advanced_tools:internal_symmetry
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advanced_tools:internal_symmetry [2017/11/14 15:55] jakobadmin [Why is it interesting?] |
advanced_tools:internal_symmetry [2019/01/24 10:19] (current) jakobadmin [Intuitive] |
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| - | ====== Internal Symmetries ====== | + | ====== Internal Symmetry ====== |
| - | <tabbox Why is it interesting?> | + | //see also [[basic_tools:symmetry]] and [[advanced_tools:gauge_symmetry]]// |
| + | |||
| + | <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. | <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 | + | 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 | 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 mo- | + | ( 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 |
| - | tion of the ball, at least if air resistence is neglected. But then you ask your | + | |
| friend: “As the ball was traveling toward you, was it spinning?” “Not a fair | 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 | question”, he responds. After all, the ball is perfectly spherical and perfectly | ||
| Line 18: | Line 19: | ||
| changes in the internal spin state of the ball would have to be accounted for | 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 | 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 interac- | + | 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 |
| - | tions 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 | 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 | the ball, not relevant to its motion through space, but conceivably relevant | ||
| in other situations. | in other situations. | ||
| + | |||
| The phase of a charged particle moving in an electromagnetic field (e.g., | 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. | a monopole field) is quite like the internal spinning of our ping-pong ball. | ||
| Line 34: | Line 35: | ||
| analogue of the room’s atmosphere, which is the agency (“force”) responsible | analogue of the room’s atmosphere, which is the agency (“force”) responsible | ||
| for any alteration in the ball’s internal spinning. | 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. | + | **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 | 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” | single particle, differing only in the value of an “internal quantum number” | ||
| Line 47: | Line 49: | ||
| and have determinant one). A bundle is built in which to “keep track” of | 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 | the particle’s internal state (generally over a 4-dimensional manifold which | ||
| - | can accomodate the particle’s “history”) | + | 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> | <cite>page 22ff in Topology, Geometry and Gauge fields by Naber</cite> | ||
| </blockquote> | </blockquote> | ||
| - | <tabbox Layman> | ||
| Line 57: | Line 64: | ||
| <cite>Deep Down Things by Schumm></cite></blockquote> | <cite>Deep Down Things by Schumm></cite></blockquote> | ||
| - | <tabbox Student> | + | <blockquote>„What the heck is an internal space?” you ask. Good question. The best answer I have is “useful.” It’s what we invented to quantify the observed behavior of particles, a mathematical tool that helps us make predictions. |
| + | “Yes, but is it real?” you want to know. Uh-oh. Depends on whom you ask. Some of my colleagues indeed believe that the math of our theories, like those internal spaces, is real. Personally, I prefer to merely say it describes reality, leaving open whether or not the math itself is real. How math connects to reality is a mystery that plagued philosophers long before there were scientists, and we aren’t any wiser today. But luckily we can use the math without solving the mystery. | ||
| + | |||
| + | <cite>Lost in Math by Sabine Hossenfelder</cite> | ||
| + | </blockquote> | ||
| + | |||
| + | |||
| + | <tabbox Concrete> | ||
| <note tip> | <note tip> | ||
| Line 63: | Line 77: | ||
| </note> | </note> | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| <note tip> | <note tip> | ||
| Line 70: | Line 84: | ||
| | | ||
| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | + | Internal symmetries are powerful that we use, for example, to derive the correct Lagrangians describing fundamental interactions. | |
| - | --> Example1# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| - | + | ||
| - | --> Example2:# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| - | + | ||
| - | <tabbox FAQ> | + | |
| - | + | ||
| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
advanced_tools/internal_symmetry.1510671349.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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