Dear students!
In this course we want to go through the second part of the book Introduction to Elementary Particle Physics by David Griffiths. The first lecture will remind you of the first course and cover the concept of bound states, which you can find in chapter 5. Many of the topics in chapter 5 can be expanded at will, but this is up to the personal interest. I can recommend to pick parts for presenting it to your colleagues in the course to earn points for presentations.
David Griffiths follows a very didactic approach: When you present something, you have to practice it, too. But you should not present too much, before you practice it. Therefore he structures the task of calculating the cross section or the decay rate for a selected process into two parts: first calculating the amplitude from the Feynman diagram and second inserting the squared amplitude into Fermi’s Golden Rule to obtain the wanted cross section or decay rate. As the second step is needed all the time and gives interesting insights on its own, Griffiths starts with it in chapter 6.
Chapter 6 introduces the framework for the calculation: how to combine the kinematics of the process to arrive at the differential cross section. As an example, it also introduces a toy model, that allows to practice calculations without the need to pay attention to the space-times structure of participating fields, as all fields are scalars. This simplifies the vertices and the propagators. But it also illustrates the basis of Quantum Field Theory: QFT deals with degrees of freedom, not necessarily with particles. Many particles can be combined to exhibit few degrees of freedom, which is the case in solid state physics or statistical physics. But single particles can have also more than one degree of freedom, like spin, polarization or isospin. Then the separation of the QFT effect from the “multiple degrees of freedom” effect is not only pedagogically good, but shows the structure of our world.
In the next three chapters David Griffiths extends the conceptual simple Feynman Rules of the toy model step by step to the Feynman Rules of the Standard Model. The order of complications goes together with the list of interactions in the Standard Model.
Chapter 7 introduces Quantum Electro Dynamics (QED). Still today QED is the simplest physical QFT. As the new concept compared to chapter 6 appears the spinor as the simplest particle containing two degrees of freedom. Discussing the mathematical and conceptual background of combining several degrees of freedom into single objects is at the heart of group theory, but goes far beyond the scope of this course.
One important message from the discussion of the Dirac equation and its solutions is, that one has to take care of Lorentz covariance, which severely restricts the possibilities of writing down field combinations and their interactions. One consequence is, that fermion fields never appear alone, but only in Lorentz covariant bilinears.
Lorentz covariance is sometimes not enough to reduce possibilities. This can be seen when discussion the photon as a gauge boson. The discussion of gauge theories is limited to the introduction of the Lorentz condition and the Coulomb gauge. (I plan to at least shortly introduce the concept of gauge invariance of amplitudes.)
On the calculational part Griffiths shows the implementation of completeness relations, for spinors known as “Casimir’s trick”, and ends the chapter with several examples of important QED processes.
Again following his didactic approach Griffiths starts the chapter about QCD (chapter 8) not with QCD itself, but first with applying the gained knowledge to objects of QCD, to quarks. This additionally allows him to introduce more applications of Lorentz covariance and discuss kinematic functions. Only after these concepts he discusses color as a degree of freedom and introduces the Feynman rules for QCD. In the calculation of the color-factors he can rely again on the already introduced concept of completeness relations. The discussion of examples of quark scattering leads to handwaving arguments for asymptotic freedom in the end of the chapter.
Chapter 9 introduces the weak interactions. The starting point are the processes mediated by the charged W-boson, using immediately the correct chiral form of the interaction vertex to leptons and quarks. But the first application focuses on the most important part for low energies: the kinematics. And to apply the same type of diagrams not only to quarks, but also to baryons and mesons, Griffiths again uses the arguments of Lorentz covariance. This allows to discuss the historical developments and show, how the understanding of the basic calculational tools allowed the development of the theory.
For the last part of the course, the construction of the Standard Model, I will not rely so much on Griffiths, but I will try to present the construction from a more theoretical angle.