Dynamics on Graphs 20
This is an introduction to the eight-fold way of particle physics, which led to the theory of quarks by Gell-mann and Zweig around 1964 and is at the heart of the current Standard Model.
Our aim is to eventually show how the numerology of this theory, meaning the positioning of particles with respect to quantum numbers connecting with the representation theory of the compact Lie group SU(3), has ultimately a much simpler description in terms of the Mutation and Numbers games on the graph A2, along with some associated elementary geometry and algebra.
As an introduction, in this video we give a quick overview of the Standard Model as it arose in the 1960's, including a discussion of quantum numbers and terminology covering Fermions, Bosons, Hadrons, Baryons, Leptons and Mesons. Then we present three important families of particles: the Baryon Multiplet or Octet, the Meson Nonet and the Baryon Decuplet. Each of these will be exhibited in a novel way that should be of interest to physicists.
The ongoing series on Dynamics on Graphs and Exceptional Structures in Maths and Physics can be found at our sister channel Wild Egg mathematics courses, and is available to Members of that channel at https://www.youtube.com/playlist?list...
These videos are also accessible to Patreon supporters. Please consider becoming a supporter at https://www.patreon.com/njwildberger
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0:00
Hello everyone I'm Norman Wildberger thanks for joining me on this adventure we're looking at
0:04
Dynamics on graphs and I'm wanting to apply what we've learned the mutation game the numbers game
0:10
Ade graphs to the standard model of physics, In particular the numerology the connections with
0:19
certain patterns of numbers that was discovered in the 1960s and I want to show you first of all a
0:28
non-standard way of looking at this so this will I think be interesting to physicists but also we're
0:35
going to setting ourselves up for being able to explain this story this aspect of the standard
0:41
model without resort to the representation theory of su(3), which is the usual method that's done
0:50
which requires all kinds of prior understanding rather sophisticated knowledge of representations
0:55
and Lie groups Etc. So we're going to sort of finesse that and have another way of a much
1:01
simpler way of looking at this and basically all what's really involved is this mutation game and
1:08
numbers game and associated geometry in algebra. Okay so to motivate the discussion around the
1:14
1960s the physicists sort of discovered that the large array of particles that they had
1:22
discovered in the accelerators and Cloud Chambers and such could be organized in certain fashions:
1:28
at least some of them could be by looking at certain quantum numbers and arranging these
1:33
particles in certain configurations, and prominent amongst them is something called the baryon
1:38
multiplet or sometimes baryon octet, which gives rise to the Eightfold way. This was initiated by
1:44
Gell-man and Ne'eman around the early 1960s so there are eight particles here prominently up
1:52
here is the neutron and the proton which are the standard sort of basic building blocks of matter
1:58
along with the electron and Photon perhaps and then there's some other particles which are much
2:04
more amorphous they only last for a split second like a millisecond or less in in a cloud chamber
2:10
or particle accelerator so they decay very quickly but uh still there or there and uh there's three
2:17
Sigma particles there's a uh I guess a Lambda particle and two side particles and it turns out
2:24
that they form this um sort of eight-fold pattern with six around the outside and two in the middle
2:31
when we organize them in terms of the charge q and another quantum number called strangeness s
2:39
which is has having values zero minus one and minus two. So one of the ideas that I'm going
2:46
to advocate here is that we should actually replace strangeness with another quantum number
2:50
I'll explain that later it's called hypercharge and that happens to have values: Y equals one,
2:56
Y equals zero and Y equals minus one okay and then that gives us an opportunity of
3:05
reconfiguring what this octet really looks like in this sort of configuration space
3:12
with charge in one direction and hyper charge in another so I'm going to reposition that usual
3:18
picture of the baryon multiplied or octet by introducing the q1 to be the charge axis and
3:26
Q2 to be the hypercharge axis and then when we plot our eight particles we have a neutron here
3:32
the proton is then here the sigma particles the Lambda particle and the these two xsi particles
3:41
arranged now in this sort of elongated hexagon which maybe looks a little bit familiar if you
3:47
remember the root system for the simple graph A2 okay so here are the the roots there are six roots
3:56
