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of electromagnetism, and indeed two of them can be identified with the ordinary electric and magnetic fields. In other words, they describe the field of the photon. The remaining Yang-Mills fields can also be taken in pairs and interpreted as electric and magnetic fields, but the photons they describe differ in a crucial respect from the known properties of the photon: they are still massless spin-one particles, but they carry an electric charge. One photon is negative and one is positive.
    The imposition of an electric charge on a photon has remarkable consequences.
The photon is defined as the field quantum that conveys electromagnetic forces from one charged particle to another. If the photon itself has a charge, there can be direct electromagnetic interactions among the photons.
To cite just one example, two photons with opposite charges might bind together to form an "atom" of light. The familiar neutral photon never interacts with itself in this way.
    The surprising effects of charged photons become most apparent when a local symmetry transformation is applied more than once to the same particle. In quantum electrodynamics, as was pointed out above, the symmetry operation is a local change in the phase of the electron field, each such phase shift being accompanied by an interaction with the electromagnetic field. It is easy to imagine an electron undergoing two phase shifts in succession, say by emitting a
photon and later absorbing one. Intuition suggests that if the sequence of the phase shifts were reversed, so that first a photon was absorbed and later one was emitted, the end result would be the same. This is indeed the case. An unlimited series of phase shifts can be made,
and the final result will be simply the algebraic sum of all the shifts no matter what their sequence.
    In the Yang-Mills theory, where the symmetry operation is a local rotation of the isotopic-spin arrow, the result of m ultiple transformations can be quite different. Suppose a hadron is subjected to a gauge transformation, A, followed soon after by a second transformation,
B; at the end of this sequence the isotopic-spin arrow is found in the orientation that corresponds to a proton. N ow suppose the same transformations were applied to the same hadron but in the reverse sequence: B followed by A. In general the final state will not be the same;
the particle may be a neutron instead of a proton. The net effect of the two transformations depends explicitly on the sequence in which they are applied.
    Because of this distinction quantum electrodynamics is called an Abelian theory and the Yang-Mills theory is called a non-Abelian one. The terms are borrowed from the mathematical theory of groups and honor Niels Henrik Abel, a Norwegian mathematician who lived in the early years of the 19th century.
Abelian groups are made up of transformations that, when they are applied one after another, have the commutative property; non-Abelian groups are not commutative.
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    EFFECTS OF REPEATED TRANSFORMATIONS distinguish quantum electrodynamics, which is an Abelian theory, from the Yang-Mills theory,
which is non-Abelian. An Abelian transformation is commutative: if two transformations are applied in succession, the outcome is
the same no matter which sequence is chosen. An exmple is rotation in two dimensions. Non-Abelian transformations are not commutative,
so that two transformations will generally yield different results if their sequence is reversed. Rotations in three dimensions exhibit
this dependence on sequence. Quantum electrodynamics is Abelian in that successive phase shifts can be applied to an electron field
without regard to the sequence. The Yang-Mills theory is non-Abelian because the net effect of two isotopic-spin rotations is generally
different if the sequence of rotations is reversed. One sequence might yield a proton and the opposite sequence a neutron.

    Illustration by Allen Beechel
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    Commutation is familiar in arithmetic as