Quantum Electrodynamics (QED) - Exam Details and Review

1. Announcements • Exam Details: • Pencil, calculator, paper • Handout resembling inside front cover • 200 points total • 10 short answer (5 each) = 50 points • ~ 8 calcuulation (variable) = 150 points • Today: Problems 6.6, 6.7 • Friday: Problems 6.10, 6.11 • Monday: Read 7A – 7D Exam Tuesday Location TBA 3:30 – 5:30 10/8

2. Quantum Electrodynamics (QED) Fine Structure Constant • Units in electromagnetism are confusing • Everyone agrees on a constant called the fine structure constant • In SI units, it is given by: • In our units: • Always write answers in terms of 

3. Electromagnetism Review • Classical E&M works with electric field E and magnetic field B • In quantum mechanics, we must work with the vector potential A • Note that the EM fields are unaffected by a gauge transformation

4. Electromagnetic Waves • Maxwell’s equations are given by • We want no source: • We want to write in terms of A’s: • We want plane waves: • It looks like k and the polarization vector  are parallel: • But if you assume this, easy to see that: • This isn’t a plane wave, it is nothing (pure gauge) • The only alternative is to demand

5. Electromagnetic Waves and Gauge Choice • Photons will automatically be massless • We still have some freedom to redefine the polarization vector • Coulomb gauge: Make 0 = 0. • Transverse waves:

6. The energy density from the waves • We will need E and B - fields • Now we need some sophisticated stuff from E & M with complex fields • Now integrate this over all space:

7. Normalizing one Photon • Properly normalized photon: • Relativistically normalized photon:

8. Announcements • Exam Details: • Pencil, calculator, paper • Handout resembling inside front cover • 200 points total • 10 short answer (5 each) = 50 points • 8 calculation (variable) = 150 points • Today: Problems 6.10, 6.11 • Monday: Read 7A – 7D Exam Tuesday Olin 103 3:30 – 5:30 10/12

9. Problem 2.9 For any process where two particles of momenta p1 and p2 collide to make two final particles with momenta p3 and p4, define the Mandelstam variables as below.Show that s + t +u is a constant, and determine it in terms of the masses mi2 = pi2.

10. Problem 3.5 Define the current density four-vector as in the green box, where  is a solution to the Dirac equation in the presence of electromagnetic fields, eq. (3.51). Show that the red box is true.

11. Problem 4.4 Consider the matrix elements of the form below, where W- is the W-boson, and qi and qj are any of the six quarks. Use the internal symmetry of electric charge to argue that of the 36 possible matrix elements, only nine of them can be non-zero. Which ones? In fact, all nine of them are non-zero.

12. A New Problem • Suppose there are three types of particles, 1, 2, and 4, all spin zero, each distinct from their anti-particles, which have charges +1, +2, and +4 respectively. • What are all possible renormalizable basic matrix elements of the form <0| H |X>, where X has three or more particles? • Which of the interactions in (a) are real? • Make up a notation, and use it to give me all vertex Feynman rules for this theory • Find the tree-level amplitude for the decay 4 211

13. Problem 6.16 Calculate the differential and total cross section for the process below in the theory with coupling described in the figure. Let the initial energies be E1 and E2, and their common momenta p. Keep all three masses arbitrary.

14. Announcements • Exam Details: • Pencil, calculator, paper • Handout resembling inside front cover • 200 points total • 10 short answer (5 each) = 50 points • 8 calculation (variable) = 150 points • Today: None • Wednesday: Read 7A – 7C • No class on Wednesday • Monday: Read 7D – 7F Exam Tomorrow Olin 103 3:30 – 5:30 • Lunch Wednesday? • On campus • Off campus 10/15

15. Errors on Problems (1) • One trace per grouping

16. Errors on Problems (2) • In the cross-section formula, E1and E2 and p1 and p2 are all initial particles • In the D(two) formula, p is the momentum of either of the final state particles in the cm frame.

17. Errors on Problems (3) • Almost any formula can be not-integrated over angles • The angular integral is defined as: • Because of the way we do things, the  integral almost always gives 2 • If there is any cos dependence, you have to work out the integral • If there is no cos dependence, you just get 4

18. Polarization Vectors • There is more than one way to describe the same EM wave • Gauge choice • We chose Coulomb gauge • We found we are forced to choose • We found that for a relativistically normalized plane wave, we need: • There are two linearly independent vectors that satisfy these conditions • These are the two possible polarization vectors • Label them  = 1,2

19. Polarization Vectors • There are multiple ways to pick them • Most common is to use real polarizations • With real polarizations, the *’s have no effect • It is sometimes convenient to use circular polarization • These are helicity eigenstates • Relativistically normalized eigenstates: • Much like spins, they need to besummed or averaged over

20. The Vertex Associate with fermions lines Associate with photon lines • The vertex rule is simply:

21. The Diagrammatics • Fermions and anti-fermions as arrows: • We often have to label them because there are multiple fermions in the standard model • Photons as wiggly lines • External lines will have factors associated with them: • There will also be propagators • For the fermion • For the photon

22. Summing over Polarizations • Just like fermions, we often have to sum polarizations over spins • We need to find a simple formula for: • This quantity is not gauge invariant (yuck!) • This quantity is generally not even Lorentz invariant (yuck!) • There is a proof that you can never go wrong if you assume:

23. Feynman Rules: External Lines The Vertex Propagators: