Feynman Diagrams

Type of physical science: Atomic physics

Field of study: Relativistic quantum mechanics

Feynman diagrams are pictorial representations of interactions among particles. They provide an invaluable computational aid in quantum electrodynamics, elementary particle theory, and other areas of physics.

Overview

The location of a particle is given by its three space coordinates, for example, x, y, z. The time t at which the particle is observed could be envisaged as a fourth coordinate. One could then show the particle's trajectory as a line in a four-dimensional space. In the theory of relativity, this line is known as the "world line" of the particle. By showing only one representative space coordinate, instead of all three, a world line can be represented on a two-dimensional plot, such as the following:

Richard P. Feynman introduced such space-time pictures in his formulation of quantum electrodynamics in the late 1940's. In this context, they are now known as "Feynman diagrams" (or Feynman graphs). Quantum electrodynamics, sometimes abbreviated QED, is the fundamental theory of interacting photons and electrons. An electron standing still would be represented by the following Feynman diagram:

Note that the time advances while the space coordinate remains constant. Here is an example of a moving electron:

High-energy processes are likely to involve positrons, the positively charged antiparticles of the electron. The preceding two diagrams have obvious analogs for positrons. A simplification is achieved if a positron is regarded as an electron traveling backward in time. A positron in motion would be represented as follows:

Analogously, the usual convention for electric current is a flow of imaginary positive charge moving opposite to the actual motion of the electrons.

According to quantum electrodynamics, electromagnetic radiation, including light, consists of photons, designated g. Photons also carry the electromagnetic force among charged particles. A photon is represented by a wavy line, as shown below:

Photons in a vacuum travel at the speed c = 3 x 10⁸ meters per second, usually known as the speed of light.

Next, particles interacting with one another will be considered. Note that the space and time axes and the labels e and gamma will be omitted. Following is the Feynman diagram for an electron in an atom absorbing a photon as it goes into a higher-energy level:

If the electron is ejected from the atom, this corresponds to the photoelectric effect. The rest of the atom, the nucleus and the other electrons, are omitted from the diagram but their actual presence is essential so that the total energy and momentum are conserved. The inverse process, an electron emitting a photon as it falls to a lower-energy level, is illustrated as follows:

When an electron and a positron collide, they mutually annihilate each other, in accordance with Albert Einstein's famous relation (special relativity) for the interconversion of matter and energy: E = mc². The energy is carried off by a gamma-ray photon. Recall that a positron will resemble an electron moving backward in time. A Feynman diagram for pair annihilation is shown below:

There must be other particles involved to conserve total energy and momentum. (In fact, pair annihilation usually produces two photons.) Following is the inverse process of pair creation:

The previous four diagrams have in common a photon line meeting two electron lines at a "vertex," one coming into the vertex, the other going away from it. This configuration represents the essence of the electromagnetic interaction: the absorption or emission of photons by electric charges. The four processes represented above can be regarded as fundamentally equivalent, differing only in the observer's perspective in space-time.

Processes involving two photon-electron vertices, such as Rayleigh scattering, are responsible for the blue color of the sky and involve the momentary absorption of a photon by an atom and its subsequent reemission, most likely in a different direction. Raman scattering is similar except that the emitted photon has a different energy, and hence a different wavelength, from that of the incident photon. If the outgoing electron is knocked out of the atom, this is known as Compton scattering. These scattering processes can be represented by the following diagrams:

In the second version of the process, the scattered photon is actually emitted before the incident one arrives. Nevertheless, quantum mechanics allows such things to happen.

According to quantum electrodynamics, the electromagnetic interaction between two charged particles is mediated by a continuous exchange of photons. Following is a simple approximation for two electrons interacting or "scattering," as physicists tend to refer to the process:

The electrons are exchanging what is known as a "virtual" photon; since it has only a momentary existence during the course of the interaction. Virtual particles are represented by internal lines in Feynman diagrams; they are not observable particles. A more accurate representation of electron-electron scattering must take account of contributions from additional diagrams, for example:

The first two diagrams above represent the exchange of two photons, while the last diagram features a virtual electron-positron pair.

