Dynamics on Graphs 22
77 views Apr 23, 2023
We show how the special numerology that in the early 1960's connected families of elementary particles (Baryons and Mesons) with representations of the Lie group SU(3) can be alternatively presented in terms of the much more elementary geometry of mutations and dual mutations for the graph A_2, consisting of just two vertices and a single edge. The Cartan matrix plays an important role, as well as the mutation matrices s_1 and s_2 defined earlier in this series, as does the Tits quadratic form on the space of populations on the graph, as well as a new variant of that we introduce called the dual Tits quadratic form on the space of weights. We introduce the canonical correspondence Psi from populations to weights, along with its inverse from weights to populations, and then use that to compose both mutation and dual mutation pictures on the same graph, resulting in a geometry that reveals the Baryon octet and decuplet, as well as the two families of quarks up down and strange, along with their antiparticles. There is a lot of content in this lecture which ought to be of considerable interest to students of Lie theory, representation theory and particle physics. The material is presented using notation and terminology that naturally and pleasantly extends to the much more general situation of ADE graphs, and in fact some of it also extends to general graphs (with the ADE tilde graphs something of an exception).
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hello everyone I'm Nolan welberger thanks for joining me on this exploration of exceptional structures in
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mathematics and physics Dynamics on graphs the last few videos we've been talking about the standard model and what I
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lightly call the numerology of the standard model is represented by the Eightfold way introduced in the early
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1960s by Marie Gelman and Yuval neyman
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and this is a way of connecting the classification of elementric
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particles which had been observed in the accelerators and Cloud Chambers up to that time with certain patterns that
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occurred in an area of mathematics called the representation theory of Lee groups in particular there are
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representations of a particular group called su3 and this is in some sense the
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birth of the standard model of particle physics what I want to do now is to
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show you an alternate way of structuring that mathematical environment in which
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this sort of numerology of the physics can be encoded
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and that's by avoiding the rather sophisticated theory of the representation theory of su3 which by
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the way I think is very interesting and I want to explain that much further at some other stage but it really is a very
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involved topic but replacing that with something that's much more Elementary namely the story of the combinatorics
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and the geometry associated with the mutations and numbers games on the graph
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A2 so of our Ade graphs this A2 graph is
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the second simplest one after the A1 graph the A1 graph sort of encodes the
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the story of su2 which is also crucial in the standard model that's the sort of
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symmetry group Associated to the weak Force while A2 Associated to su3
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Associated to the strong nuclear force but what we're going to do here is just
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step back and consider the graphs as fundamental that's the point of view of this entire series and we want to
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understand how the mutation story creates a geometrical framework in which
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we can find these baryon multiplets decouplets Architects
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Etc also the mizon story uh in in a way that avoids the complexity of representation
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Theory so we're going to get on to that at some later point But ultimately it's a really simple kind of uh geometry but
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it's interestingly an unfamiliar geometry because it's not really euclidean geometry okay so I'm also
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going to ask you to rethink the story of the su3 root system weight system story
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in a different way which I think is is more true to the intrinsic geometry
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that's going on here so there's going to be a lot in this lecture if you're an advanced undergraduate or a beginning
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graduate student interested in representation Theory and Lead groups this will be really important and really
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valuable and if you're a physics student especially interested in particle physics or theoretical physics it will
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also be really important but you know I am presenting things in in a somewhat novel way so it's important to go slowly
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lots of content in this video okay so let's start with A2 okay it's a graph
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and we're going to be thinking about two different games the mutation game and
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the numbers game on these graphs and the the games themselves are actually played on the functions on the graphs and it
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turns out that there are sort of two different versions The mutation version and the numbers version and so we find
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it convenient to separate the functions into two different kinds populations which are given by row
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vectors thinking of that as a function on the vertices and weights which are
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represented by column vectors also represent as a function on the vertices but now with uh the graph written this
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way with open nodes rather than with closed nodes and here is the adjacency Matrix for the
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the graph 0 1 1 0 and here is the closely associated carton Matrix which
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is two times the identity minus the adjacency so it's this two by two Matrix here that plays a crucial role and as
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we'll see also the inverse of this Matrix plays a crucial role and this thing is invertible because its
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determinant is a 4 minus one which is three the number three plays an
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important role okay and a lot of what I'm saying I'm using a language which
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actually generalizes very naturally to the general Ade situation so I'm
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choosing my language very carefully here and it's the invertability of this
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Matrix C which is a standard assumption so once we assume that then actually
