# Feynman Diagrams

# Feynman Diagrams

American physicist Richard Feynman’s (1918–1988), work and writings were fundamental to the development of quantum electrodynamic theory (QED theory). With regard to QED theory, Feynman is perhaps best remembered for his invention of what are now known as Feynman diagrams, to portray the complex interactions of atomic particles. Moreover, Feynman diagrams allow visual representation and calculation of the ways in which particles can interact through the exchange of virtual photons and thereby provide a tangible picture of processes outside the human capacity for observation. Because Feynman diagrams allow physicists to depict subatomic processes and develop theories regarding particle interactions, the diagrams have become an indispensable and widely used tool in particle physics.

Feynman diagrams derive from QED theory. Quantum electrodynamics (QED), is a fundamental scientific theory that is also known as the quantum theory of light. QED describes the quantum properties (properties that are conserved and that occur in discrete amounts called quanta) and mechanics associated with the interaction of electromagnetic radiation (of which visible light is but one part of an electromagnetic spectrum) with matter. The practical value of QED rests upon its ability, as set of equations, to allow calculations related to the absorption and emission of light by atoms and thereby allow scientists to make very accurate predictions regarding the result of the interactions between photons and charged atomic particles (e.g., electrons).

Feynman diagrams are form of shorthand representations that outline the calculations necessary to depict electromagnetic and weak interaction particle

processes. QED, as quantum field theory, asserts that the electromagnetic force results from the quantum behavior of the photon, the fundamental particle responsible for the transmission electromagnetic radiation. According to QED theory, particle vacuums actually consist of electron-positron fields and electron-positron pairs (positrons are the positively charged antiparticle to electrons) are created when photons interact with these fields. QED accounts for the subsequent interactions of these electrons, positrons, and photons. Photons, unlike the particles of everyday experience, are virtual particles constantly exchanged between charged particles. As virtual particles, photons cannot be observed because they would violate the laws regarding the conservation of energy and momentum. QED theory therefore specifies that the electromagnetic force results from the constant exchange of virtual photons between charged particles that cause the charged particles to constantly change their velocity (speed and/or direction of travel) as they absorb or emit virtual photons. Accordingly, only in their veiled or hidden state do photons act as mediators of force between particles and only under special circumstances do photons become observable as light.

In Feynman time-ordered diagrams, time is represented on the x axis and a depicted process begins on the left side of the diagram and ends on the right side of the diagram. All of the lines comprising the diagram represent particular particles. Photons, for example, are represented by wavy lines. Electrons are denoted by straight lines with arrows oriented to the right. Positrons are depicted by a straight line with the arrow oriented to the left. Vertical y axis displacement in Feynman diagrams represents particle motion. The representation regarding motion is highly schematic and does not usually reflect the velocity of a particle.

Feynman diagrams depict electromagnetic interactions as intersections (vertices) of three lines and are able to describe the six possible reactions of the three fundamental QED particles (i.e., the electron, positron, and the photon). Accordingly, the diagram can depict the emission and absorption of photons by either electrons or positrons. In addition, is possible to depict the photon production of an electron-positron pair. Lastly, the diagrams can depict the collisions of electrons with positrons that results in their mutual annihilation and the production of a photon.

Feynman diagrams presuppose that energy and momentum are conserved during every interaction and hence at every vertex. Although lines on the diagram can represent virtual particles, all line entering or leaving a Feynman diagram must represent real particles with observable values of energy, momentum and mass (which may be zero). By definition, virtual particles acting as intermediate particles on the diagrams do not have observable values for energy, momentum or mass.

Although QED theory allows for an infinite number of processes (i.e., an infinite number of interactions) the theory also dictates that interactions of increasing numbers of particles becomes increasingly rare as the number of interacting particles increases. Correspondingly, although Feynman diagrams can accommodate or depict any number of particles, the mathematical complexities increase as the diagrams become more complex.

Feynman developed a set of rules for constructing his diagrams (appropriately named Feynman rules) that allow QED theorists to make very accurate calculations that closely match experimental findings. The Feynman rules are very simple, but as greater accuracy is demanded the diagrams become more complex. Feynman diagrams derive from the Feynman path integral formulation of quantum mechanics. Using asymptotic expansions of the integrals that describe the interactions, physicists are able to calculate the interactions of particles with great (but not unlimited) accuracy. The mathematical formulae associated with the diagrams are added to arrive at what is termed a Feynman amplitude, a value that is subsequently used to calculate various properties and processes (e.g., decay rates).

The development of QED theory and the use of Feynman diagrams allowed scientists to predict how subatomic processes create and destroy particles. Over the last half-century of research in particle physics, QED has become, arguably, the best-tested theory in science history. Most atomic interactions are electromagnetic in nature and, no matter how accurate the equipment yet devised, the predictions made by modern QED theory hold true. Some tests of QED, predictions of the mass of some subatomic particles, for example, offer results accurate to six significant figures or more. Although some predictions can be made using one Feynman diagram and a few calculations, others may take hundreds of Feynman diagrams and require the use of high speed computers to complete the necessary calculations.

## Resources

### BOOKS

Feynman, R. P. *QED: The Strange Theory of Light and Matter. Princeton.* NJ: Princeton University Press, 1985.

Gribbin, John, and Mary Gribbin. Q *is for Quantum* Touchstone Books, 2000.

Johnson, G.W., and M.L. Lapidus. *The Feynman Integral and Feynman’s Operational Calculus.* Oxford, England: Oxford University Press, 2000.

Mattuck, R. D. A *Guide to Feynman Diagrams in the Many-Body Problem,* 2nd ed. New York: Dover, 1992.

### PERIODICALS

Feynman, R. P. “Space-Time Approaches to Quantum Electrodynamics.” *Phys. Rev.* 76 (1949):769-789.

K. Lee Lerner

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**Feynman Diagrams**