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advanced_tools:internal_symmetry [2018/04/15 09:38] ida [Why is it interesting?] |
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====== Internal Symmetry ====== | ====== Internal Symmetry ====== | ||
+ | //see also [[basic_tools:symmetry]] and [[advanced_tools:gauge_symmetry]]// | ||
<tabbox Intuitive> | <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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<tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
+ | Internal symmetries are powerful that we use, for example, to derive the correct Lagrangians describing fundamental interactions. | ||
</tabbox> | </tabbox> | ||