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what I'm saying here Works in in quite a very broad generality even beyond the Ade story
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okay so the starting point then is the mutation matrices so these are two by two matrices who
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square is the identity and here's S1 minus one zero one one and S2 1 1 0
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minus one and these matrices act both on population vectors which are row vectors
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and on weight vectors which are column vectors on these ones they act on the right on these ones they act on the left
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so you just do the multiplication you get this thing here P1 P2 is set to minus P1 plus P2 comma P2 and the effect
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of that is to so this is the mutation at one so the effect is to leave the population at two unchanged and to
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replace the population here with the sum of its negative plus the population of
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the adjacent neighbor now on the other side it's a different action because now
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we're multiplying by the same matrices but now on the left so the effect is to
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take W1 W2 to minus W1 W1 plus W2 so
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it's like the firing at vertex one as opposed to the mutation at vertex one and what the firing at vertex one does
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is two things it changes the value at the vertex one two it's negative and
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also it adds the former value at one to the to the neighbors in this case just
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to W2 okay so these are the two different games or interpretations of
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these mutation matrices and uh it's really the interaction of
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these two stories which we'll see reveals the interesting patterns that uh
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the the physicists find so interesting in that are at the heart of the the standard model and the Eightfold way we
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want to bring these two stories together but before we do that we have to understand the connection between them
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and the associated geometry that's implicit in both of them so it's a very
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rich story okay so to review some of what we just said so here's how we represent the population P values P1 and
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P2 at these two vertices as a row Vector is how we represent the weight w
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w1w2 in terms of this different kind of graph these are the weights so we have a
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two-dimensional space of populations we have a two-dimensional space of weights and you should at this stage think of
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them as different Now The crucial sort of theoretical result that we're going to be using over
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and over again is what I call the carton Matrix invariance theorem which tells us that this carton Matrix
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that we introduced is unchanged if we multiplied by S I on the left and by S I
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transpose on the right for I equals one and two and this is a general result
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that holds for General graphs okay and in this case here this has also
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an equivalent formulation if we take the inverse of both sides which we can do because we know that c is invertible
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so C is not always invertible there's some really important graphs which we've already talked about the ad tilde graphs
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for which C is not invertible so then this story has to be sort of modified or changed but if C is invertible then we
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can just sort of take this equation and invert it to get S I transpose C inverse s i equals c
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inverse because when you take the inverse of something like this you get the inverse of the individual things in the opposite order and inverse commutes
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with transpose okay and the inverse of s i is s i itself because s i squared
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equals the identity okay so that's how you get from here to here okay so then we introduced the tits
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quadratic form which is providing geometrical structure to our
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two-dimensional space of populations okay and it's this geometry that really
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plays a crucial role as we'll see so what is it so give it a population P
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like this one here the quadrants of P is p times c times P transpose row
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Vector 2 by 2 Matrix column Vector that is a one by one Matrix or a number
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so that's the quadrants so we're really working in the world of rational trigonometry here we don't have
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to know a lot of rational trigonometry but enough to understand hopefully that you know it's not distance or length
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which is fundamental in euclidean Geometry it's the quadrants which is uh usually x squared plus y squared but in
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this case it's non-euclidean it's a variant so this is not just x squared plus y squared it's something a little
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bit more General but in any case this is providing the geometrical structure
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and there is a dual version of that which I haven't talked about yet which
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will now introduce which is a quadrants on the weight side so on this side here
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in the space of weight vectors also two-dimensional there is also a geometrical structure
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given by a similar kind of equation but crucially not involving C but rather C
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inverse so if we take a column Vector W like this one here
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then we can form W transpose times C inverse times w this becomes a row Vector that's a two
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by two Matrix that's a column Vector so that's again a one by one Matrix or a
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number so we'll call this the Dual tits quadratic form
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and then an important property of these things is that
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they are invariant under multiplication by these
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sis so if we act on a population P by s i and here I is either one or two so
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we're talking about these two mutation matrices that I just defined if we multiply by a mutation Matrix in other
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words we perform a mutation then the quadrants doesn't change the vector that
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we get changes but its quadrants doesn't change and that's ultimately going to tell us
