Internal Symmetry [The Physics Travel Guide]

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 <tabbox Why is it interesting?>
-<blockquote>You are sitting in a room with a friend and a ping-pong ball (perfectly spherical and perfectly white— the ping-pong ball, not the friend). The conversation gets around to Newtonian mechanics. You toss the ball to your friend.
-Both of you agree that, given the speed and direction of the toss, F = m A
-and the formula for the gravitational attraction at the surface of the earth
-( F = −mg k, if the positive z-direction is up), you could calculate the motion of the ball, at least if air resistance is neglected. But then you ask your
-friend: “As the ball was traveling toward you, was it spinning?” “Not a fair
-question”, he responds. After all, the ball is perfectly spherical and perfectly
-white. How is your friend supposed to know if it’s spinning? And, besides,
-it doesn’t matter anyway. The trajectory of the ball is determined entirely
-by the motion of its center of mass and we’ve already calculated that. Any
-internal spinning of the ball is irrelevant to its motion through space. Of
-course, this internal spinning might well be relevant in other contexts, e.g., if
-the ball interacts (collides) with another ping-pong ball traveling through the
-room. Moreover, if we believe in the conservation of angular momentum, any
-changes in the internal spin state of the ball would have to be accounted for
-by some force being exerted on it, such as its interaction with the atmosphere
-in the room, and we have, at least for the moment, neglected such interactions in our calculations. It would seem proper then to regard any intrinsic
-spinning of the ball about some axis as part of the “internal structure” of
-the ball, not relevant to its motion through space, but conceivably relevant
-in other situations.
-The phase of a charged particle moving in an electromagnetic field (e.g.,
-a monopole field) is quite like the internal spinning of our ping-pong ball.
-We have seen that a phase change alters the wavefunction of the charge
-only by a factor of modulus one and so does not effect the probability of
-finding the particle at any particular location, i.e., does not effect its motion
-through space. Nevertheless, when two charges interact (in, for example, the
-Aharonov-Bohm experiment), phase differences are of crucial significance to
-the outcome. The gauge field (connection), which mediates phase changes
-in the charge along various paths through the electromagnetic field, is the
-analogue of the room’s atmosphere, which is the agency (“force”) responsible
-for any alteration in the ball’s internal spinning.
-**The current dogma in particle physics is that elementary particles are
-distinguished, one from another, precisely by this sort of internal structure.**
-A proton and a neutron, for example, are regarded as but two states of a
-single particle, differing only in the value of an “internal quantum number”
-called isotopic spin. In the absence of an electromagnetic field with which to
-interact, they are indistinguishable. Each aspect of a particle’s internal state is
-modeled, at each point in the particle’s history, by some sort of mathematical
-object (a complex number of modulus one for the phase, a pair of complex
-numbers whose squared moduli sum to one for isotopic spin, etc.) and a group
-whose elements transform one state into another (U (1) for the phase and, for
-isotopic spin, the group SU (2) of complex 2 × 2 matrices that are unitary
-and have determinant one). A bundle is built in which to “keep track” of
-the particle’s internal state (generally over a 4-dimensional manifold which
-can accommodate the particle’s “history”). Finally, connections on the bundle
-are studied as models of those physical phenomena that can mediate changes
-in the internal state. Not all connections are of physical interest, of course,
-just as not all 1-forms represent realistic electromagnetic potentials. Those
-that are of interest satisfy a set of partial differential equations called the
-Yang-Mills equations, developed by Yang and Mills [YM] in 1954 as a
-nonlinear generalization of Maxwell’s equations.
-<cite>page 22ff in Topology, Geometry and Gauge fields by Naber</cite>
-</blockquote>
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