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a property of addition and multip lication, where for any numbers A and B it can be stated that A + B = B + A and A X B = B X A. How the principle can be applied to a group of transformations can be ill ustrated with a familiar example: the group of rotations. All possible rotations of a two-dimensional object are commutative, and so the group of such rotations is Abelian. For instance, rotations of + 60 degrees and
-90 degrees yield a net rotation of -30 degrees no matter which is applied first.
For a three-dimensional object free to rotate about three axes the commutative law does not hold, and the group of three-dimensional rotations is nonAbelian.
As an example, consider an airplane heading due north in level flight. A 90-degree yaw to the left followed by a 90-degree roll to the left leaves the airplane heading west with its left wing tip pointing straight down.
Reversing the sequence of transformations,
so that a 90-degree roll to the left is followed by a 90-degree left yaw,
puts the airplane in a nose dive with the wings aligned on the north-south axis.
    Like the Yang-Mills theory, the general theory of relativity is non-Abelian:
in making two succe ssive coordinate transformations, the order in which the y are made usually has an e ffect on the outcome. In the past 10 years or so several more non-Abelian theories have been devised, and even the electromagnetic interactions have been incorporated into a larger theory that is non-Abelian.
For now, at least, it seems all the forces of nature are governed by non-Abelian gauge the ories.
    The Yang-Mills theory has proved to be of monumental importance, but as it was originally formulated it was totally unfit to describe the real world. A first objection to it is that isotopic-spin symmetry becomes exact, with the result that protons and neutrons are indistinguishable;
this situation is obviously contrary to fact. Even more troubling is the prediction of electrically charged photons. The photon is necessarily massless because it must have an infinite range. The existence of any electrically charged particle lighter than the electron would alter the world beyond recognition.
Of course, no such particle has been observed. In spite of these difficulties the theory has great beauty and philosophical appeal. One strategy adopted in an attempt to fix its defects was to artificially endow the charged field quanta with a mass greater than zero.
    Imposing a mass on the quanta of the charged fields does not make the fields disappear, but it does confine them to a finite range. If the mass is large enough,
the range can be made as small as is wished. As the long-range effects are removed the existence of the fields can be reconciled with experimental observations.
Moreover, the selection of the neutral Yang-Mills field as the only real long-range one automatically distinguishes protons from neutrons. Since this field is simply the electromagnetic field, the proton and the neutron can be distinguished by their differing interactions with it, or in other words by their differing electric charges.
    With this modification the local symmetry of the Yang-Mills theory would no longer be exact but approximate,
since rotation of the isotopic-spin arrow would now have observable consequences.
That is not a fundamental objection: approximate symmetries are quite commonplace in nature. (The bilateral symmetry of the human body is only approximate.) Moreover, at distance scales much smaller than the range of the massive components of the Yang-Mills field, the local symmetry becomes better and better. Thus in a sense the microscopic structure of the theory could remain locally symmetric,
but not its predictions of macroscopic,
observable events.
    The modified Yang-Mills theory was easier to understand, but the theory still had to be given a quantum-mechanicalinterpretation. The problem of infinities turned out to be severer than it had been in quantum electrodynamics, and the standard recipe for