Dear all,
You can collect your graded answersheets from Priyadarshi (or Manish if Priyadarshi gives them to Manish). Please bring any discrepancy to my attention by Monday, after which the grades will be finalized.
A few pointers on the expected answers.
1) In Q1, note that _{in} <k_1, k_2 | k_3, k_4>_{out} = (_{out} <k_3, k_4| k_1, k_2>_{in})^* The two terms you need to add are complex conjugates of each other. The tree-level contribution cancels. The one-loop contribution is finite since the UV divergent parts cancel. This finite part can be computed through "cuts" of Feynman diagrams. Up to leading delta functions and theta functions, the answer is also given in 7.17 of https://arxiv.org/pdf/2405.20326
2) For the Feynman path integral question on the density matrix, most people got part 1 right. See:
http://www.hartmanhep.net/topics2015/5-UnruhHawking.pdf
for a nice discussion about part 2.
3) Many people got the decay question, right. One just needs to work out the kinematically allowed region. The second answer is 6 times the first answer.
4) Many people got part 1 of the scalar QED problem right. The cross-section was probably too hard. So I have given full points to anyone who simplified the answer, wrote the formula cross-section and, most importantly, realized the need to do polarization sums. You can read about this near Page 159 of Peskin and Schroeder in the subsection titled "Photon polarization sums"
regards,
Suvrat