Dynamics on Graphs 21
We see how the simple two dimensional geometry obtained by imbedding the Baryon and Meson Multiplets into the Charge / Hypercharge space gives us simple rules for decomposing these particles into constituent quarks (up down and strange). However there are some wrinkles here.
The mysteries of Quantum Mechanics end up figuring in the explanations for some of the Mesons, where superposition of particle states or wave functions is reflected in the decomposition formulas.
A more refined picture and explanation will arise in our next video where we show how the Mutation and Numbers Games on the very simple graph A2 leads to very similar diagrams and decompositions.
0:00
hello everyone I'm Norman walburger
0:02
thanks for joining me on this adventure
0:04
we're looking at the mutation game and
0:06
the numbers games and seeing how they
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can
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perhaps give us some new insight into
0:12
the numerology of the standard model so
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I'm just having a quick overview of some
0:19
of the early features
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developed around the 1960s of the
0:23
standard model and the uh the sort of
0:26
framework of quarks to describe baryons
0:30
and mesons so I want to give you some
0:32
Quark decompositions now of some of the
0:34
multiplets that I discussed in my last
0:37
video starting with this baryon octet so
0:40
I remind you that there were these eight
0:41
particles that formed this
0:44
pattern that Murray Gelman termed the
0:47
Eightfold way
0:49
connection with Buddhism okay and the
0:51
particles were the neutron the proton
0:54
three sigma particles a Lambda particle
0:57
also and two PSI particles
1:01
and we saw that we could put them on a
1:03
two-dimensional diagram if we use the
1:06
axis of charge say we call it q1 and in
1:10
this direction
1:11
hypercharge
1:13
usually denoted by Y which is close to
1:15
connected to strangeness but it's a
1:17
little bit different so all these
1:19
particles turn out to have spin one half
1:21
and what we want to do is explain these
1:24
particles in terms of these sort of
1:26
three more primary quarks
1:29
which were postulated to exist and which
1:32
we only really have indirect evidence of
1:34
so the three quarks up down and strange
1:37
and they're anti-particles up bar here
1:40
down bar here and strange bar here
1:43
okay so interestingly the the various
1:47
decompositions which are described here
1:49
can be read off immediately from the
1:52
geometry of this diagram once we
1:54
position things as I've done here so
1:57
what we're really doing is we're using
1:58
the vectors associated with these three
2:00
points so we think of U as a vector okay
2:02
and D is a vector and S is Vector then
2:05
these Expressions really just correspond
2:08
to writing these various positions in
2:11
terms of these vectors so for example
2:13
the neutron is up down down so there's
2:17
an up there's a down Direction so we go
2:19
up down down then we get to the
2:21
Neutron's position the proton is up up
2:25
down the sigma minus particle over here
2:29
is down down and then
2:33
strange the sigma zero particle is well
2:38
some combination which makes zero and
2:41
that combination is unique it's up down
2:44
strange that corresponds to the fact
2:46
that in this diagram here this point
2:48
here is actually the centroid of this
2:49
triangle it's the center of mass of this
2:52
little triangle formed by these three
2:54
quarks okay that's a manifestation of
2:58
the vector relation that this Vector
3:00
plus this Vector plus this Vector is
3:02
zero and the other particle that's down
3:05
here in this Central position the Lambda
3:07
particle is also described in this
3:10
fashion as a combination of up down and
3:12
strange
3:13
and then over here we have the sigma
3:15
plus which is up up and then strange
3:19
and then we have the two side particles
3:23
PSI minus which is down strange strange
3:28
and PSI zero which is up
3:32
strange strange
3:34
okay so uh very pleasant and so we see
3:38
that all of these baryons are made out
3:41
of three quarks
3:43
so then the fact that they're all spin
3:46
half particles is reflecting the fact
3:47
that they're fermions okay but there's a
3:50
little bit something strange here
3:52
certainly something interesting is
3:53
happening right at zero because we have
3:55
these two particles which on the face of
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it things uh look like they're sort of
3:59
the same the sigma zero and the Lambda
4:01
particle how do we explain that so what
4:04
distinguishes the sigma Zero from the
4:07
Lambda particle since we can see from
4:09
here that they're both made up of one up
4:12
Quark one down quark and one strange
4:14
quark well if we look on the list of
4:17
baryons which you can find in Wikipedia
4:19
it's quite interesting table lots of
4:21
entries these are just some of them
4:24
it's around 1960s right in the 1970s the
4:28
further quarks like charm
4:31
bottom and ultimately uh top
4:34
were discovered and that makes the
4:38
picture much more complicated but around
4:40
this time
4:41
there's a list of baryons which includes
4:44
these ones here and we see that the iso
4:46
spin which is another quantum number I
4:48
discussed very briefly last time the iso
4:50
spin of the sigma zero particle is one
4:52
but the iso spin of the Lambda particle
4:54
is zero so that's one way of saying how
4:56
