ϵ}

⟳Fourier-transform the time-ordered two-point function

⟨ 0∣ T ψ (x) \overset{ˉ}{ψ} (y) ∣0 ⟩

; the

+ i ϵ

encodes the Feynman boundary condition (positive frequencies forward in time, negative backward).

Photon propagator (covariant gauge):

D_{μν} (k) = \frac{- i}{k ^{2} + i ϵ} [g_{μν} - (1 - ξ) \frac{k _{μ} k _{ν}}{k ^{2}}]

⟳Invert the gauge-fixed photon quadratic operator

(- k^{2} g^{μν} + (1 - \frac{1}{ξ}) k^{μ} k^{ν})

; without gauge fixing this operator is singular because of gauge redundancy.

Vertex factor:

Γ^{μ} = - i e γ^{μ}

⟳Read off the coupling from

i L_{int} = - i e \overset{ˉ}{ψ} γ^{μ} ψ A_{μ}

; the

- i e

is the single coupling that generates all of QED.

Feynman Rules for QED Amplitudes

The invariant amplitude $i M$ for a given process at fixed order is obtained by:

Draw all topologically distinct, connected, amputated diagrams with the correct external legs.
For each internal fermion line, write $S_{F} (p)$ ; for each internal photon, $D_{μν} (k)$ .
For each vertex, write $- i e γ^{μ}$ and impose 4-momentum conservation.
For external lines, attach wavefunction factors: $u^{s} (p)$ (incoming $e^{-}$ ), $\overset{u}{ˉ}^{s} (p)$ (outgoing $e^{-}$ ), $v^{s} (p)$ (outgoing $e^{+}$ ), $\overset{v}{ˉ}^{s} (p)$ (incoming $e^{+}$ ), $ϵ_{μ} (k)$ (incoming photon), $ϵ_{μ}^{*} (k)$ (outgoing photon).
Integrate over each undetermined loop momentum: $\int \frac{d ^{4} ℓ}{( 2 π ) ^{4}}$ .
Include $(- 1)$ for each closed fermion loop and for each interchange of external fermion lines (Fermi statistics).
Read fermion lines against the arrow direction, contracting Dirac indices.

⟳Apply Wick's theorem to [2], assign each contraction a propagator and each

H_{int}

a vertex; the

1/ n!

is cancelled by the

n!

orderings of vertices, leaving a symmetry-factor-corrected sum over distinct topologies.

Why It Holds / Derivation

From Dyson Series to Diagrams

Step 1 — Expand the S-matrix. Insert

H_{int} = e \overset{ˉ}{ψ} \neq A ψ

into [2]. At order

n

we have

n

spacetime integrations and a time-ordered product of

n

vertex operators, each containing one

\overset{ˉ}{ψ}

, one

ψ

, one

A_{μ}

. Step 2 — Apply Wick's theorem. A time-ordered product equals the sum of all normal-ordered products with all possible contractions:

T {ϕ_{1} \dots ϕ_{n}} = : ϕ_{1} \dots ϕ_{n} : + single \sum ϕ_{i} ϕ_{j} contraction : \dots : + \dots

⟳Prove by induction on

n

, splitting

T {ϕ_{1} X}

and commuting

ϕ_{1}^{(+)}

through using

[ϕ^{(+)}, ϕ^{(-)}] =

c-number; each commutator becomes a Feynman propagator.

Step 3 — Identify contractions as propagators. Each field contraction

⟨ 0∣ T ϕ (x) ϕ (y) ∣0 ⟩

is precisely a propagator [3] or [4]. Internal lines = contractions of two fields at distinct vertices; external lines = fields left uncontracted that act on the asymptotic states. Step 4 — Cancel the $1/ n!$ . The

n

vertices can be permuted in

n!

ways producing identical diagrams; this cancels the

1/ n!

in [2] except where a diagram has a nontrivial automorphism, leaving the symmetry factor. Step 5 — Transform to momentum space. Each

\int d^{4} x_{i}

over a vertex produces a momentum-conserving delta function

(2 π)^{4} δ^{4} (\sum p)

. One overall delta survives as the total momentum conservation; the rest fix internal momenta, leaving

L = I - V + 1

unconstrained loop integrals. Step 6 — Read off $i M$ . Strip the overall

(2 π)^{4} δ^{4} (p_{i} - p_{f})

and the external-state normalization; what remains is

i M

as in the theorem.

