## The best theory for explaining the subatomic world got its start in 1928 when
theorist Paul Dirac combined quantum mechanics with special relativity to explain
the behavior of the electron. The result was relativistic quantum mechanics,
which became a major ingredient in quantum field theory. With a few assumptions
and ad hoc adjustments, quantum field theory has proven powerful enough to form
the basis of the Standard Model of particles and forces.

“Even so, it should be remembered that the Standard Model is not a final
theory of all phenomena, and is therefore inherently incomplete,” says
Dmitry Budker, a staff scientist in the Nuclear Science Division of the U.S.
Department of Energy’s Lawrence Berkeley National Laboratory and a
professor of physics at the University of California at Berkeley.

Budker has long been interested in testing widely accepted underpinnings of
physical theory to their limits. In the June 25 issue of Physical Review Letters,
he and his colleagues report the most rigorous trials yet of a fundamental assumption
about how particles behave on the atomic scale.

Why we need the spin-statistics theorem

“We tested one of the major theoretical pillars of quantum field theory,
the spin-statistics theorem,” says Damon English, Budker’s former
student and a postdoctoral fellow in UC’s Department of Physics, who led
the experiment. “Essentially we were asking, are photons really perfect
bosons?”

The spin-statistics theorem dictates that all fundamental particles must be
classified into one of two types, fermions or bosons. (The names come from the
statistics, Fermi-Dirac statistics and Bose-Einstein statistics, that explain
their respective behaviors.)

No two electrons can be in the same quantum state. For example, no two electrons
in an atom can have identical sets of quantum numbers. Any number of bosons
can occupy the same quantum state, however; among other phenomena, this is what
makes laser beams possible.

Electrons, neutrons, protons, and many other particles of matter are fermions.
Bosons are a decidedly mixed bunch that includes the photons of electromagnetic
force, the W and Z bosons of the weak force, and such matter particles as deuterium
nuclei, pi mesons, and a raft of others. Given the pandemonium in this particle
zoo, it takes the spin-statistics theorem to tell what’s a fermion and
what’s a boson.

The way to tell them apart is by their spin – not the classical spin
of a whirling top but intrinsic angular momentum, a quantum concept. Quantum
spin is either integer (0, 1, 2…) or half integer, an odd number of halves
(1/2, 3/2…). Bosons have integer spin. Fermions have half integer spin.

“There’s a mathematical proof of the spin-statistics theorem, but
it’s so abstruse you have to be a professional quantum field theorist
to understand it,” says Budker. “Every attempt to find a simple
explanation has failed, even by scientists as distinguished as Richard Feynman.
The proof itself is based on assumptions, some explicit, some subtle. That’s
why experimental tests are essential.”

Says English, “If we were to knock down the spin-statistics theorem,
the whole edifice of quantum field theory would come crashing down with it.
The consequences would be far-reaching, affecting our assumptions about the
structure of spacetime and even causality itself.”

In search of forbidden transitions

English and Budker, working with Valeriy Yashchuk, a staff scientist at Berkeley
Lab’s Advanced Light Source, set out to test the theorem by using laser
beams to excite the electrons in barium atoms. For experimenters, barium atoms
have particularly convenient two-photon transitions, in which two photons are
absorbed simultaneously and together contribute to lifting an atom’s electrons
to a higher energy state.

“Two-photon transitions aren’t rare,” says English, “but
what makes them different from single-photon transitions is that there can be
two possible paths to the final excited state – two paths that differ
by the order in which the photons are absorbed during the transition. These
paths can interfere, destructively or constructively. One of the factors that
determines whether the interference is constructive or destructive is whether
photons are bosons or fermions.”

In the particular barium two-photon transition the researchers used, the spin-statistics
theorem forbids the transition when the two photons have the same wavelength.
These forbidden two-photon transitions are allowed by every known conservation
law except the spin-statistics theorem. What English, Yashchuk, and Budker were
looking for were exceptions to this rule, or as English puts it, “bosons
acting like fermions.”

The experiment starts with a stream of barium atoms; two lasers are aimed
at it from opposite sides to prevent unwanted effects associated with atomic
recoil. The lasers are tuned to the same frequency but have opposite polarization,
which is necessary to preserve angular momentum. If forbidden transitions were
caused by two same-wavelength photons from the two lasers, they would be detected
when the atoms emit a particular color of fluorescent light.

The researchers carefully and repeatedly tuned through the region where forbidden
two-photon transitions, if any were to occur, would reveal themselves. They
detected nothing. These stringent results limit the probability that any two
photons could violate the spin-statistics theorem: the chances that two photons
are in a fermionic state are no better than one in a hundred billion –
by far the most sensitive test yet at low energies, which may well be more sensitive
than similar evidence from high-energy particle colliders.

Budker emphasizes that this was “a true table-top experiment, able to
make significant discoveries in particle physics without spending billions of
dollars.” Its prototype was originally devised by Budker and David DeMille,
now at Yale, who in 1999 were able to severely limit the probability of photons
being in a “wrong” (fermionic) state. The latest experiment, conducted
at UC Berkeley, uses a more refined method and improves on the earlier result
by more than three orders of magnitude.

“We keep looking, because experimental tests at ever increasing sensitivity
are motivated by the fundamental importance of quantum statistics,” says
Budker. “The spin-statistics connection is one of the most basic assumptions
in our understanding of the fundamental laws of nature.”

“Spectroscopic test of Bose-Einstein statistics for photons,” by
Damon English, Valeriy Yashchuk, and Dmitry Budker, appears in the June 25 issue
of Physical Review Letters and is available online. The research was supported
by the National Science Foundation.