of A2 and they're given by these these vectors and these are exactly the vectors or positions
4:01
that we see here on the defining this hexagon outside so that's a first indication that maybe
4:08
there's some connection between this elementary particle story and this A2 graph so to set the
4:17
stage let me just give you a very very quick overview of some of the basic ideas of the
4:22
standard model which is the the current theory that explains or tries to explain uh the the
4:28
zoo of Elementary particles that we currently have okay so there are four important forces in physics
4:34
there's gravity that's the most sort of obvious one it's not directly relevant for the standard
4:39
model at least not currently although people are working very hard to try to incorporate it into
4:44
the standard model but that's a huge challenge but in the standard model itself there are three
4:49
forces that are at work there's electromagnetism which involves electrons and photons and then
4:55
there's the weak Force which is much harder to see it's responsible for a decay of particles and
5:02
there are some some particular particles called w and z bosons which sort of create this weak force
5:09
and then there's the strong force which binds the nucleons together that is the new neutrons
5:14
and protons that form the nucleons of atoms and this is where the corks and come in and also the
5:22
particles that sort of mediate this force called gluons okay so the standard model is concerned
5:29
with these three forces and these days these are described in terms of what's called gauge
5:36
gauge groups so there's a U1 for electromagnetism and su(2) for the weak Force an su(3) for the
5:43
strong force and we'll talk about these things you know later so this is a very interesting and
5:48
elaborate story (I don't really know that much about it) but we're going to be concentrating
5:53
on trying to understand the particles in the context of these three forces okay so there's
5:59
a whole range of particles and most of them are just lasting a very brief amount of time
6:05
but still physicists want to try to understand the nature and so part of that story is to try
6:10
to explain these particles and physicists have categorized these so there's certain terms that
6:16
really you need to have some basic familiarity with and this is my attempt at sort of roughly
6:22
sketching what these various classifications are so this is not meant to be entirely accurate
6:29
um probably I'm leaving some things out and maybe I've put some things in the wrong places but it
6:33
roughly the idea is something like this a first subdivision of the particles is into fermions
6:38
and bosons so this column is the fermions this column is the bosons and that has to
6:44
do with distinguishing the type of spin that these Elementary particles have now this is a little bit
6:50
in somehow analogous to a top spinning around but it's something different it's a different kind of
6:56
spin okay that's a little bit hard to visualize but in any case some particles have integer value
7:02
of spin so spin is a number like you know one spin of one spin of two spin of three Etc but you could
7:08
also have half integer spin like spin of half or spin of minus a half and the fermions are the
7:13
ones that have half integer Spin and the bosons are the ones that have integer spin now there's
7:19
a very important distinction because it turns out that these particles obey sort of different rules
7:25
we'll call statistics so these ones obey what are called Fermi Dirac statistics and these ones Bose
7:33
Einstein statistics and The crucial difference is that the fermions obey the Pauli Exclusion
7:39
Principle which prohibits two particles of the same type from occupying the same state from
7:46
being together you know at the same time okay now there's another subdivision corresponding to the
7:53
rows here that roughly corresponds to whether the particles feel the strong force or don't feel the
7:59
strong force okay so the hadrons are the particles that feel the strong force and the other ones I
8:06
don't know if there's actually a name for them but they're the ones that don't feel the strong force
8:10
okay so the fermions which do feel the strong force are called baryons and the fermions which
8:16
don't feel the strong force are called leptons so that's the basic subdivision and in some
8:20
sense these are sort of the the main classes of particles that actually contribute to to matter
8:26
you know to the stuff around us now uh in this baryon box we see first of all prominently the
8:35
quarks although technically speaking um probably hadron means a composite particle so maybe quarks
8:42
are actually officially excluded but okay so quarks are so really Elementary kind of
8:48
particles uh that come in six different flavors up down strange charm top and bottom okay we'll talk
8:57
about that later and also in three they come in three colors blue red and green so there's
9:02
