This wee we will finish the introduction of the standard model by using the gauge transformations of

SU(3)color X SU(2)_{L} weak isospin X U(1)
hypercharge

to build the QCD and Elctroweak forces.

The form of the interactions will allow us to display much of the detail in a diagrammatic fashion. Feynam diagrams will be introduced as a pictorial format. The underlying mathematical machinery will not be covered.

Review our goals:

Find elementary or fundamental particles.

- Classically this might be a point particle
- QM
choose P
^{μ}, J^{ 2}, J_{z}. Label Momentum Energy and Spin

Ignore the mathematical justification and use the normal spin multiplets from QM

Jè 0,1/2,1,3/2…

Throughout our identification of fundamental particles we will be looking for multiplets.

Multiplets are labeled by observables that don’t change and therefore id the multiplets. For spin states this is the total spin. Multiplets are also labeled by observables that can be mixed. The z-projection of spin can change for a spin S particle.

Multiplets

spin many example in QM books.

flavor SU(3) è
Two labels within a multiplet are plotted on and x-y plane with x(T_{3})
axis and y(Λ_{8}). The graphical representation is more appealing
than the labels and one builds triangles, octets, decuplets as the multiplets.
This is only an approximate symmetry but the multplet structure is important in
building light baryons and mesons

color SU(3) è Same structure but now an exact symmetry. Normally the color structure of SU(3) is not manifest because the gluonic forces require configurations of color singlets.

weak isospin SU(2) è Here we find particles
as doublets with W-isospin as “½” è (u,d) (e,ν_{e})
… and singlets W-isospin = 0.

Now we add interactions to the Standard Model by making the symmetries local. First we can examine the typical introduction of the E&M interaction.

Normally one defines a vector
potential **A** and a scalar potential φ. One can put them together as
A^{μ} = (**A,** φ) as a four vector. For the hydrogen atom one add qφ to the
Hamiltonian. For more general interactions involving magnetic fields on adds J^{μ}A_{μ}
as the interaction.

Demonstrate how SU(2) and U(1) generate the weak and QED interactions. Note that

QED based on Q as the generator

U(1) electro weak will be Y the hypercharge.

The theory is made a bit more complex by the Higgs mechanism that provides mass for the weak field particles W,Z.

Introduced Feynman diagrams.