QFT A Problem Solution Set #0 (by C. Zeoli)
(1) Using Einsteins summation convention and assuming all the indices are Lorentz indices, explain why each of the following expressions makes no sense. Also, for each case,
suggest a correct form of the express
QFT A Take-Home Final Exam
(Due Thursday, December 13, 5:00 pm)
(Put your exam in my mailbox on the 3rd oor)
[WARNING] This is an EXAM! You must work entirely on your own. You are not permitted
to discuss the exam with anyone (not even with your grandma),
PHY 5667 Problem Set no. 12
solution
Problem 1
T
L = ( )( ) m2 + iL L iL 2 L + i L 2 L
(i) The U(1) charge of L has to be 1 . The kinetic term of the lefthanded
2
spinor remains unchanged with under any U(1)-charge (because it contains L
and L , the rst c
PHY5667 Problem Set #12 (Last HW!)
(10 points total, due in two weeks on Tue Nov 23)
(no late HW accepted)
(1) Consider the following QFT for a complex scalar and a left-handed spinor L :
T
L = ( )( ) m2 + iL L L i 2 L + L i 2 L ,
where is a real constant
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PHY5667 Problem Set #11 (due Tue Nov 9)
(1) The gamma ve 5 is dened by 5 i 0 1 2 3 .
(i) In the basis of the Dirac spinor and -matrices used in class, i.e.,
=
L
R
=
,
0
0
,
show that 5 can be used to project onto the left-handed or right-handed spinor
PHY 5667 Problem Set no. 10
solution
Problem 1
(i) The Feynman rule obtained from
Lint = = a ab b
is
a
= iab
b
(ii) To get the diagram, we need to contract the eld with creation and annihilation operators in the intial and nal state.
ak,s
k
a
b
k
p
a
,p
a
PHY 5667 Problem Set no. 4
solution
Problem 1
Amp. = 0|aF 2 aF 1
T
1
2
2!
dtd3 x(i)F F B
dt d3 x (i )F F B
a 1 a 2 |0
F
F
= (i)(i )
dt d3 x dt d3 x aF 2 aF 1 F B F a 1 a 2
F
F
BF
F
+ aF 2 aF 1 F B F a 1 a 2
F
F
BF
F
+ aF 2 aF 1 F B F a 1 a 2
F
F
BF
F
=
PHY5667 Problem Set #4 (due Tue Sep 21)
(1) To understand how the conservation of energy and momentum at each vertex in Feynman
diagrams arise, lets look at one of the examples discussed in the lecture, where we had a fermion
F and a boson B with an inter
PHY5667 Problem Set #2 (due Tue Sep 7)
(1) In quantum mechanics, the operators x and p satises a commutation relation [, p] = i
x
(in the = 1 units as usual). Then, in the basis spanned by the eigenstates of x (dened as
x|x = x|x ), the representation p =
PHY 5667 Problem Set no. 1
solution
Problem 1
(a)
[A, BC ] = ABC BCA
= ABC BAC + BAC BCA
= (AB BA)C + B (AC CA)
= [A, B ]C + B [A, C ]
(b)
[A, BC ] = ABC BCA
= ABC + BAC BAC BCA
= (AB + BA)C B (AC + CA)
= cfw_A, B C + B cfw_A, C
Problem 2
This identity i
PHY5667 Problem Set #1 (due Tue Aug 31)
(If you are taking a qualifying exam this week, just email me and you will be allowed to
hand in this HW by Thursday, Sep 2, without any penalty.)
(1) Prove the following identities for the commutator [ , ] and the