Feynman Diagrams Explained: What the Little Squiggles Actually Mean

Feynman diagrams look like doodles: a few straight lines, a wiggle, maybe a loop. They are also one of the most useful tools in physics, the way particle physicists turn quantum theory into numbers that can be checked in a lab. This post builds on our explainer What Is Quantum Measurement?, because the diagrams compute exactly the kind of probabilities that measurement reveals. It is drawn by our Richard Feynman sticker and the Feynman Diagram hoodie, whose slogan, “two in, one photon, two out,” is the simplest diagram there is.

What is a Feynman diagram?

A Feynman diagram is a picture of one way a particle process can happen, with a rule that turns the picture into maths. Richard Feynman introduced the method in two 1949 papers in Physical Review, “The Theory of Positrons” (Feynman 1949a^[1]) and “Space-Time Approach to Quantum Electrodynamics” (Feynman 1949b^[2]). Quantum electrodynamics (QED) is the quantum theory of light and electric charge. As Frank Wilczek puts it, “The first published Feynman diagram appeared in Physical Review in 1949: Two electrons exchange a photon” (Wilczek 2016^[3]).

The key move is the rule Feynman attached to each drawing. In Wilczek’s words, “The formula expresses the likelihood of the process the diagram depicts.” Strictly, each diagram gives a complex number called an amplitude. You add the amplitudes from every allowed diagram, then square the size of the total to get a probability. That is the same add-then-square rule behind the double-slit experiment.

How do you read the lines and squiggles?

In the usual drawing, time runs from left to right (some books run it bottom to top). Then:

• Straight lines with arrows are electrons (or other matter particles).
• Wavy lines are photons, the particles of light that carry the electromagnetic force.
• A vertex, the point where two electron lines and one photon line meet, is where a charge emits or absorbs a photon. Every QED diagram is built from that one junction.
• Outer lines are the particles that go in and come out, the ones a detector can see. Inner lines link vertices and are never seen directly. Wilczek’s figure labels the exchanged photon a “virtual quantum.”

So the hoodie’s diagram reads: two electrons come in, one photon passes between them, two electrons go out. That single picture gives the familiar repulsion between like charges, the Coulomb force, at lowest order. Wilczek adds that “It is also possible to exchange two or more photons,” and those extra diagrams slightly modify the classical force law.

The shapes are not trajectories. The diagrams are “‘topological’ constructions, reflecting quantum uncertainty,” and it does not matter how you bend the lines “as long as you get the connections right” (Wilczek 2016^[3]).

Why does the positron’s arrow point backwards in time?

A positron is the antiparticle of the electron. In a Feynman diagram it is drawn as an electron line with its arrow pointing backwards in time. Feynman explained the idea’s origin in his Nobel lecture: “I remembered Wheeler’s old idea about the positron being, maybe, the electron going backward in time.” In his theory, “positrons could simply be represented as electrons going from the future to the past in a back section of their world lines. That, I stole!” (Feynman 1965^[4]).

This is a bookkeeping trick, not time travel: it lets one line describe both particles. Turn the basic vertex on its side and it becomes an electron and a positron meeting and turning into a photon. With two photons out (e⁻ + e⁺ → γ + γ), that is the annihilation we covered in Particle Annihilation Explained.

Where does the number 1/137 come in?

Each vertex carries the strength of the electromagnetic coupling, measured by the fine-structure constant:

α = e²/(4πε₀ħc) ≈ 1/137.036

Here e is the elementary charge, ε₀ is the vacuum electric permittivity, ħ is the reduced Planck constant and c is the speed of light. α has no units. The CODATA 2022 value is α = 7.297 352 5643(11) × 10⁻³ (NIST^[5]), or α⁻¹ = 137.035 999 177(21) (NIST^[6]).

Because α is small, diagrams with more vertices usually contribute less. Wilczek calls this the regime of “weak coupling,” where “each additional complicating line is relatively rare,” and says “That is almost always the case for photons in quantum electrodynamics” (Wilczek 2016^[3]). In practice, physicists sum the simplest diagrams first and add more complicated ones until the answer is precise enough.

What did Feynman diagrams actually predict?

The classic test is the electron’s magnetism. The simplest diagram, with no loops, gives the electron g = 2, where g is the dimensionless number linking its magnetic moment to its spin. Diagrams in which the electron briefly emits and reabsorbs a photon (a loop) shift g a little above 2. Physicists measure that shift as the anomaly aₑ = (g − 2)/2.