these things are distinguished
4:59
a slightly more sophisticated perhaps
5:02
however waffling answer is that all of
5:04
this has to be taken into
5:07
the in the context of quantum mechanics
5:10
okay so these are not classical
5:12
particles they should all be viewed
5:13
Quantum mechanically and that's uh
5:16
that's a hard thing to get one's head
5:18
around unless one has a degree in
5:20
physics but so roughly speaking when we
5:22
go to Quantum Mechanics we want to shift
5:25
from a particle view to more a wave
5:28
function point of view
5:30
and another way of maybe expressing that
5:32
is that instead of thinking of sets of
5:36
particles like sets of objects where
5:39
rather working in linear spaces
5:43
so instead of having three particles and
5:45
saying oh this is a set with three
5:46
elements quantum mechanics it seems that
5:49
nature is more interested in a vector
5:52
space formed with three these three
5:55
particles as a basis it's a linear space
5:58
okay that allows more possibilities
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because instead of just having this
6:02
particle or this particle or this
6:03
particle we can have combinations of
6:05
them okay that's really a crucial aspect
6:08
of of quantum mechanics that does
6:10
involve getting your ones head around so
6:13
there's a little bit of a parallel here
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with our mutation game story so we've
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been talking about graphs here's a graph
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so it has four vertices which are very
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explicit things but we've seen that
6:25
actually from the point of view of the
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mutation in numbers games the
6:28
populations on the graphs are more
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capable of interesting sort of structure
6:34
right the populations are functions on
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the graphs vertices that's in this case
6:38
a four-dimensional space so there's more
6:40
possibilities because you can take
6:42
weighted sums of these various things so
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in that direction when we write down up
6:48
down strange we're positing some
6:50
combination of these three things but
6:52
you know maybe there's multiple possible
6:54
meanings to exactly how we could
6:56
interpret such a combination from this
6:59
quantum mechanical point of view okay
7:01
that's a bit waffley but we will see a
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little bit more of this um just a little
7:05
bit later in the lecture right so now
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let's have a look at the Meson Nonet
7:09
pattern of nine mesons so mesons are
7:13
boson so that means they have integer
7:15
spin in this case these ones all have
7:17
spin zero and we're plotting them in the
7:20
same framework as previously with charge
7:23
in this direction and hypercharge in
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this direction
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so our up down and strange quarks are
7:29
exactly in the same position and they're
7:30
anti particles also and now we still
7:34
have the same kind of eight-fold pattern
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with the chaos k0 K plus a k minus which
7:40
happens to be the anti-particle of this
7:42
one k zero barge is the anti-particle
7:44
this one
7:45
and then Pi minus pi plus these are
7:48
anti-particles and then inside here at
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the zero position there's actually three
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particles this time okay not two three
7:55
and they are the pi zero and then
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ETA and an ETA Prime particle
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okay so again what happens is that the
8:04
geometry of this diagram
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uh pretty well
8:08
pretty well determines the decomposition
8:11
of these various particles into the
8:13
constituent quarks and their
8:15
anti-particles and
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it's a little bit different from the
8:20
baryons because now what's involved are
8:23
pairs of things because by the way the
8:26
the spins work we went spin zero so it
8:30
has to be sort of some cancellation here
8:32
and also these are these are pairs so
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they're
8:36
um yeah they don't obey the poly
8:38
Exclusion Principle okay so let's start
8:40
up here the k0 this K on okay
8:44
um how do we construct it from the
8:46
various vectors of these three quarks
8:49
and their their anti-particles their
8:50
anti-particles of course have vectors
8:52
which are just the negatives of the ones
8:53
corresponding the quarks so the the
8:55
vector for uh S Bar is just the negative
8:58
of the vector for S Etc okay how do we
9:00
get to k0
9:02
well we can do a down that Vector there
9:05
and uh an S Bar okay so if we were just
9:09
using two of these things that's the way
9:12
we would get to this point we'd take
9:13
this vector and this Vector telling us
9:15
that the k0 is a combination of the down
9:19
quark and the S Bar Quark the anti-quark
9:23
the anti-particle to the strange
9:26
K plus over here is up
9:30
together with S Bar okay so sum of these
9:33
