COMPUTING SCIENCE
g-OLOGY
Brian Hayes
g-Whiz
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.
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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