The course notes here were taken by Alissa Crans. Garett Leskowitz has kindly scanned them in and made PDF files out of them. Some homework assignments have been made into web pages, while others still need to be scanned in.
Overview of possible course topics: gauge theory, categorified gauge theory, and the algebraic design of physics. Clifford algebras, spinors and the Dirac equation.
A little category theory: categories, monoids and monoidal categories. On April 6, Alissa Crans lectured on Lie 2-algebras.
Groups associated to a vector space V with metric g:
- the rotation/reflection group O(V,g)
- the rotation group SO(V,g)
- the spin group Spin(V,g)
- and the pin group Pin(V,g)
How the rotation/reflection group is generated by reflections. Pinors and spinors. Calculating the Clifford algebras Cp,q.
Clifford algebras and normed division algebras. Proof that the only normed division algebras are:
- the real numbers, R
- the complex numbers, C
- the quaternions, H
- the octonions, O
Classification of compact Lie groups and their relation to normed division algebras. For more on all this, see my introduction to the The Octonions.
Bott periodicity for complex Clifford algebras. Toby Bartels on quaternionic Hilbert spaces. For more on this, see Toby's discussion of quaternionic functional analysis. Quantum theory unitary group representations. The Poincare group IO(n,1), its connected component IO0(n,1), and their double covers IPin(n,1) and ISpin(n,1).
Clifford algebras as *-algebras. Elementary particles as representations of ISpin(n,1) x G where G is some "internal" symmetry group. The massive spin-0 particle and massive spin-1/2 particle as unitary representations of the Poincare group. Left- and right-handed massless spin-1/2 particles as a unitary representation of the Poincare group. Direct sums and tensor products of group representations. Miguel Carrion-Álvarez on the Weyl algebra and Fock space.
Patterns in the particle content of the Standard Model. The SU(5) and SO(10) grand unified theories. Interactions as intertwining operators between group representations. Describing interactions using Feynman diagrams. A brief history of hadron physics. Heisenberg's theory of the strong interaction in terms of isospin SU(2) symmetry: the nucleon and pion as representations of isospin SU(2).
Heisenberg's theory of the strong interaction in terms of isospin SU(2). Neutrinos and the weak interaction. The eight lightest mesons and Gell-Mann's theory of the strong interaction in terms of SU(3): the Eightfold Way.
Jeffrey Morton on Jordan algebras. The Eightfold Way. Hypercharge (Y) and its relation to charge (Q) and the third component of isospin (I3).
Quarks and the Standard Model. Weak isospin SU(2) and color SU(3). What people usually consider the internal symmetry group of the Standard Model:
SU(3) x SU(2) x U(1).
The Z6 subgroup that acts trivially on all particles. Embedding the "true"internal symmetry group of the Standard Model:
(SU(3) x SU(2) x U(1))/Z6 = S(U(3) x U(2))
into SU(5). The SU(5) grand unified theory: all known fermions and their antiparticles fit neatly into the representation of SU(5) on the exterior algebra of C5.
Enjoy! If for some strange reason you want the original LaTeX files for some of the above material, you can get them here. However, the authors reserve the rights to all this work.