two vectors is that point there so
9:35
that's K plus it's a combination of up
9:37
and S Bar
9:40
the uh
9:42
the Pion down here Pi minus okay that's
9:46
going to be a combination of down and up
9:49
bar
9:50
okay those two things it's a little
9:52
parallelogram and similarly pi plus
9:56
which is the anti-particle of Pi minus
9:58
now you get that just by taking the
10:00
anti-particles of the constituent quarks
10:03
so the antiparticle of D is D Bar the
10:05
anti-particle of up bar is is up so that
10:08
sort of follows from that and similarly
10:09
the K minus here which is the
10:11
anti-particle of K plus is just obtained
10:13
by taking the the essentially the
10:15
negation of the corresponding vectors so
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we get S U Bar and similarly the k0 bar
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which is the anti-particle of this thing
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uh there it is there so at the end of
10:25
particle of this one it becomes S D Bar
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now what about these three inside here
10:34
that's an interesting question so I
10:36
might like to think about what you might
10:38
think those things are
10:40
so if we're following the pattern then
10:41
we should guess that they're composed of
10:44
a quark and an anti-cork and the sum of
10:47
these vectors should correspond to the
10:49
zero Vector in this case
10:51
well hey there's sort of three obvious
10:53
combinations that do that there's the
10:55
combination of up with up bar
10:58
combination of down with down bar and
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the combination of s with S Bar so a
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natural guess would be that that's what
11:06
these three things are one of them is up
11:08
up bar the other one's down down bar and
11:11
the other one is s s bar
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but which ones are they and is that
11:15
actually correct
11:17
well it turns out that nature is more
11:20
complicated than we might at first guess
11:22
and in fact that's a sort of a general
11:24
principle that has unfortunately Arisen
11:27
over and over again in this story of
11:29
particle physics in the last uh in the
11:32
last 50 60 years in fact one has to feel
11:35
a little bit sorry or at least have some
11:37
sympathy for particle physicists because
11:39
the story is really complicated and a
11:42
lot of these particles after all they
11:44
only last for a fraction of a second
11:45
like 10 to the minus 10 seconds some of
11:48
them much less than that like 10 to the
11:50
minus 15 or maybe even minus 20 seconds
11:53
so they're almost you know not there
11:56
but somehow we still have to explain
11:59
them and
12:01
they're not really contributing to our
12:03
world in any significant way that we
12:04
know but they are theoretically a huge
12:08
problem because they're so complicated
12:09
and there's lots of them okay
12:12
um so there's there's more than but just
12:13
what we're showing you here there's also
12:15
various resonances of all these
12:16
particles so it gets very very
12:18
complicated this is a big story and so
12:21
the physicists you know have to struggle
12:23
with with this I think Enrico Fermi the
12:26
famous Italian physicist um perhaps
12:28
somewhat jokingly said that if he had
12:30
known what the story was going to be he
12:32
would have become a botanist instead of
12:34
a physicist but anyway so here's here's
12:37
the answer that we find in the list of
12:40
misons on Wikipedia for how to express
12:43
these three mesons which are at the
12:47
center of our diagram with charge and
12:49
hypercharge so the pi zero turns out to
12:52
be up up bar minus down down bar over
12:55
root two
12:59
what does that mean well that's one of
13:01
these quantum mechanical combinations
13:03
that I was telling you about you know
13:04
the fact that
13:06
in quantum mechanics since things tend
13:09
to be in terms of linear spaces rather
13:11
than just sort of solid objects things
13:13
can come in linear combinations and
13:16
here's a prime example of that
13:18
and what's the root 2 doing down here
13:20
well of course for me the root 2 is
13:22
immediately suspect because I'm thinking
13:25
wait a minute is is nature actually
13:27
making a computation of root 2 here that
13:30
sounds very suspicious to me but anyway
13:33
the the point here is to make this a
13:35
sort of a unit Vector in sort of natural
13:37
coordinates
13:39
the Ada particle is even a little bit
13:41
weirder it's up up bar plus down down
13:44
bar minus 2 times s s bar all over root
13:48
- again you think of this as a
13:50
normalization to sort of make it a unit
13:52
length and the ETA Prime is up up bar
13:56
plus down down bar plus s s bar over
13:59
root three again a renormalization
14:01
and it's not too hard to stare at this
14:03
and to see that these are sort of
14:04
linearly independent in the space
14:07
three-dimensional space span by U U Bar
14:10
D D Bar s s bar
14:13
but you know what kind of theory would
14:16
let you predict a particular
14:19
combinations like this it's it's
14:21
disconcerting and it also strengthens
14:23
just a little bit one suspicions that
14:26
you know I mean are these quarks really
14:28
real or are they mathematical constructs
14:31
that are are very convenient for us
14:35
you know I mean I think the majority of
14:37
physicists think that they're real but I
14:39
think we're allowed to have a little bit
14:41