Feynman diagrams can also be used to represent strong and weak interaction processes. The best-known weak interaction is beta decay, in which a neutron, either free or part of a nucleus, is transformed into a proton, along with an electron and an antineutrino: n → p + e + ῡ . . The neutron has the quark makeup "udd," while the proton has the makeup of "uud." Thus, beta decay is fundamentally a d-to-u quark transformation, as represented by the diagram:

The weak interaction is mediated here by a W-particle. Other weak processes involve W+ or Z to the power of 0 particles.

The strong force is, at the most fundamental level, an interaction among quarks that act on their "color," a property that plays a role analogous to electric charge in quantum electrodynamics. The strong force is carried by gluons, of which there are eight varieties. The simplest diagram for quark-quark interaction resembles the analogous one in QED:

In contrast to photons, which themselves carry no electric charge, gluons do carry color. Gluons therefore also interact among themselves, exchanging other gluons. Diagrams, such as the one below, show how complicated things can get in quantum chromodynamics (QCD), the color-force analog of QED.

Applications

Feynman diagrams do not ordinarily impinge on everyday life, except for theoretical physicists, so this section must appeal to some very indirect connections. Any advance in the knowledge of the subatomic world does eventually yield dividends, witness nuclear power and positron emission tomography (PET). Diagram techniques have become a useful theoretical tool in solid-state physics (see diagram below) and, as such, will ultimately lead to the development of new materials and processes.

Feynman originally invented his diagrams as a shorthand representation for algebraic quantities used to compute probabilities of elementary-particle interactions. By what are known as Feynman's rules, each vertex, each external line, and each internal line contributes a specific function to a mathematical expression for the probability. Some interactions require a sum over graphs, for example, the electron-electron scattering considered above. In this case, an infinite number of graphs are necessary, which can be represented by the following pictorial:

In QED, under the most favorable circumstances, the first few graphs converge rapidly to an adequate approximation so that further contributions can be neglected. This convergence results from the fact that the intrinsic strength of the electromagnetic interaction is such that each vertex contributes a factor 1/137 to the scattering probability. In applications to strong interactions, the corresponding factor is close to 1 and the analogous series does not converge; therefore, other mathematical techniques must be used.

Feynman diagrams have also become an invaluable aid in the study of a many-particle system, as for example in solid-state physics and statistical mechanics. Consider, for example, the many-electron system in a crystalline solid. In the electronic ground state, the orbitals are filled from the lowest energy upward, with two electrons per orbital (one of each spin), in accordance with the Pauli exclusion principle. This ground state would be the state of the crystal at absolute zero. The state of the system is thought of as a vacuum. If one of the electrons is excited to a higher-energy orbital, it can be thought of as creating a particle where none existed before. At the same time a "hole" is left in the orbital vacated by the electron, which can be thought of as the corresponding "antiparticle." It will have the opposite charge to the particle since the absence of a negative charge is equivalent to a positive charge. The force carriers (analogous to photons or gluons) take the place of the complex electromagnetic interactions among the electrons and nuclei in a crystal.

Context

Quantum mechanics, the fundamental theory of matter that succeeded Newtonian or classical mechanics, was developed during 1925-1926 principally by Werner Heisenberg, Erwin Schrodinger, and Paul Adrien Maurice Dirac. Heisenberg's formulation, known as matrix mechanics, was based on arrays of numbers (matrices) which represented dynamical variables. Schrodinger's formulation, known as wave mechanics, made use of operators and wave functions. Most quantum-mechanical computations are now based on the Schrodinger equation. Dirac's transformation theory provided a unified synthesis encompassing both matrix and wave mechanics as special cases. Modern physicists routinely use ideas and notations taken from all three versions of quantum mechanics.

