Dynamics on Graphs| Looking ahead: A2, quarks, and the Standard Model of particle physics
The usual numerology of the Standard Model of particle physics, which arose in the 1960's from the work of M. Gell-Mann and Y. Ne'eman connects the eight-fold way to the representation theory of the compact Lie group SU(3) and eventually to the theory of quarks and chromodynamics.
However we are going to see that the mathematical / numerological side of this story is somewhat surprisingly captured by a much simpler theory that side-steps the need for representations and Lie groups, and in fact follows pretty directly from the geometry and associated algebra of the Mutation and Numbers games played on the second simplest graph: that with only two vertices and a single edge joining them: A2. This is a simplification that ought to be familiar to physicists.
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hello everyone I'm Norman wobberger
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thanks for joining me on this adventure
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we're looking at Dynamics on graphs and
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exceptional structures in mathematics
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and physics
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so we've been talking mostly about
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mathematics today I want to move towards
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some Physics applications in particular
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having a look at what I call the
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numerology of the standard model of
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particle physics which is touted as one
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of the most successful and accurate
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theories in science which in some sense
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it is but also there's a lot of
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questions and sort of open aspects to
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this subject what has a feeling this is
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not at all a final theory that is really
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waiting for some deeper understanding
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some some more thorough
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realization of what's going on
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so
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in very broadly and of course I'm a
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mathematician not a physicist so this is
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a very rough overview from from outside
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okay the physicists have done remarkable
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things experimentally in the last 50 60
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years
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and discovered a whole range of
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fascinating
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particles in their accelerators and such
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and these particles suggest that the the
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subatomic world is a lot more
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complicated than we first might appeared
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not just electrons protons and neutrons
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and so on there's a whole range of
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particles and there's uh you know
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hadrons and baryons and mesons and
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leptons and fermions and so on various
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classifications of these particles so in
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the early 1960s this landscape was
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somewhat cataloged by uh by physicists
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and and which revealed um underlying
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structures having to do with with quarks
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and so the emergence of a quarks side of
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the story ended up uh strongly
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connecting with a branch of mathematics
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called Lee Theory and representation
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theory in particular having to do with a
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certain group sort of lead group called
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su3 it's a compact leg group
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now these days the standard model has
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been enlarged so not just the the strong
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force having to do with quarks but also
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the weak Force
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um and of course electromagnetism and so
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it sort of encapsulated by a a group
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theoretical point of view involving S1
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and su2 su2 for the weak force and su3
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for the strong force okay
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um
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now as associated with this there's
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there's various gauge sort of theory
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models feel Theory so all of this is
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sort of
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um theoretically
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um constructed in terms of uh you have
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space time and you have uh bundles over
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space time and and these particles are
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excitations of certain fields that uh
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that are that emerge as sort of sections
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of these bundles so what I want to do is
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I'm I'm interested in the purely
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mathematical side of the of the su3
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stories but also the su2 story so it
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turns out that the Dynamics on graph's
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story that we've been developing here
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has direct applications to both the su2
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and the su3 and I'm going to start with
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the su3 which is um you know sort of
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immediately interesting and connects
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with the the graph A2 the su2 connects
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with the graph A1 so in our Pantheon of
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of graphs we've identified the Ade graph
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that's particularly important and except
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and so E8 is sort of the biggest and
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most exceptional one of those but on the
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other side of the spectrum the simplest
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ones are A1 and A2 you know and then
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there's A3 and
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from a sort of interesting point of view
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they're also exceptional they're
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exceptional because they're so small and
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and so basic and so Elementary so they
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also play actually a distinguished role
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in the theory so in a kind of a
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complementary way to to E8 and so I want
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you to think about A1 and A2 as also
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kinds of exceptional structures and
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we're going to to look at A2 especially
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and it will turn out that this is a very
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very rich story okay but I want to try
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to avoid all the Lee Theory so usually
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physicists who want to understand su3
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they have to learn about lead groups and
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they have to learn about Lee algebras uh
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there's various decompositions
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subgroups uh there's vile groups there's
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heart measures there's representation
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Theory there's highest weights there's
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Associated harmonic analysis with L2
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spaces etc etc Peter vile theorem et
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cetera so um we want to somehow you know
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avoid having to spend several years
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learning all of that stuff and so I want
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to give you a way of thinking about the
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numerology of the standard model
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having to do with quarks and the various
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um
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like octets and decouplets that
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physicists have found to organize their
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particles I want to give you a way of
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thinking about that in a very Elementary
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way that's a direct consequence of the
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kinds of thinking that we've been doing
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in this series up till now okay so it's
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just a continuation of what we're doing
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but we're basically going to zero in on
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this A2 story which is such a simple
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it's really the second simplest example
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A1 is even simpler and A1 is actually
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also really important we're gonna have
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to think about that but A2 is a good
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place to start because it's already
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pretty rich and we'll be able to see a
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lot of this Quark story The emerging
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purely from this graph theoretical point
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of view so I think this is really novel
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and it also suggests to physicists that
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the the usual attendant um you know
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theoretical framework that depends uh on
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on su3 you know bundles over space time
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there might be alter to this okay
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because I mean if you can
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create some this this underlying
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numerology from a graph theoretical
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point of view that it might be possible
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to to rethink the the the basic physics
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framework about what's really going on
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um in a fundamental way so I think
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that's it's going to be really exciting
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and I think it's going to be really
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interesting okay so it's just a broad
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overview of where we're going to go and
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and one of the things that we're going
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to have to do is we have to develop some
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some techniques um some algebraic and
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analytic techniques having to do with
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measures so we have to look at measure
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theory in a slightly different way okay
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because the usual measure Theory and
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from my point of view doesn't really
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make sense
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um so we have to um we have to think
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about things in a novel way but it's
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going to be really concrete and based on
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on A2 and such an incredibly Rich story
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and and I think you'll you'll be really
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excited by it