SU(3) from three belts?

  • #1

Main Question or Discussion Point

Dear PFers,

I am looking for a way to visualize SU(3). I have heard from a friend
(who heard it as a rumor) that like SU(2) can be
visualized with a (Dirac) belt, also SU(3) can be visualized, but
with three belts, because SU(3) has three independent copies of
SU(2) as subgroups.

I found no material on this on the internet.
Is there anybody who can help me with more details? Thank you!

Franca
 

Answers and Replies

  • #2
I am looking for a way to visualize SU(3). I have heard from a friend
(who heard it as a rumor) that like SU(2) can be
visualized with a (Dirac) belt, also SU(3) can be visualized, but
with three belts, because SU(3) has three independent copies of
SU(2) as subgroups.
Hi Franca,

the only place I have seen something similar is the last manuscript
on http://www.motionmountain.net/research [Broken]
where Schiller uses the triple belt trick to model the strong
interaction. Does this help you?

François
 
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  • #3
tom.stoer
Science Advisor
5,766
160
what's the Dirac belt?
 
  • #4
apeiron
Gold Member
2,013
1
Penrose used the single belt in a book to illustrate SU2 and spinor symmetry on around p200 of his Road to Reality from memory. Maybe your friend saw those pictures?

But not come across a three belt illustration of SU3.

The 1 belt trick is also illustrated on p21 of this paper...
http://website.lineone.net/~cobble6/Clifford Report.pdf
 
  • #5
the only place I have seen something similar is the last manuscript
on http://www.motionmountain.net/research [Broken]
where Schiller uses the triple belt trick to model the strong
interaction.
François
Thank you, that preprint is very interesting! I wanted the three
belts to understand nuclei, but that manuscript does something
much more ambitious: it tries to derive QED, QCD and QAD from
topologocal arguments. Incredible!

Franca

P.S. I very much like this (single) belt trick applet:
http://gregegan.customer.netspace.net.au/APPLETS/21/21.html
 
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  • #6
tom.stoer
Science Advisor
5,766
160
thanks - now I remember
 
  • #7
the only place I have seen something similar is the last manuscript
on http://www.motionmountain.net/research [Broken]
where Schiller uses the triple belt trick to model the strong
interaction. Does this help you?
In the meantime, I asked a bit around. The three belt idea is not
well-known yet. Searching for "three belts" and "SU(3)" gives
no hits anywhere. the manuscript above is a bit terse on
the topic, probably because the main topic is unification.

The main idea is that SU(3) has three copies of SU(2)
that are linearly independent. Each SU(2) can be modelled
by one belt. So three copies need three belts.
Then the three belts are connected by joints.

The 8 generators of SU(3) are explained as rotations by 180
degrees; and their products are said to be concatenations.
But though I can deduce some of the products, I fail
for others.

Did anybody else try?

Franca
 
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  • #8
The 8 generators of SU(3) are explained as rotations by 180
degrees; and their products are said to be concatenations.
But though I can deduce some of the products, I fail
for others. Did anybody else try?
Franca
I have tried a few products, and it seems to work out. The rest of the
file is more intriguing: the guy is going for the holy grail
of physics... It is quite a change in approach to what is seen around.
Unification in 3 d , unusual SU(2) symmetry breaking proposal, no GUT, non Susy - that is not the usual stuff...

François
 
  • #9
The rest of the
file is more intriguing: the guy is going for the holy grail
of physics... It is quite a change in approach to what is seen around.
Unification in 3 d , unusual SU(2) symmetry breaking proposal, no
GUT, non Susy - that is not the usual stuff
I asked about this on usenet. It seems to me that the approach has a chance.
I will post more on this approach to unification once I have made up my mind.

Franca
 
  • #10
I asked about this on usenet.
On sci.physics.research, Heinz posted this assessment:

--------

Franca, tell me if I write too much or too little. Here is what I get
from that text. It has 2 claims.
One is mathematical. It claims that the first Reidemeister move (that
is a standard way to deform knots)
is a generator of U(1), the second move(s) gives the generators of SU
(2), and the third Reidemeister
moves gives the generators of SU(3). See the wikipedia entry
http://en.wikipedia.org/wiki/Reidemeister_move .
*IF* the connection is correct, then the answer to
your question is that the gauge groups can indeed be related/tied to 3
dimensions.
The opposite is true if the claim is wrong. But even if the claim is
correct,
this does not mean that the physical gauge groups (as opposed to the
mathematical groups)
really are due to 3 dimensions. Other explanations are possible:
string theory.
So the answer to your question is: "maybe."

The other claim is physical: Schiller claims that particles are
tangles, and that
Reidemeister moves model gauge interactions. That can only be tested
against experiment. Schiller says that tangles lead to the Dirac
equation, and
gives a published paper from 1980 as a proof, plus an unpublished text
by himself.
*IF* the reasoning is correct, test with experiment would not be
necessary: it is
known that the Dirac equation is very precise.
The opposite is true if the reasoning is wrong. The 1980 paper appears
to
have almost no citations. So the answer to this claim is "hmm".
Does this answer your question?

Heinz

------

I will read the 1980 paper and more on Reidemeister things and come back soon.

Franca
 

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