the world of quarks whatever they are and then there are other families of baryons and it turns
9:07
out that they form sort of patterns just like the one that I've showed you that baryon multiplet or
9:13
baryon octet so there's a picture of that baryon octet okay and then there's another picture um
9:20
baryon the couplet like that that has a bunch of particles that form a triangular pattern like this
9:27
the leptons are also fermions so they also obey the Pauki Exclusion Principle and the most notable
9:34
one there is the electron and then there are other versions sort of more of massive versions of the
9:39
electron called the Muon and the Tau particles and then there's also neutrinos which are sort of
9:44
weird things that don't seem to pay any attention to anything and they just zip through everything
9:48
okay on the boson side the things with integer spin things that don't obey the power Exclusion
9:53
Principle we have the mesons so these are the ones that are basically somehow associated with
9:59
quarks and so examples of them are gluons and okay there's also multiplets that are associated
10:08
with M so here's a meson nonet it's called which we'll talk about shortly and then mesons which
10:16
don't feel the strong force well there's the photon and then the w and z bosons which are
10:22
involved with the weak force and then there's this sort of remarkable relatively new particle,
10:27
called the Higgs particle, which was discovered only less than 10 years ago and um okay so very
10:35
roughly this is sort of a classification of course lots of particles in in these boxes
10:40
typically especially in in these ones here um but this is a rough classification so fermions and
10:46
bosons important and then there's the baryons and the leptons they're types of fermions and there's
10:51
also mesons which are bosons but they share the property of the variants that they're basically
10:56
built from quarks roughly so they they're involved in the strong interaction and then these these
11:01
other bosons are involved with other interactions either electromagnetism or the weak Force
11:08
okay very rough overview okay and the other important ingredient that we need to talk about
11:13
are quantum numbers so these are numbers that one Associates to Elementary particles that are
11:20
conserved in particle interactions so if there's a collision then you total up the uh the total of
11:28
the this whatever quantum number you're interested in before and after and those two numbers should
11:33
agree so the most famous such quantum number is charge denoted by q and then there's something
11:42
called ISO spin which is denoted by I and then there's sort of a z coordinate of the
11:48
isospin called I_Z and I should say that there's some sensitivity here in terms of what kind of
11:55
particle interactions we're talking about okay so some of these quantum numbers are invariant under
12:00
all interactions and some of them only for strong or perhaps only for weak interactions so you know
12:06
the situation is a little bit more complicated but I'm just giving you a very brief overview and
12:10
this is roughly you know around the 1960s that the picture was like this since then there are
12:16
more quantum numbers that have been discovered so baryon number is an important number that's
12:22
just the number of baryons so the ones in that box that I showed you okay so if you count the number
12:28
of baryons at the beginning of some interaction and the number of baryons at the end they should
12:33
match up and similarly with the leptons so you know the actual particles can change their nature
12:39
but the total number of leptons to begin with and the total number of leptons at the end should also
12:44
match in fact there's further subdivisions here but we'll just leave it at that and then there's
12:48
this thing called strangeness which I believe was introduced by Murray Gell-mann going back into
12:53
the 50s and they noticed that certain particles that we thought were associated with the strong
13:01
force also some somehow were affected by the weak force in in a perhaps strange way and so this was
13:09
called strangeness and this quantum number is a sort of a measure of uh the extent to
13:14
which particles do that and um and then there's something called hypercharge and maybe one way
13:21
of thinking about hypercharge is it's the sum of the strangeness and the baryon number and this is
13:27
quite important for us because this Y is the same one that I introduced when I was talking about the
13:32
baryon multiplet as an alternative alternative scale instead of using strangeness we were using
13:41
the hypercharge and then there's a famous relation due to Gell-mann and Nishijima that relates the
13:49
charge the Z component of the isospin and the hypercharge it's Q equals I_Z plus one half Y
13:57
so these things are all you know connected and let me state again this is only very rough.