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that this is an isometry and ultimately we'll see that it's actually a reflection in a suitable sense okay so
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this ends up having a geometrical significance and over here there's a corresponding
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result but over here since we're having column vectors the two by two matrices S I have
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to act on the left okay you act on the left so quadrants of s i w is quadrants of w so again this
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dual tits quadratic form also is being maintained by the action of the mutation
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matrices firings as we would say in terms of the numbers game terminology so this here is
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a nice exercise for you to do and you can you know use this thing here perhaps
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to help you so we have some geometry here so let's now see how the vial Group which is the
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group generated by these two mutations S1 S2 acts on this two-dimensional space
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in this case first of populations okay so this is going to give us vile
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group orbits on this two-dimensional space so the vial group abstractly it's just
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the group generated by these two matrices that means you you take all possible products you know that you like
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of these matrices but you find that there's not a an unbounded number there's only six because the square of
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any one of these two is the identity so you get the identity and you get these two themselves and then you get the
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product S1 S2 also the product S2 S1 those are different because this is a non-commuted situation and then there's
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only one additional one which is S1 times S2 times S1 and importantly that's
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the same Matrix that you get by taking S2 times S1 times S2 so this is a
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general relation that's true in this mutation game world it's called the brain
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relation we've talked about that so there's only six elements in this group so abstractly as a group it's like the
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symmetric group S3 but it acts on this two-dimensional space in a in a geometrical way so I
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want to illustrate that with an example so let's start with a population P which is three two so it's the function on the
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vertices three at the first vertex two at the second vertex and here's our axes X and Y
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so over three and up two there's the 0.32 that we're starting with
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so let's think about you know what happens when we apply these various
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elements to it and it's nice to sort of organize things in this way that we sort
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of start from here and we perform an S1 in this direction or at S2 in this direction so what that means is that we
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take this thing here and we multiply it on the right by S1
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okay so if you just do that multiplication you'll see you get minus 1 2 which corresponds to that point
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there minus one two so the S1 operation takes this thing to this thing
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on the other hand if we apply S2 instead that's multiplying on the right by S2
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Matrix then you go from three two to three one so it's going from here to here
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now this may not look like a reflection at this point it is a reflection but it's of somewhat unusual sort of
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non-euclidean kind of reflection okay then we can keep on going so from this point here we've just applied S1 we
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can apply S2 and then you can check that you get minus 1 minus three you get this one
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here okay and if we start here and we do S1
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we've just done S2 but if we do S1 now we get minus 2 1 over in this direction
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and now from here if we do an S1 we get to minus 2 minus 3. and similarly from
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over here from minus two comma 1 if we do an S2 then we get also minus two
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minus 3. so note that the S1 actions sent this
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one to this one that was one S1 also sent three one to minus two one over
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here and also sent uh minus one minus 3 2 minus two minus three so in each case
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this S1 mutation uh geometrically maps from a
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point on on this horizontal line to another point on this horizontal line now same thing here it
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goes here and here and here to here and if you had to describe this as a
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reflection you might come up with the idea that it's actually a reflection in that line there which is obtained by
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taking the midpoint of this segment the midpoint of this segment and also the midpoint of this segment you can compute
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that they all actually lie online so there's actually some sort of implicit line there which this reflection S1 is
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sort of reflecting through in some sort of strange way
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and similarly if we look at the S two actions we went from here to here there
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we went from -1 to to minus one my three so we went from here to here and we also
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went from minus two one to two minus two minus three we went from here to here so in each of those cases the S2 is
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keeping us on the same vertical line and the same kind of argument shows that there's actually uh uh sort of a line of
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symmetry through the midpoints of those segments and in some sense S2 is a reflection in
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this line here so we get this interesting hexagon and hexagons play an integral important role
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in this su3 story or this A2 story and interestingly these six points
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um lie on iconic okay and in fact geometrically we know that because the tits quadratic form
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is maintained by the action of these sis so if we compute the tits quadratic form
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on this initial Vector 3 2 it's has the value 14. please check that and that
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means that correspondingly the tits quadratic form on all of these other ones also has to have the value of 14.
