✎ Internal Symmetry [The Physics Travel Guide]

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====== Internal Symmetry ======

<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>
<tabbox Layman>

<blockquote>For the case of regular [[basic_notions:spin|spin]], we had to take spin-space seriously because it was associated with a concrete, measurable, physical quantity—angular momentum. This was only mildly uncomfortable because, although spinspace has the somewhat hard-to-stomach property that you have to turn all the way around twice to get back to your original condition, it’s otherwise pretty much like regular space. Isospin space, however, is completely abstract; it bears no relation whatsoever (other than through analogy) to anything we can grasp with our faculties of perception. How could rotations in such a space possibly have anything to do with the physical world? And yet the physical manifestation of the invariance of the strong force with respect to rotations in this space, the conservation of isospin, is a solidly established fact in the world of experimental science. So, what then is isospin-space from a physical point of view? Physicists usually describe it as an internal symmetry space, but what’s that, really? It’s your old buddy again, telling you that your car’s carburetion system “works on a vacuum principle.” How’s that going to help you to understand and fix the thing? It isn’t. Regarding the physical interpretation of the notion of isospin space, again your guess is as good as mine. Perhaps its experimental manifestations are hinting at some new and deeper truth about the universe that lies just beyond the current limits of our comprehension. Perhaps not. But one thing, however, is true: The introduction of the idea of internal symmetry spaces, of which isospin space was the first example, was an essential step forward in our understanding of the universe and the nature of the laws that govern it."
<cite>Deep Down Things by Schumm></cite></blockquote>

<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
</note>
 
<tabbox Researcher>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Example1#

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--> Example2:#

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</tabbox>

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