14:03
Actually later on there were more quantum numbers associated with other quarks in fact
14:10
and this equation has to be modified etc but very roughly this is the idea
14:14
and this is a really important one for us because we're going to be changing
14:17
our point of view instead of strangeness we're going to be focusing on hypercharge
14:22
okay so the next little grouping that we want to talk about is called the Meson Nonet non-ed
14:28
for nine and it looks a lot like the baryon octet that I'm showing you before except that there are
14:35
three mesons at in that Central position and the quantum numbers that we're interested in are the
14:42
same ones as previously the charge here minus one zero one and the strangeness one zero minus one
14:52
now remember I said that I want to convert things to hypercharge so I have to also tell you what
14:58
the hypercharge is and to get the hypercharge we take the strangeness and we add the baryon number
15:04
barion number okay so what are these things these are mesons so baryons and mesons are
15:09
both hadrons they're both particles that feel the strong force but these ones are bosons
15:15
the other ones were fermions so the bosons are not baryons so these guys all have baryon numbers of
15:22
zero which means that their hypercharge actually equals their strangeness so y equals one y equals
15:29
zero and Y equals minus one so actual make no difference in this case here okay so this is the
15:36
usual picture that one sees in texts and papers of uh the mesons at least some of the mesons
15:45
okay so here's our straightened view of this Meson nonet our axis here of charge which we
15:53
might call it q1 and here is hypercharge this Y which is the same as Q2 and I've just plotted
15:58
the various points now and we get something very similar looking as we did for the baryon multiplet
16:05
okay zero K plus pi minus Pi zero pi plus and there's also an eight and a to Prime and
16:11
a k minus and a k bar zero okay so there's the Meson knotted in this sort of straight
16:18
and form where we don't have 60 degree angles we've made everything rectangular
16:23
there's some really good advantages to that as we'll see later on when we connect with
16:28
the mutation game and the numbers game and Associated geometry okay and now
16:33
we come to another pattern this time also of baryons and it's called the Baryon Decouplet
16:39
because there's 10 particles in this little family or grouping and there are four Delta particles
16:48
okay Delta minus Delta zero Delta plus Delta double plus and then some Sigma particles but
16:54
Sigma star minus Sigma star zero Sigma Star Plus two PSI particles PSI star minus x i star zero
17:03
and then another one down here at the bottom quite a different name
17:07
Omega minus this one's actually quite important in this story because Gelman famously predicted
17:14
the existence of this particle before it actually had been discovered by observing
17:20
that these particles created this pattern when we plot them in the familiar way with charge along
17:27
these lines here charge going -1 0 1 and 2. and strangeness zero minus one minus two minus three
17:39
and so he noticed that this natural pattern had something missing here and so he said oh there
17:44
should be a particle there and he was able to predict uh you know what kind of characteristics
17:49
it had and then the experimentalists were able to go out in the year or two later they discovered
17:54
this particle in their accelerators and this was a huge confirmation of the essential validity
17:59
of this eight-fold way and this approach to the classification of Elementary particles so these
18:05
are all baryons okay they're baryons so when we want to convert the strangeness to hypercharge
18:12
remember this the hypercharge is strangest plus baryon number these are all baryons so we should
18:18
add one so hypercharge equals one hypercharge equals zero hypercharge equals minus one and
18:26
Hyper charge equals minus two okay so when we plot these 10 particles in the same context as before
18:33
charge in this direction hypercharge in this direction so now the Delta particles go from from
18:41
a charge of -1 to a charge of two but their y value is is one
18:49
not zero as it was before and here's the sigma star particles PSI star particles and our
18:55
Omega minus down here so it's quite an attractive view we get a nice triangle and notice that it's
19:01
sort of a uniform triangle in the sense that the multiplicity of the inside here is the same
19:06
so it's sort of like flat multiplicities okay so once the physicists had put these
19:12
families of particles together on nice patterns two-dimensional patterns corresponding to these
19:18
quantum numbers they realize that this suggested to them that there were actually underlying more
19:25
Elementary particles that could be used as building blocks for this the zoo of baryons
19:31
and mesons that they had found and these new particles became known as quarks and they are
19:37
now sort of the central objects in some sense of the standard model and the way this comes about is
19:44
through this diagram so here again we have charge and hypercharge with the same taxes before and
19:50
I've labeled the points where we've already had some particles so we can superimpose all three
19:57
diagrams the baryon octet the Meson Nonet and the baryon the couplet all on the same diagram but it