■

Richard P. Feynman | 1918–1988

At the 1948 Pocono Conference, Feynman presented his diagrams to a skeptical audience including Bohr and Dirac; his space-time pictures, born of an operational "sum over paths" philosophy, initially baffled the establishment. Feynman conceived a positron as an electron traveling backward in time — encoded by the direction of the arrow on a fermion line — and treated virtual photons as carriers of momentum without an energy-momentum shell constraint. Freeman Dyson's 1949 papers proved the equivalence of Feynman's intuitive scheme to the operator-based formalisms of Schwinger and Tomonaga, granting the diagrams rigorous standing. Feynman shared the 1965 Nobel Prize with Schwinger and Tomonaga. The diagrams are arguably the most consequential calculational tool of 20th-century theoretical physics, now ubiquitous far beyond QED.

Building Intuition

Think of a Feynman diagram as a flowchart for momentum and charge. Electrons march along solid lines in the direction of their arrows; positrons are arrows pointing "the wrong way." At every vertex a photon is emitted or absorbed and momentum is conserved exactly — but energy and momentum together need not put the internal particle on its mass shell, which is why internal lines are virtual. The amplitude is a sum over all the ways a process can happen: the electron-electron scattering happens both by exchanging a photon "directly" and by an exchange where the final electrons are swapped, and quantum mechanics demands we add these complex amplitudes before squaring. Higher-order diagrams (loops) are quantum corrections — the electron is perpetually surrounded by a cloud of virtual photons and electron-positron pairs, and each loop costs another factor of $α / π \approx 0.0023$ , explaining why the leading diagram dominates.

Tree-level Møller and Compton diagrams: the two basic QED processes

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Worked Examples

Electron–Muon Scattering at Tree Level

Given: Compute

∣ M ∣^{2}

(spin-averaged) for

e^{-} μ^{-} \to e^{-} μ^{-}

at lowest order, the simplest QED scattering with a single

t

-channel photon exchange. Let incoming/outgoing electron momenta be

p, p^{'}

and muon momenta

k, k^{'}

, with

q = p - p^{'}

. Step 1 — Write the amplitude. A single diagram contributes (electron and muon are distinguishable, so no exchange term):

i M = \overset{u}{ˉ} (p^{'}) (- i e γ^{μ}) u (p) \frac{- i g _{μν}}{q ^{2}} \overset{u}{ˉ} (k^{'}) (- i e γ^{ν}) u (k)

⟳Apply the Feynman rules — two vertices (

- i e γ

), one photon propagator in Feynman gauge (

ξ = 1

), external spinors for each fermion line.

Step 2 — Square and average over initial spins, sum over final.

\overline{∣ M ∣^{2}} = \frac{e ^{4}}{4 q ^{4}} Tr [(\neq p^{'} + m_{e}) γ^{μ} (\neq p + m_{e}) γ^{ν}] Tr [(\neq k^{'} + m_{μ}) γ_{μ} (\neq k + m_{μ}) γ_{ν}]

⟳Use the spin-sum completeness

\sum_{s} u^{s} \overset{u}{ˉ}^{s} = \neq p + m

; the factor

1/4

averages over the 2×2 initial spin states.

Step 3 — Evaluate traces (neglecting masses, ultrarelativistic limit) using

Tr [γ^{μ} \neq a γ^{ν} \neq b] = 4 (a^{μ} b^{ν} + a^{ν} b^{μ} - g^{μν} a \cdot b)

\overline{∣ M ∣^{2}} = \frac{8 e ^{4}}{q ^{4}} [(p \cdot k) (p^{'} \cdot k^{'}) + (p \cdot k^{'}) (p^{'} \cdot k)]

⟳Contract the two traces; cross terms with odd numbers of gamma matrices vanish.