of skepticism about this issue and and
14:44
formulas like this certainly
14:47
um maybe enhance that skepticism a
14:49
little bit okay so our final thing that
14:52
we want to explain is the baryon
14:54
decouplet where we have this pattern of
14:56
10
14:57
particles row four row three row of two
15:01
row of one and again we're looking at
15:03
the same framework this two-dimensional
15:04
framework and again here's up down and
15:07
strange and there are anti-particles U
15:10
Bar D Bar at S Bar
15:12
okay and again it's the same story very
15:15
pleasantly it's just a question of
15:16
taking these things here which are
15:19
baryons okay so we're expecting them to
15:21
be composed of three quarks not two okay
15:24
three quarks and basically you know
15:27
figuring out which combinations of these
15:29
basic simple vectors will get us to
15:32
these various points so for example if
15:34
we want to get to this Delta plus plus
15:37
particle and we want to use these
15:39
vectors and we want to use three of them
15:40
well you're going to have to use up up
15:44
so where is Delta plus Delta plus plus
15:46
is up up Delta minus is down down down
15:51
and Delta 0 is up down down Delta plus
15:56
is up up down and then the sigma star
16:01
minus okay is down down and then strange
16:06
this one here which is the only one at
16:09
zero this time is again a combination up
16:13
down strange
16:14
if we want to get zero that's the way
16:16
you get the zero vector and then Sigma
16:18
Star Plus is up up then strange
16:23
this PSI star minus is down strange
16:29
strange and this PSI star 0 is up
16:33
strange strange and finally the
16:37
Omega minus particle is strange strange
16:41
strange
16:42
that's sort of special because Marie
16:45
Gilman famously predicted the existence
16:48
of this thing here by analyzing the
16:50
other nine elements in this decoupling
16:52
and realizing that there should be a
16:54
tenth particle to complete the picture
16:57
now there's another special reason why
17:00
this particular one is interesting along
17:02
with this one here
17:04
and uh and this one here so these ones
17:07
here are all made out of
17:09
three quarks which are the same
17:12
so it's actually the these particular
17:15
combinations that gave rise to this
17:18
additional
17:19
kind of uh quantum number is called
17:22
color
17:24
um and that's arising because the quarks
17:27
are fermions that means they obey the
17:29
poly exclusion principle that means that
17:31
you you're not allowed to have two
17:33
particles which are exactly the same
17:35
occupying the same spot now
17:38
you know in the baryon uh octet that we
17:40
started off with there were some
17:42
duplications like an up up down or
17:45
whatever
17:46
um but that can be explained because
17:49
these um
17:51
these uh particles could have they have
17:55
a sort of iso a spin Direction so the
17:57
spin could be plus and minus so it's
17:59
possible to have one up uh spin up Cork
18:03
and another downspin up Quark Etc so you
18:07
know the the existence of pairs of of
18:10
similar labeled quarks can be sort of
18:13
explained through through some spin
18:15
argument so that the power Exclusion
18:17
Principle is not being directly
18:19
contravened but with these three there's
18:23
no such easy out okay here we have three
18:26
s's which are all together and
18:30
the spin will only allow us sort of two
18:33
degrees of freedom so we have to explain
18:35
how are these three fermions occupying
18:38
the same same position essentially and
18:41
it's because of these three things that
18:44
they had to introduce this new concept
18:47
of color so basically introduce the idea
18:49
well there's something else in addition
18:51
to all of this stuff called color the
18:54
color being blue
18:56
red and green and you know these three
19:00
strange quarks end up having different
19:02
colors this one's the blue one this
19:04
one's the green one or maybe the red one
19:06
and this is the green one
19:07
that way they're not contradicting the
19:11
power Exclusion Principle okay but that
19:13
does involve yet another three-fold
19:16
symmetry that enters the story
19:18
and this is the the origin of uh what's
19:21
called uh
19:23
chromodynamics you know um
19:26
like Quantum chromodynamics or like
19:29
Quantum electrodynamics but this
19:31
three-fold symmetry in in the color
19:33
space that has to be introduced to sort
19:36
of explain these three things these
19:37
vertices of this sort of triangle that
19:40
we get
19:42
so these are very important examples and
19:45
um
19:45
they they're remarkable on on in many
19:49
ways the fact that the geometry of these
19:52
two-dimensional diagrams really is
19:54
providing a lot of powerful information
19:57
but if we want to go further than this
19:59
the usual story is in terms of as I
20:02
mentioned several times in terms of Lee
20:04
theory in terms of the representation
20:06
theory of su3 okay and the physicists
20:09
are are very well versed in this subject
20:13
nowadays they've had to learn a lot of
20:15
representation Theory to try to make
20:16
sense of uh sort of finer details about
20:19
this and it's that aspect of things and
20:22
this basic numerology that I want to
20:24
give you an alternate view on okay so to
20:27
try to avoid all that complexity by just
20:30
going right back to a simple graph A2
20:33
with just two vertices and trying to
20:35
explain how a lot of this numerology
20:37
really is already residing there in this
20:40
graph with two elements once we know how
20:43
to play the mutation in numbers games