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On 07/31/2007 07:16 AM, Dan Crowe wrote:
On a more serious note, how many types of color charge are there?
Three.
John
argues that there is only one type of electric charge, just as there is
only one type of mass charge, because each can be described by one
independent variable. The only difference is that mass charge is
limited to non-negative values, but electric charge can be positive,
zero, or negative. How many independent variables are needed to
describe color charge?
That's a good question. The conventional and obvious answer is three.
In this case, the conventional and obvious answer is correct.
Folks who are not interested in details should stop reading now.
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The question is, how many kinds of color charge are there? This is
not a trivial question. At first glance, the answer is obviously
three. On a second look, it seems like the answer might be less
than three. However, on a third look, the original obvious answer
is seen to be correct after all.
A second look is required because there is a constraint involved.
The constraint says that ordinary things like baryons and mesons
are color-neutral. You might think that three dimensions subject
to one constraint leaves you with only two dimensions, but no,
it's not that sort of constraint.
There are two ways of seeing that three is the right answer.
1) First, the physics argument:
The aforementioned constraint (called quark confinement) applies
only to the macroscopic situation; it does not apply inside
things like baryons. You may have heard of the "bag model" of
quark confinement. Inside the bag it's a free-for-all. That is,
on the sub-baryon length scale, there are really, truly, and
unconstrainedly three types of color charge.
So I'm sticking with my original answer: Three kinds of color
charge.
2) There's also a perfectly good mathematical answer:
The symmetry of the color charge is given by the special unitary
group, SU(3). In general, SU(N) is a constrained version of the
unitary group U(N). The U(N) symmetry is definitely N dimensional;
the group has N^2 generators and can be represented using NxN
matrices.
You should not think that the constraint lowers SU(N) to having
only (N-1)^2 generators. Instead, the right answer is N^2 - 1.
In the case of SU(3) we have 3^2-1 = 8, which is the same 8
as shows up in MGM's expression "eightfold way". The matrix
representation for SU(3) uses 3x3 matrices, and there is no
representation using 2x2 matrices, so you are quite safe in
saying the SU(3) involves three colors, not two.
You can get more information by googling
http://www.google.com/search?q=color-charge
and
http://www.google.com/search?q=SU(3)
Which leads to nice things including
http://www2.slac.stanford.edu/VVC/theory/colorchrg.html
http://curious.astro.cornell.edu/question.php?number=650
http://en.wikipedia.org/wiki/Special_unitary_group
http://www.math.ucr.edu/home/baez/physics/ParticleAndNuclear/gluons.html
I particularly like the Baez article. It is easy to read and
emphasizes the physics. The math is correct but sketchy, which
is OK because you can get the mathematical details from the
other references.
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