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so we're staying on the conic quadrants with P equals 14. and that's iconic it's
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given by a quadratic equation and actually you can check it that what you actually get is the explicit conic x
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squared minus X Y plus y squared equals 7. and I've roughly given you an idea of
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what that conic is it's an ellipse that goes through those six points let me remind you that in general Five
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Points determine iconic section so if you have six points which lie on iconic section it's something a little bit
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exceptional random six points do not lie on conics okay but these six points do they all
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lie on this Central conic whose Center is at the origin so this is the analog of the uh the
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circle it's like a circle in this geometry now I have to say that you're not going
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to find this picture in any texts or or papers on the subject almost invariably
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the story associated with su3 is in terms of equally spaced or equilateral
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so there's a line like this and then a 60 degrees and 120 degrees so it's that that picture that you're usually going
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to to see and unfortunately that's not the best picture this one's actually algebraically and geometrically more
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natural even though at first it might appear a little bit more complicated but as we'll see it turns out to be hugely
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advantageous so if you're a physicist great to start thinking about shifting your point of view okay from this
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equilateral triangle kind of orientation of su3 and then the roots and weight story to this more rectangular story
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where it's the vertical and horizontal lines which are providing the structure
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for the reflections now I'm going to look at an example where the weight that we're considering
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is minus 4 minus 1. I haven't chosen that at random in fact it turns out that
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exactly that weight is the the weight that corresponds to the population three two so it turns out that there's a
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correspondence between populations and weights and we've talked about this before but I will review it but um so I
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want to show you this is an analogous picture to the example that I've just shown you in some sense it's a kind of
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the the Dual version of that story that I've just told you so we're going to start with minus four minus 1 which we
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can think of as a function on the although the weight A2 like this okay
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and um so here it is there and now we are acting by S1 and S2 in the
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usual way but now on the the left okay and here's what ends up happening so
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we're starting with minus four minus one so that's over here so it's a plane with
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axis say u and v this direction in this direction representing uh you know the unit basis
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vectors in the column sense so this is the vector one zero this is the vector zero one and I might also mention that
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we should perhaps give these names so one standard thing we could do is we
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could call that Lambda 1 and that Lambda 2. all right so like these are you think of these as basis vectors of this space
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and so um any the point that we're going to get if we
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start with with this one here with integral coordinates is also going to be on the
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integral lattice generated by by these ones here so the integral lattice you
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know would be some lattice that contains all these integer multiple points okay
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in all directions okay so what do we actually get well if we apply S1
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okay from here we apply S1 we get four minus five that's down here
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if we apply S2 on the other hand we get minus 5 1 which is over here so S1 takes
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us from here to here while S2 takes us from here to here from here if we apply S2 now okay we get
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to -1 5 minus one for up here going up there notice that this is
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parallel to to here okay that's S2 on the other hand if we have minus 5 1 over here and we apply S1
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we get 5 minus 4 which is down here and notice that that is also parallel to
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this and then from -1 5 if we do an S1 we get 1 4 which is over here again that's also
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parallel to to these and similarly from five minus 4 to to 1 4 we get that also
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by applying S2 okay so we get a different kind of
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a different kind of hexagon sides which are sort of in
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different directions but still having the property that these three lines here are parallel and these three lines here
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are also parallel so the quadratic form over here which is
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maintained and there is a quadratic form the Dual tits quadratic form okay also
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has an ellipse in this case given by the equation quadrants of w equals 14. and
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let me remind you that the the Dual tits quadratic form dependent on The Matrix C inverse okay
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so if C is the Matrix two minus one minus one two the carton Matrix that we started with then the inverse of this is
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two-thirds one-third one-third two-thirds so this is a rather important
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a matrix in this situation okay and the quadratic form is in terms
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of that and I hope you'll check that it also has this alternate a form in terms
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of the u and v coordinates U squared plus UV plus v squared equals 21. and here is my attempt at drawing it it's
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also an ellipse going through these six points so we have these two pictures
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one is the while group orbit where W is acting on the right on row vectors and
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this other picture where W is acting on the left on column vectors so let's
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describe how this correspondence between the population story and the weight
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story with the action of the vial group on both of them in different ways how we
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can connect these with what I call the canonical correspondence and we'll use
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the Greek letter PSI to denote that so we're going to think about PSI as being a mapping
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from populations on A2 which are like row vectors to weights on A2 which are like column
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vectors and to to motivate that uh let's have a look at a similar story