20:04
gets a little bit confusing but I just put some of them here so there's the neutron there and
20:10
the proton there's k0 K plus those are mesons and here are a few of the Delta particles Delta minus
20:19
Delta plus plus and a few others there's Omega minus down there there's Pi minus okay what are
20:25
the new particles that are suggested so they realize that there were three other kinds of
20:33
particles that you couldn't observe directly but that had sort of an indirect existence because we
20:40
could build up these other particles in terms of these three actually and their anti-particles so
20:45
there's actually three new particles three three new quarks and they're anti-particles so they're
20:51
called up down and strange and here are where they are on this diagram okay so up is at the
21:00
point with coordinates two-thirds and one-third when we're measuring with respect to charge and
21:07
hypercharge okay crucially not strange hypercharge the dial cork is minus one-third charge and plus
21:15
one-third hypercharge and the strange is minus one-third charge and minus two-thirds
21:23
um hypercharge so now they all have anti-particles and the antiparticles
21:27
are sort of opposite in pretty well every sense so the anti-particle of up is is down
21:33
here so this is a two-thirds a one-third so we have to go to minus two-thirds minus one-third
21:39
um I guess that would be about right there okay so that's a kind of an anti-particle
21:45
two to up so this is called up bar it's the anti particle the anti-cork to up similarly there's an
21:54
anti-particle to down there it is there so that's D Bar and there's an anti-particle to S that's
22:03
um we have to go uh one-third two-thirds up here
22:12
so that's S Bar and those three anti-particles
22:16
also form a a kind of a triangle so let's make dotted lines between them
22:25
okay so there's our two triangles of quarks here there's up down strange and the anti
22:30
particles of those up bar down bar strange bar and now a miraculous thing happens that just
22:39
by looking at the geometry of this diagram we can sort of decompose these various particles
22:48
that we previously had thought were sort of Elementary we can now decompose these into these
22:55
perhaps somewhat ephemeral quarks and antiquarks so let me show you how that's
23:00
done so the neutron for example how do you get the neutron so the neutron is a baryon
23:07
which has half integral spin and these quarks all have half integral spin also as it turns out so
23:14
um we need three of these quarks or antiquarks to make these baryons so this baryon here is
23:23
um let's see one up and two downs so you see that sort of vector think of this Vector it's like the
23:29
vector d That's the vector U if you like okay so to get to Neutron we go up and then two downs
23:40
that's telling us that the neutron is composed of one up and two downs how about the proton well
23:49
of course there's three particles here so the the decomposition here is not actually unique but
23:55
um okay uh it's sort of probably unique if you restrict yourself in some reasonably natural
24:03
way so the proton turns out to be two ups so go up up here and then a down so this is up up down
24:14
so let's stop here and pause so this is a remarkable thing that we're postulating the
24:18
existence of these new particles which actually are not visible right no one's actually observed
24:24
a cork or an antiquark directly they seem to always be bound together with other quarks they
24:31
they're not appearing in nature by themselves okay for various reasons so they're almost
24:37
impossible to verify their existence actually but so we have a lot of indirect evidence
24:43
maybe not entirely conclusive in my mind I mean I think one should maintain a bit of skepticism
24:48
here but at least mathematically it's interesting to think of these as sort of constituents of these
24:54
more elaborate particles so for example this Delta plus plus that's a baryon uh
25:03
so it's also made out of three three of these quarks and not too hard to
25:10
see that it's got to be three of these ups okay so there's up up up
25:16
and uh simulator this Omega minus down here for example uh that's going to be uh one two three
25:23
three S's strange strange strange. The the pi minus how are we going to get that one so we want
25:31
to cook this this is a Meson okay mesons have integral spin so we should take two quarks or
25:40
two quark slash anti-quarks okay so what we can do here if we want to
25:46
um to get to this one what we could do is we could go have a down and then an up bar
25:54
okay that's like the up bar vector there okay the negative of the up Vector so there it is there if
26:01
we combine that with this one we get the pi minus so that's the combination of a down and anti up
26:09
so quarks and anti quarks can combine to also make particles so then it's an interesting exercise to
26:18
say okay well what about all of them can we do this sort of consistently in a nice fashion and
26:22
I think that's a really important exercise to to illustrate what's going on and that's
26:26
what I'm going to do in my next video so I want to carry on with this and show you uh exactly
26:30
how we can get the presumed decomposition of all these baryons and mesons that I've
26:38
shown you today in terms of the three fundamental quarks up down strange and their antiparticles.