In 1948, Feynman proposed yet another formulation of quantum mechanics, called the "path-integral approach." The basic idea was that a quantum-mechanical particle moving from point one to point two could be represented by a summation over all possible classical paths between the two points. Schematic space-time representations of such processes, known as Feynman diagrams, were an outgrowth of this approach. The path-integral formulation was spectacularly successful in applications to quantum electrodynamics, earning Feynman the 1965 Nobel Prize in Physics with Shin'ichiro Tomonaga and Julian Seymour Schwinger.

Feynman diagrams are exploited in several other branches of physics, including elementary particle theory, statistical mechanics, solid-state dynamics, and molecular collision dynamics.

Principal terms:

GLUON: the carrier of the strong force between quarks

PHOTON: according to quantum electrodynamics, a particle of electromagnetic radiation; also carries the electromagnetic force between charged particles

POSITRON: the antiparticle of the electron, with the same mass but the opposite electric charge

QUANTUM ELECTRODYNAMICS (QED): the quantum theory of the electromagnetic field and its interactions with charged particles

QUARKS: elementary particles that combine in threes to make up protons and neutrons

SCATTERING: a generic term for interactions between particles

SPECIAL RELATIVITY: Albert Einstein's theory unifying the concepts of space and time; as a consequence, matter and energy become equivalent according to the relation E = mc to the power of 2

STRONG INTERACTION: the force that binds the protons and neutrons in nuclei; on a more fundamental level, the forces between quarks

WEAK INTERACTION: one of the four fundamental forces, responsible for the radioactive beta decay of nuclei

Bibliography

Crease, Robert P., and Charles C. Mann. THE SECOND CREATION. New York: Macmillan, 1986. An absorbing popular account of how the modern theory of particles and fields was created. Personal glimpses of the scientists involved. Chapters 5-8 describe the conceptual difficulties of quantum electrodynamics and how they were overcome.

Feynman, Richard P. QED: THE STRANGE THEORY OF LIGHT AND MATTER. Princeton, N.J.: Princeton University Press, 1985. Based on a series of four popular lectures given by Feynman at the University of California at Los Angeles. Explains QED using pictures rather than formulas. The third lecture introduces Feynman diagrams.

Feynman, Richard P. QUANTUM ELECTRODYNAMICS. New York: W. A. Benjamin, 1961. Lecture notes and reprints of Feynman's PHYSICAL REVIEW papers on QED are included.

Feynman, Richard P., and A. R. Hibbs. QUANTUM MECHANICS AND PATH INTEGRALS. New York: McGraw-Hill, 1965. A graduate-level textbook on the path-integral formulation of quantum mechanics.

Feynman, Richard P., and Ralph Leighton. SURELY YOU'RE JOKING, MR. FEYNMAN! New York: W. W. Norton, 1985.

Feynman, Richard P., and A. R. Hibbs. WHAT DO YOU CARE WHAT OTHER PEOPLE THINK? New York: W. W. Norton, 1988. A collection of anecdotes and reminiscences about Feynman's personal and scientific life.

Mattuck, Richard D. A GUIDE TO FEYNMAN DIAGRAMS IN THE MANY-BODY PROBLEM. 2d ed. New York: McGraw-Hill, 1976. A lively and readable introduction to the applications of diagram techniques in solid-state physics and statistical mechanics. Contains the famous pinball machine analog to a particle propagator.

Weidner, Richard T., and Robert L. Sells. ELEMENTARY MODERN PHYSICS. 2d ed. Boston: Allyn & Bacon, 1973. An undergraduate textbook on modern physics, covering atomic, molecular, solid-state, and elementary particle physics. Chapter 11 makes use of Feynman diagrams in connection with the electromagnetic and strong interactions.

world line

electron standing still

moving electron

moving positron

photon

electron absorbing photon

electron emitting photon

pair annihilation

pair creation

Raman and Compton scattering

two electrons scattering

electron-electron scattering

beta decay

quark-quark interaction

QCD interaction

multi-graph interaction

Electrons and Atoms

Leptons and the Weak Interaction

Quarks and the Strong Interaction