Step 4 — Mandelstam form. With

s = (p + k)^{2}

t = (p - p^{'})^{2} = q^{2}

u = (p - k^{'})^{2}

\overline{∣ M ∣^{2}} = \frac{2 e ^{4}}{t ^{2}} (s^{2} + u^{2}) = \frac{32 π ^{2} α ^{2}}{t ^{2}} (s^{2} + u^{2})

(dimensionless, with

e^{2} = 4 π α

). This is the QED analogue of Rutherford scattering with the characteristic

1/ t^{2} \sim 1/ sin^{4} (θ /2)

forward peak.

The Electron Anomalous Magnetic Moment (Schwinger Term)

Given: The one-loop vertex correction to the electron-photon coupling shifts the magnetic moment from the Dirac value

g = 2

. Estimate the leading anomaly

a_{e} = (g - 2) /2

. Step 1 — The diagram. The order-

α

correction is a single vertex diagram: the electron emits a virtual photon, interacts with the external field, then reabsorbs the photon. The vertex function takes the form

\overset{u}{ˉ} (p^{'}) Γ^{μ} u (p) = \overset{u}{ˉ} (p^{'}) [γ^{μ} F_{1} (q^{2}) + \frac{i σ ^{μν} q _{ν}}{2 m} F_{2} (q^{2})] u (p)

⟳Gauge invariance (Ward identity) and Lorentz covariance restrict the most general vertex to these two form factors;

F_{1} (0) = 1

fixes the charge,

F_{2} (0)

is the anomaly.

Step 2 — Identify the anomaly. The Gordon decomposition shows

F_{2}

multiplies the spin-magnetic coupling, so

a_{e} = F_{2} (0)

⟳Compare the

σ^{μν} q_{ν}

term to the nonrelativistic Pauli interaction

- μ \cdot B

; the coefficient is precisely the moment anomaly.

Step 3 — Evaluate the loop integral (after Feynman parameterization and regularization, the famous Schwinger result):

a_{e} = \frac{α}{2 π} \approx \frac{1}{2 π \cdot 137.036} \approx 0.0011614

(dimensionless). The measured value

a_{e}^{exp} = 0.00115965218...

agrees with the full QED calculation (through 5 loops) to better than

1 0^{- 12}

— the most precise confirmation of any theory in physics.

Common Misconceptions

Virtual particles are not real

Internal lines do not represent physical particles fleetingly popping into existence. A virtual particle is a propagator — a Green's function — carrying off-shell four-momentum ( $p^{2} \neq = m^{2}$ ). It has no associated wavefunction, no definite energy, and cannot be detected. The "energy-time uncertainty" hand-wave ( $Δ E Δ t ≳ ℏ$ ) is at best heuristic; rigorously, the off-shellness is just the analytic structure of the integrand. Likewise, a single diagram is not a physical history; only the coherent sum of all diagrams at all orders is physical, and individual diagrams are gauge-dependent.

Diagrams are perturbative and the series is asymptotic

The Feynman expansion is a power series in $α$ that is asymptotic, not convergent (Dyson's 1952 argument: a theory with $- α$ would be unstable, so the radius of convergence is zero). For QED this is harmless because $α$ is tiny, but it means summing "all orders" is formally ill-defined and non-perturbative effects (e.g., Schwinger pair production in strong fields) are invisible to any finite number of diagrams.

Forgetting the relative sign of exchange diagrams

In Møller scattering $e^{-} e^{-} \to e^{-} e^{-}$ there are two tree diagrams (direct and exchange of identical final electrons). A common error is to add their amplitudes with a $+$ sign. Fermi statistics demands a relative minus sign:

M = M_{t} - M_{u}

Omitting it gives the wrong cross section and violates antisymmetry. The sign arises from the anticommutation of fermion operators in the Wick contractions — precisely the

(- 1)

per fermion-line interchange in the Feynman rules.

