### Testing the Spin-Statistics Theorem

**The image shows two UC Berkeley physicists Dima Budker and Damon English. Dima Budker (**

*left*) is in a fermionic state, occupied by only one of himself. Many copies of the bosonic Damon English (*right*) occupy the same state at once. [image credit: Roy Kaltschmidt and Damon English /University of California, Berkeley and Lawrence Berkeley National Laboratory]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 [1], 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.”

**Two opposed laser beams, identical except for polarization, attempt to excite forbidden two-photon transitions in a beam of barium atoms. [image credit: Damon English]**

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.”

**References**

**[1]**Damon English, Valeriy Yashchuk, Dmitry Budker, “Spectroscopic test of Bose-Einstein statistics for photons”, Phys. Rev. Lett. 104, 253604 (2010). Abstract. arXiv:1001.1771.

**[2]**M.G. Kozlov, Damon English, and Dmitry Budker,“Symmetry-suppressed two-photon transitions induced by hyperfine interactions and magnetic fields,” by M.G. Kozlov, Damon English, and Dmitry Budker, Phys. Rev. A80, 042504 (2009). Abstract. arXiv:0907.3727.

[This report is written by Paul Preuss of Lawrence Berkeley National Laboratory]

Labels: Elementary Particles 2

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