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to what I showed you in the special case 3-2 but now more generally so we can sort of see what's happening with
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variables rather than specific numbers so suppose that I have an arbitrary population say x y and I still perform
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these sort of sequences S1 S2 S1 and also S2 S1 S2
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so if we apply S1 then this thing negates and we have to add the neighbors we get minus X Plus y y we perform Y2 we
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have to negate this one and add what's over here so we have minus y plus minus
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X Plus right there's a total of minus X this doesn't change then we apply S1 to this we have to negate this to get x
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minus y and then add this one so we just get minus y minus X and over here we do
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S2 to this we get x x minus y we do y S1 we get minus y x minus y and we do S2
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and we get the same thing minus y minus X so notice that we're going from a population X Y to the other one at the
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Other Extreme minus y minus X okay now at at each stage
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here for example we had sort of two possible directions to go in fact even in this stage here we have two possible
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directions to go we can either apply S1 which will get us here or we can apply S2 which will get us back there
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so in this case here if we apply S1 we get to here and the effect of that is to
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augment the the first entry by how much well the difference
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between this one and this one is minus X Plus y minus X or a total of minus 2x
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plus y so this entry here is telling us how much the first entry is going to
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change if we mutate at that first vertex while this second entry here is
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telling us how much the second entry is going to change if we mutate at the second vertex
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we go from y to x minus y the difference this minus this is x minus 2y so this
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Vector here this column Vector is encoding the different expected changes that we would get to
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the population in question depending on whether we either mutated at the first Vertex or at the second vertex and
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that's a way of thinking about what this PSI mapping does is it tells us for a
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given population the expected changes that we get if we apply either mutation S1 or the mutation
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S2 so this is the corresponding picture on
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the weight side so the correspondence to X Y is minus 2x plus y x minus 2y
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if we then look at this thing here and say okay well what are the expected changes if we apply either S1 or S2 if
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we apply S1 then we would be going here and expect to change in the first variable would be x minus this for a
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total of 2x minus y well if we applied S2 the expected change in the Y value would be minus x
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minus y that's why we're getting this value here for this entry here
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so we're applying the same formulation so for each one of these six things here we can compute the corresponding
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potential changes the PSI Vector for them
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and then we know if you've watched the earlier lectures that you know that
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there's an alternate way of thinking about this story here purely in terms of the numbers game another way of going
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from here to here without any reference to what's going on on the population side is to just say What's Happening
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Here is that we're applying S1 so this thing here changes in sign so instead of
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having minus 2x plus y we now have 2X minus y and subsequently also we have to
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add this original value here to the direct neighbor so we add this to this we get minus x minus y and that happens
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in each of the cases right so we can generate this picture here without any reference to the
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original populations just by playing the numbers game on this side so the mutation game on this side corresponds
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to the numbers game on this side and the link between them is this PSI map the
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canonical correspondence which we have going from populations these guys to
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weights these guys over here and in particular as a special case if
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we look at the vector the basis Vector 1 0. I didn't give that a name earlier but we
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might I might just make a note that the vector 1 0 in the population story we
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could call that say Delta one that's consistent with what we've done before and the other basis Vector 0 1 we could
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call that Delta 2. so these are two basis vectors for the population space just like
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Lambda 1 and Lambda 2 where basis vectors for the weight space and
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when we apply this PSI to 1 0 well put in one zero here we're just going to get minus 2 1 and we'll put in 0 1 we're
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going to get one minus two and note that this is a linear map okay
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it's a linear map so it's a very simple kind of map and this is the map that's giving us a correspondence between these
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two pictures okay so now it turns out that there's another way of thinking about this PSI canonical correspondence
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since it's a linear map it can also be given essentially by multiplication by a matrix and it turns out that that Matrix
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is essentially minus C so another way of saying what PSI of the row Vector p is
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it's what you get when you take the column Vector P transpose make it a column vector and multiply it on the
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left by minus c c is the course of the carton Matrix
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and there's an inverse to this which I might mention so if we call this W because
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that's in the weight space then it's easy to sort of invert that and express p as a function of w so it turns out
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that that's minus W transpose times C inverse so the inverse
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correspondence going from weights to populations just undoing what PSI does
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is given by this equation so this one involves C this one involves C inverse
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another reason why C inverse is quite important for us okay so this map has
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two important properties first of all it's an isometry okay it's mapping populations to weights
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we want to show that the quadrants of PSI of p is the same as the quadrants of P so over here we're in the space of