⟳Track the ordering of

ψ

operators in the two contraction patterns; swapping two adjacent fermion fields introduces a minus sign.

Connections

Feynman diagrams are the diagrammatic face of Perturbation Theory in Quantum Field Theory and emerge directly from Wick's Theorem applied to the Dyson Series. The propagators are Green's Functions of the free field equations, and the photon propagator's gauge dependence is controlled by Gauge Fixing and the Faddeev–Popov Procedure. Loop integrals require Regularization and Renormalization to tame ultraviolet divergences, and the gauge-invariant cancellations they exhibit are guaranteed by Ward–Takahashi Identities. The same machinery, with non-Abelian vertices and ghost loops, generalizes to Quantum Chromodynamics and the Electroweak Theory. The path-integral derivation links diagrams to Generating Functionals and Connected Correlators.

Real-World Applications

Electron $g - 2$ measurement: The most stringent test of QED; theoretical predictions require thousands of multi-loop diagrams (Kinoshita and collaborators computed the 5-loop coefficient, ~12,672 diagrams).
Lamb shift: The $2 S_{1/2}$ – $2 P_{1/2}$ splitting in hydrogen, explained by self-energy and vacuum-polarization diagrams, historically launched QED renormalization.
Particle collider cross sections: Drell–Yan, Bhabha scattering ( $e^{+} e^{-} \to e^{+} e^{-}$ ) used for luminosity calibration at LEP and the LHC.
Vacuum polarization / running coupling: The fermion-loop correction to the photon propagator makes $α$ energy-dependent, measured to $α (M_{Z}) \approx 1/128$ .
Positronium decay rates: Para- and ortho-positronium lifetimes computed from annihilation diagrams.
Delbrück scattering: Light-by-light scattering via electron loops (the box diagram), observed in the Coulomb field of heavy nuclei.
Muon $g - 2$ anomaly: Precision QED + hadronic diagrams underpin the Fermilab muon $g - 2$ experiment probing physics beyond the Standard Model.

Historical Development

1927–1932: Dirac formulates the relativistic electron equation and quantizes the electromagnetic field, founding QED but with intractable divergences.
1947: Lamb and Retherford measure the hydrogen fine-structure shift; Bethe computes a non-relativistic estimate on a train, signaling renormalization works.
1948 (Pocono Conference): Feynman presents his diagrams; Schwinger presents his operator method; reception is mixed.
1948: Schwinger calculates $a_{e} = α /2 π$ , the first loop result.
1949: Feynman publishes "Space-Time Approach to Quantum Electrodynamics" with the diagrammatic rules.
1949: Dyson proves the equivalence of the Feynman, Schwinger, and Tomonaga formulations and organizes the renormalization program.
1950: Tomonaga's covariant formulation (developed independently in wartime Japan) becomes widely known.
1952: Dyson argues the perturbation series is asymptotic, not convergent.
1965: Feynman, Schwinger, and Tomonaga share the Nobel Prize in Physics.

Subtleties & Limits

Several deep issues remain. The asymptotic nature of the QED series means it cannot be summed to arbitrary precision; understanding the role of renormalons and instanton-like contributions in resumming the series is incomplete. Infrared divergences from soft and collinear photons cancel only when one sums over degenerate final states (Bloch–Nordsieck / KLN theorem), and constructing a fully infrared-finite S-matrix with proper asymptotic charged states remains subtle. The Landau pole — the apparent divergence of the QED coupling at ultra-high energy — suggests QED is not ultraviolet-complete on its own (it is "trivial" in isolation), and its resolution requires embedding in a larger theory. Finally, efficiently summing the factorially growing number of high-order diagrams motivates ongoing work on amplitude techniques (BCFW recursion, unitarity methods, the amplituhedron) that bypass diagrams entirely.