In December 1947, in a five-paragraph letter to Physical Review, Julian Schwinger worked out the first correction (CERN Courier^[7]; original paper: Schwinger, Phys. Rev. 73, 416, 1948, DOI 10.1103/PhysRev.73.416). He gave the value as α/2π = 0.00116 and called it “the simplest example of a radiative correction.” In modern numbers:

aₑ ≈ α/2π ≈ 0.001161

The CODATA 2022 value of the anomaly is aₑ = 1.159 652 180 46(18) × 10⁻³ (NIST^[8]). Schwinger’s single loop is already within about 0.15% of it. The remaining gap is filled by higher-order diagrams; Toichiro Kinoshita has carried the calculation “up to the tenth order” (CERN Courier^[7]).

The measurements kept pace. In 2023, Gerald Gabrielse’s group reported g/2 = 1.001 159 652 180 59(13), with an uncertainty of 0.13 parts per trillion, 2.2 times more accurate than the value that had stood for 14 years, testing the Standard Model “to 1 part in 10¹²” (Fan et al. 2023^[9]).

The 1965 Nobel Prize in Physics went jointly to Sin-Itiro Tomonaga, Julian Schwinger and Richard Feynman “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles” (NobelPrize.org^[10]).

Are Feynman diagrams real, or just a calculation trick?

They are an approximation method, and a very good one. Feynman himself proved, in Wilczek’s words, “that his diagram method is not a true alternative to the field approach, but rather an approximation to it” (Wilczek 2016^[3]). The underlying theory is quantum fields; the diagrams are the terms of a series that approximates it.

The series also needed fixing. Loop diagrams can give infinite answers, and the cure, renormalization, absorbs those infinities into the measured charge and mass of the electron. Feynman was blunt about it in his Nobel lecture: “I think that the renormalization theory is simply a way to sweep the difficulties of the divergences of electrodynamics under the rug” (Feynman 1965^[4]).

Where the method stops working: the series only works when the coupling is small. For the strong force, which binds quarks with gluons, Wilczek notes that at low energies “their coupling is not weak,” so physicists compute the quantum fields directly, “carried out on banks of supercomputers,” to get the masses of protons and neutrons (Wilczek 2016^[3]). That is why you will not find simple Feynman-diagram explanations in Why Can’t Quarks Be Isolated?.

FAQ

Do Feynman diagrams show the paths particles take?

No. They show which particles connect at which vertices. The exact shapes and angles mean nothing; only the connections count.

What is a virtual particle in a Feynman diagram?

It is an internal line, one that starts and ends at a vertex inside the diagram and is never detected directly. It stands for a term in the calculation, not a particle you can catch.

Why is the fine-structure constant important for Feynman diagrams?

Each vertex brings in the electromagnetic coupling, and α ≈ 1/137 is small. That makes diagrams with more vertices smaller, so a handful of simple diagrams gives very accurate answers in QED.

Who invented Feynman diagrams?

Richard Feynman, who published them in Physical Review in 1949. He shared the 1965 Nobel Prize with Tomonaga and Schwinger for their work on quantum electrodynamics.

References

1. R. P. Feynman (1949), “The Theory of Positrons,” Phys. Rev. 76, 749–759 (DOI 10.1103/PhysRev.76.749). https://journals.aps.org/pr/abstract/10.1103/PhysRev.76.749
2. R. P. Feynman (1949), “Space-Time Approach to Quantum Electrodynamics,” Phys. Rev. 76, 769–789 (DOI 10.1103/PhysRev.76.769). https://journals.aps.org/pr/abstract/10.1103/PhysRev.76.769
3. F. Wilczek (2016), “How Feynman Diagrams Almost Saved Space,” Quanta Magazine, 5 July 2016. https://www.quantamagazine.org/why-feynman-diagrams-are-so-important-20160705/
4. R. P. Feynman, Nobel Lecture (11 December 1965), “The Development of the Space-Time View of Quantum Electrodynamics”. https://www.nobelprize.org/prizes/physics/1965/feynman/lecture/
5. NIST CODATA 2022 recommended values: fine-structure constant. https://physics.nist.gov/cgi-bin/cuu/Value?alph
6. NIST CODATA 2022 recommended values: inverse fine-structure constant. https://physics.nist.gov/cgi-bin/cuu/Value?alphinv
7. CERN Courier (25 January 2011), review of “Lepton Dipole Moments,” B. L. Roberts & W. J. Marciano (eds.), on Schwinger’s 1947 letter. https://cerncourier.com/a/lepton-dipole-moments/
8. NIST CODATA 2022 recommended values: electron magnetic moment anomaly. https://physics.nist.gov/cgi-bin/cuu/Value?ae
9. X. Fan, T. G. Myers, B. A. D. Sukra & G. Gabrielse (2023), “Measurement of the Electron Magnetic Moment,” Phys. Rev. Lett. 130, 071801 (arXiv:2209.13084). https://arxiv.org/abs/2209.13084
10. NobelPrize.org, The Nobel Prize in Physics 1965 (Tomonaga, Schwinger, Feynman). https://www.nobelprize.org/prizes/physics/1965/summary/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.