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populations over here we're in the space of Weights we have these two different quadratic forms okay one is the tits
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quadratic form on this side the other one's the Dual tits quadratic form on the other but this mapping maintains the
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quadrants so quadrants of PSI of P what is that well here's the formula for it so it's
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quadrants of minus c times P transpose and over here on the weight side what
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was the definition of the Dual tits quadratic form well we take this thing here whatever it is and we take its
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transpose we multiply by C inverse and we multiply by this thing itself
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now the transpose of this is the product of the transposes but in the opposite order so P transpose transpose is just p
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C transpose the two minus signs canceled so we can get rid of them and then we have a c inverse and then we have a c
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and then we have a p transpose and then the C and the C inverse cancel we're just left with p c p transpose which was
32:49
the quadrants of pay so that's a proof that this thing actually is an isometry
32:55
so it respects the two geometries that we have here it's saying that these two geometries are essentially the same and
33:03
you know the fact that it's uh invertible it follows from from this formula here we've we've written down an
33:10
inverse so as long as the carton Matrix is invertible this thing really is an invertible linear map and the inverse is
33:16
also a linear map the second important result is that this
33:22
PSI mapping respects the action of the mutation
33:28
matrices so the mutation Matrix acts on a population p on the right so and if we take PSI of P
33:36
of s i we get a weight vector and that turns out to be PSI of P acted on the
33:42
left by the same s i so that's a statement of this invariance
33:49
or the sort of the PSI mapping respects the action of the mutation
33:56
matrices on the respective row and column vectors okay proof of that PSI of P of s i well
34:04
PSI if anything is minus c times this thing transpose and the transpose of this thing is
34:10
we use the transposes in the other order so we get S I transpose P transpose
34:17
and then we are going to replace minus CSI transpose with minus s i c
34:24
and that's coming from the carton Matrix invariance theorem here it is again
34:30
s i times c times S I transpose equals c if we take this s i and we multiply by
34:37
its inverse which is equal to itself because s i squared equals the identity so if we multiply both sides by s i then
34:43
the sis cancel on this side we just get CSI transpose and on the right hand side
34:49
we get s i c in that order so that means that up here we have a CSI
34:54
transpose we can replace this product with this product s i times C
35:00
and so over here we see p transpose minus sign that's the same as PSI of P
35:05
so the minus c p transpose is just a PSI of p and the SI is still on the
35:11
left so that proves this theorem so we saw earlier how the
35:19
the action of the s i matrices on populations of general population sort
35:27
of transforms to corresponding things on the weight side symmetrically it's also
35:33
valuable to see how this functions on the other side so I could maybe put a dotted line here
35:43
okay so over here now I'm looking starting with a general Vector on the
35:49
weight side say UV and applying again our same family of s
35:54
i operations S1 S2 S1 or S2 S1 S2 now on
36:00
the weight side so we're playing the numbers game so from S1 we negate this one and we add this one to this one Etc
36:06
we just go on like this okay to get to the opposite one which is minus V minus
36:11
U has a very similar form to what happened on the population side and over here is the inverse so the inverse of
36:19
PSI on UV well remember it's a question of multiplying by C inverse and C
36:26
inverse was that Matrix that had the two thirds and one-thirds in in it okay so we get minus two thirds U minus
36:34
one-third V comma minus one third U minus two-thirds V in this special case for the A2
36:42
so over here we can go and look at corresponding elements so from here we
36:47
plug this in here we get this one here we take this one we plug it in we get this one here Etc okay and we see that
36:54
the story that we're getting here is really just the mutation game on this one here so for example to go from here
37:00
to here what happens is the second one doesn't change we're mutating at the first vertex so we we're taking this
37:07
thing here and we're negating it to get 2u plus v and then we have to add this one so 2u plus v adding this one we get
37:15
U minus V so we can go from here to Here by looking at PSI inverse or we can
37:22
generate this thing here just from the first one by playing the mutation game in the familiar way
37:28
so this is very analogous now we had sort of a balanced point of view we can either start with populations and move
37:33
to weights or we can start with weights and move to populations the two things are have equal Merit equal sort of
37:41
importance and similarly the basis vectors remember
37:46
these ones were the basis vectors that we called Lambda 1 and Lambda two so you plug in 1 0 in here and you see you get
37:52
the vector minus two thirds What minus one-third and you plug in 0 1 and you get minus one third comma minus
38:00
two-thirds and here already you can start to see that we're naturally
38:07
zooming in on this eight-fold way Story coming from physics so the Quark Story
38:14
the baryon multiplex stories Etc are emerging from this very Elementary
38:21
linear algebra which from this point of view is just happening in this two-dimensional world because we're
38:27
playing some Elementary games on these graphs with just two vertices su3 representation theory is nowhere in
38:35
sight here but we're going to now see that we're going to recover essentially the same pictures that are motivating
38:42
neymon and Gelman to obtain the start of particle physics through the Eightfold
38:48
way now what we're going to do is we're going to combine these two pictures into
38:53
one picture this PSI map and its inverse map allow
38:59
us to take these two pictures and sort of superimpose one on the other there's two ways of doing that we can
39:07
impose the population picture on top of the weight picture or the weight picture on top of the population picture they're
39:15
pretty well equivalent but it's much more familiar to take the weight picture
39:20
and superimpose it on the population picture so that's what this diagram shows okay
39:27
and it's very important to understand the representation theory of su3 but it
39:35
also has sort of independent interest because it's just really coming from the mutation and the numbers games now on
39:41
the same in the same diagram so these axes represent the axes of
39:47
populations and so this thing here corresponds to
39:54
um the population that we called Delta one the basis Vector with a one at vertex
40:01
one and zero the other one and here is Delta Two and in black the black dots represent
40:08
the integral span or the lattice generated by these two vectors so it's
40:14
just like the ordinary coordinate lattice in two-dimensional space
40:21
now superimposed on that is this picture in blue and crucial for that what we're
40:27
going to do is we're going to sort of expand the story so that we can clearly identify the point one-third two-thirds
40:34
on this segment similarly one-third and two-thirds in this direction and of course we could uh
40:40
extend that you know in all directions okay and it's the the reason that we're dividing by
40:46
three is exactly because of the the determinant of the C Matrix the fact
40:52
that those threes end up in the denominators uh in the Expressions involving C inverse
40:58
okay so in blue here what we have is the image of the weight diagram
41:06
under PSI inverse remember the PSI inverse takes us from the weight picture
41:12
to the population it takes a weight and subjects it or creates a corresponding
41:18
population and that's what these two things here are so this this thing here
41:24
corresponds to Lambda 1 because it's the point minus two-thirds comma minus
41:30
one-third remember that was the image of the the Lambda 1 under PSI and this one
41:36
here is Lambda 2. it's minus one third comma minus two-thirds so that's the
41:43
image of the second basis Vector in the weight space in here and then what we have is the lattice
41:50
generated by these two vectors okay so if that's Lambda 1 and that's Lambda 2 that will be Lambda 1 plus Lambda 2.
