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Brian Hayes


Calculating g from theoretical principles might seem to be far easier than measuring it experimentally. After all, the theorist can leave behind all the messy imperfections of the physical world and operate in an abstract realm where vacuums and magnetic fields are always ideal, and no one ever spills coffee on the control panel. But theory has challenges of its own, and in the saga of the g factor, 20-year-long experiments are matched by 30-year-long calculations.

Figure 1. The <em>g</em> factor setsClick to Enlarge Image Figure 2. Feynman diagramClick to Enlarge Image

What needs to be calculated is the strength of a charged particle's interaction with a magnetic field. The problem can be phrased in terms of something directly observable: Given a particle of known mass, charge and momentum, and a magnetic field of known intensity, how much will the particle's path be deflected when it passes through the field? Classical physics envisions magnetic lines of flux that induce a curvature in the particle's trajectory. Quantum electrodynamics takes a different approach. Instead of a field exerting its influence throughout a volume of space, QED posits a discrete, localized "scattering event," where an electron either emits or absorbs a photon (the quantum of the electromagnetic field); the recoil from this emission or absorption alters the electron's own motion.

A key tool for understanding such scattering events is the diagrammatic method introduced in the 1940s by Richard P. Feynman. A Feynman diagram plots position in space along one axis and time along another, so that a particle moving with constant velocity is represented by an oblique straight line. The Feynman diagram for a simple scattering event might have an electron moving diagonally until it collides with a photon coming from the opposite direction; at this "vertex" of the diagram the photon disappears and the electron reverses course.

There is more to a Feynman diagram, however, than just a spacetime depiction of particles colliding like billiard balls. As a matter of fact, in QED a particle cannot be assigned a unique, definite trajectory; all you can calculate is the probability that the particle will make its way from point A to point B. A Feynman diagram represents an entire family of possible trajectories, corresponding to collisions taking place at various positions and times. Each such trajectory has an associated "amplitude"; adding all the amplitudes and squaring the result yields the probability for the overall process.

The simplest scattering event—one electron bouncing off one photon—was the process considered by Dirac in his first computation of g in the 1920s. As noted above, Dirac got an exact result of g=2. The reason this value needs correcting is that the simplest, one-photon scattering process is not the only way for an electron to get from point A to point B. The direct route may well be the most important path, but in QED you dare not ignore detours or distractions along the way.

One such distraction is for the electron to emit a photon and then reabsorb it, somewhat like a child throwing a ball in the air and running to catch it herself. The evanescent photon is called a virtual particle, because it can never be detected directly, but its effects on g are certainly real. Adding a virtual photon to the Feynman diagram is easy enough—it forms a loop, diverging from and then rejoining the electron path—but computing the photon's effect on g is more difficult. The problem is that the virtual photon can have unlimited energy. For an accurate computation, you have to add up the amplitudes associated with every possible energy—and without an upper limit, this sum comes out infinite. These implausibly infinite answers stymied the further development of QED for two decades.

The solution was a trick called renormalization, worked out by Feynman, Julian Schwinger, Sin-Itiro Tomonaga and Freeman Dyson. In 1947 Schwinger finally succeeded in calculating the contribution of a single virtual-photon loop to the g factor of the electron. The answer was given in terms of another fundamental constant of nature, known as , which measures the electric charge of the electron and has a numerical value of about 1/137. Schwinger showed that the one-loop contribution to the "anomalous magnetic moment" of the electron is /2π, or approximately 0.00116. The anomalous magnetic moment is defined as one-half of g–2, and so the corrected value of g comes out to about 2.00232.

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