COMPUTING SCIENCE
g-OLOGY
Brian Hayes
QED
The naive mental picture of an electron is a blob of mass and
electric charge, spinning on its axis like a tiny planet. If we take
this image seriously, the moving charge on the spinning particle's
surface has to be regarded as an electric current, which ought to
generate a magnetic field. The g factor (also known as the
gyromagnetic ratio) is the constant that determines how much
magnetic field arises from a given amount of charge, mass and spin.
The formula is:
where µ is the magnetic moment, e the electric
charge, m the mass and s the spin angular momentum
(all expressed in appropriate units). Early experimental evidence
suggested that the numerical value of g is approximately 2.
In the 1920s P. A. M. Dirac created a new and not-so-naive theory of
electrons in which g was no longer just an arbitrary
constant to be measured experimentally; instead, the value of
g was specified directly by the theory. For an electron in
total isolation, Dirac calculated that g is exactly 2. We
now know that this result was slightly off the mark; g is
greater than 2 by roughly one part in a thousand. And yet Dirac's
mathematics was not wrong. The source of the error is that no
electron is ever truly alone; even in a perfect vacuum, an electron
is wrapped in a halo of particles and antiparticles, which are
continually being emitted and absorbed, created and annihilated.
Interactions with these "virtual" particles alter various
properties of the electron, including the g factor.
Methods for accurately calculating g were devised in the
1940s as part of a thorough overhaul of the theory of
electrons—a theory called quantum electrodynamics, or QED.
That the calculation of g can be honed to such a razor edge
of precision is something of a fluke. The mass, charge and magnetic
moment of the electron are known only to much lower accuracy; so how
can g, which is defined in terms of these quantities, be
pinned down more closely? The answer is that g is a
dimensionless ratio, calculated and measured in such a way that
uncertainties in all those other factors cancel out.
Experimental measurements of g benefit from another fortunate
circumstance. The experiments can be arranged to determine not
g itself but the difference between g and 2; thus
the measurements have come to be known as "g minus 2
experiments." Because g–2 is only about a
thousandth of g, the measurement gains three decimal places
of precision for free.
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