41:57
that's 2 Lambda um two that's three Lambda two
42:02
um you know Etc this one here will be uh I guess Lambda 2 plus 2 Lambda 1 Etc but
42:11
of course also the integral combinations give us also negative one so all the blue ones here and you can see that the
42:19
the blue lattice is a finer lattice it contains within it
42:24
the black lattice because this point here is just the sum of Lambda one and Lambda 2. or this one here is the sum of
42:33
um of minus Lambda 1 and minus Lambda 2. Okay so
42:39
and what about this one here for example this one is uh minus 2 Lambda 1
42:45
plus Lambda 2 Etc so now what we have is the possibility
42:50
of describing points in this space in two different ways
42:56
so for example let's look at this point up here all right this one right here so
43:01
that's three over in the X Direction in the population picture and up two
43:08
so in the population story as a row Vector
43:13
that would be three comma two now if we wanted to express it as a combination of Lambda 1 and Lambda 2
43:20
well we can do that also we can go minus one minus 2 minus 3 minus four Lambda 1.
43:26
and then minus Lambda 2. so we can use the column Vector notation
43:35
to express this point as minus four minus 1.
43:42
so the row Vector notation tells us the coordinates of a point with respect to
43:48
the population Vector with essentially with respect to the basis Delta one Delta Two
43:53
while the column notation tells us the coordinates of that same point or vector
43:59
with respect to the basis Lambda 1 Lambda 2.
44:05
and that's exactly the correspondence that we have we've seen that if we have a population of 3 2 then the
44:11
corresponding weight is minus four minus one so we're seeing both pictures here at the same time
44:19
so great so I claim that this is one way of thinking about the the mathematical
44:25
structure which underlies the standard numerology the su3 numerology in the standard model
44:34
and if history had been a little bit different than Gilman and Niemann might
44:40
have discovered this Eightfold Way by connecting the particles not with su3
44:47
representations but rather just with the geometry contained in this two-dimensional picture coming
44:54
ultimately just from the simple graph A2 okay which is at least an order of
45:00
magnitude simpler conceptually mathematically sophisticated fashion
45:06
simpler than the su3 representation story
45:11
so let me give you a brief indication of how that works by looking at how the mutations act in this story okay so the
45:21
the beauty of this using this PSI correspondence is that the PSI correspondence re respects the action of
45:28
the mutation matrices so the the the action of the mutation matrices
45:34
looks the same from both the population and the weight a point of view in this picture
45:43
so let me indicate that let's let's look at not the image of this one but something a little bit simpler let's uh
45:50
say in green let me um okay I might use this one here okay so let's look at that
45:55
point and look at what happens if we apply S1 and S2 and various combinations of those
46:02
to that so we look at the vile group orbit of that point there okay so
46:08
in the uh if we apply S1 to it okay
46:13
what's going to happen is that the the the second coordinate is going to stay
46:18
unchanged okay because we're applying a mutation and the the first coordinate is
46:24
going to change so uh right now it's 1 1 and if we do the mutation we get the the
46:29
fir the first one changes to a minus one then we have to add the other one so we get zero so again getting 0 1 which is
46:35
that point there okay so the the S1 applied to this S1 sends this starting
46:41
point let's say the double that's the starting point to this okay which is a horizontal movement and notice
46:48
importantly that the midpoint of that is turns out to be exactly on this line
46:53
here so we can think about this this action of S1 as actually reflection in
47:00
this point here but it's a reflection which is not a euclidean reflection because it was a
47:06
euclidean reflection then the image would be over here somewhere it's a reflection with respect to the tits
47:11
quadratic form and with respect to the test quadratic form we have orthogonality this way this line here
47:17
and this line here are orthogonal well maybe I'll write it like that okay and similarly this line here the
47:24
the vertical one and and this one here are orthogonal so maybe I'll Express that like this
47:30
okay those are the two kinds of orthogonalities that we have so this is actually an orthogonal
47:37
reflection in the sense of the tits quadratic form through this line This is the the axis of the of this reflection
