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--- advanced_tools:internal_symmetry [2018/04/15 09:38]
ida [Why is it interesting?]
+++ advanced_tools:internal_symmetry [2018/04/15 09:38]
ida [Intuitive]
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 <tabbox Intuitive>
+<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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