47:44
so we're going from here to here and what if we apply S2 well S2
47:50
is going to be a reflection in in this line and we're going in the S2 directions it gives us from there to
47:56
there okay what happens if we do S2 to this
48:03
thing well again we need to reflect in this line and that's uh going to be that
48:09
point there so we're going to go from here to here so the image of this thing under S2 is down here
48:15
and what's the image of this one under S1 the reflection in in this line is going to be over here
48:22
and then the reflection so this one here was uh S1 followed by S2 if we do
48:27
another S1 that's another reflection in this line that sends us over here and similarly if we do this one this was
48:33
S2 followed by S1 if we do another S2 then we get also here so the the image
48:41
of this is a hexagon which you'll recognize as the same shape
48:49
as we saw when we were talking about the baryon octet in our preferred
48:56
coordinate system with with the charge and hypercharge however in that story there was an octet
49:04
there were also there's also a weight of two so we haven't explained that weight I still have to explain that I'm going
49:09
to do that in a subsequent video but we can already see the the geometry exactly
49:15
the same geometry that we we saw before arising this way let's do another example maybe
49:23
um I'll do it in in Red so suppose that we started with this point instead I'm going to do a uh yeah Circle okay we did
49:30
a circle there so if we do an S1 then we're reflecting in here so we have to go sort of an equal amount over here so
49:36
we're going to pass through all of these we're going to go up to here okay and then if we do an S2 to that
49:43
that gives us down here and and if we do a an S2 to this one it
49:51
doesn't change okay so what ends up happening is that the the vial group
49:57
orbit in this case is just three points and the the set of
50:03
black lattice points usually called the root lattice in the usual literature that is contained in
50:11
this triangle would include this one and this one and this one and and this one
50:17
and this one and this one and this one and this one yeah we already got that
50:24
one so one two three four five six seven eight nine ten that is the baryon
50:29
decouplet that famously involved the Omega minus particle down here so that's
50:35
also arising as a vile group orbit in this in this story
50:41
and what about the quarks well the quarks I prefer you to think about
50:46
starting with Lambda 1 and Lambda 2. if we perform these same Reflections okay there's Lambda one if we sort of reflect
50:54
it we reflect it in in this line it's not going to move if we reflect it in this line we have to go over here so
51:00
maybe I'll do it in Black it could be a triangle okay triangle a triangle and then we reflect this one in in this line
51:06
then we get up to here okay so we get another
51:12
a kind of triangle but a sort of a smaller triangle and that is uh associated with one
51:20
family of quarks up down strange and if we look at the other one maybe I'll do
51:26
it upside down on a triangle and look at the orbit there so the reflection in this line doesn't change it then but
51:31
reflection in this line it's a vertical thing so we're going to get up to here and then reflection in this one here
51:37
gives us this one over here and we end up getting um maybe you know do it solid
51:42
make a little bit different so there's the the anti-cork triangle to the the original
51:48
Quark triangle so the the positioning of these
51:55
particles with respect to these quantum numbers hypercharge and charge that I
52:00
that I mentioned if we position things in the right way really coincides exactly with the geometry that's
52:07
presented here when we bring the mutation game and the numbers game
52:13
together we're considering populations and weights at the same time
52:19
however there's now a more advanced story having to do with multiplicities of interior waste in particular the the
52:26
baryon octet that we started with the green vertex here this hexagon also somehow contains
52:33
um or ought to contain a doublet of particles at zero so how do we explain
52:39
that well to do that we have to essentially look at what's called vile character
52:44
formula but again that's uh lee Theory that's quite sophisticated we're going to circumvent all of that and I'm going
52:51
to show you how that also is a beautiful consequence of just the geometry of this
52:57
situation combined with suitable algebra and it's just polynomial algebra or as
53:03
we would prefer to say poly number algebra so that's pretty exciting coming up next
53:09
time