Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509

Fall 2011
Physics 509: Relativistic Quantum Field Theory
Problem Set 3
Due Friday, 14 October 2011
Reading: Peskin 2.4, 3.13.2, 4.14.4
11. Anharmonic SHO and Feynman Diagrams I
1
Consider, as usual, the anharmonic oscillator L = 2 2 1 m2 2
2
4
,
4!
where is a
sm
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509

Fall 2011
Physics 509: Relativistic Quantum Field Theory
Problem Set 1
Due Friday, 30 September 2011
1. SHO Greens Function from Euclidean Time
1 imt
2m e
0 T (t)(0) 0 =
1 2
2
1 m2 2 . It is known that G(t) =
2
by using the Heisenberg equation (t) + m2 (t) = 0
Ph509 Problem Set 1
Wentao Fan
September 23, 2016
1.
(a)
0
(t) (t) = ei (t) with R, the Lagrangian transforms as
0
0
0
1
1 m2 ei (t)ei (t)
L(t) L (t) = 2 (t) (t) 21 m2 (t) (t) = 21 ei (t)ei (t)
2
1 2
1
= 2 (t)(t) 2 m (t)(t) = L(t),
Under the transform
Ph509 Problem Set 3
Wentao Fan
November 15, 2016
1.
(a)
Notice rst that both sides of the equation T (t)(t ) = GF (t t )+ : (t)(t ) : are invariant under permutations of cfw_t, t , so we only have to prove the equation in the
0
0
1
case where t t , since
Ph509 Problem Set 5
Wentao Fan
November 15, 2016
1.
Using the Feynman path integral, we can
write the
normalized generating function as
4
4 1
4
2
2
DeiS0 ei d x(x)j(x)
1
Z[j(x)] h0T ei d x(x)j(x) 0i =
= De
D eiS0 ei d x[ 2 ( m +i)(x)j(x)]
iS0
DeiS0
Ph509 Problem Set 6
Wentao Fan
November 15, 2016
1.
(a)
In the coordinate
space, the Dirac equation reads
i/ m (x) = 0.
So, for positiveenergy
Fourier modes of the form (x) = eipx u(p), the momentum space Dirac equation can be derived as
0 = i/ m (x)eipx
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509
Physics 509: Relativistic Quantum Field Theory
Problem Set 9
Due Friday, 16 December 2011
30. Alternative Standard Model
In this problem, we pose an alternative to the standard model, in which the Higgs boson
(as opposed to the actual Higgs eld of the sta
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509
Physics 509: Relativistic Quantum Field Theory
Problem Set 8
Due Friday, 09 December 2011
28. Unitarity and scattering longitudinal vector states.
Massive vectors have three polarization states: two transverse and one longitudinal.
However, massless vecto
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509
P & S 10.2, Renormalization of Yukawa theory
P & S 11.3, Gross Neveu model
Part d of the second problem uses a bit of one loop
technology (one loop determinant) that I will discuss on
Wednesday.
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509

Fall 2011
Physics 509: Relativistic Quantum Field Theory
Problem Set 5
Due Tuesday, 22 November 2011
23. Spontaneous symmetry breaking: scalars.
A theory with a higher amount of symmetry in the action than in the a particular
groundstate is said to undergo spontan
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509

Fall 2011
Physics 509: Relativistic Quantum Field Theory
Problem Set 5
Due Friday, 11 November 2011
20. Coulomb Scattering
Peskin, Problem 5.1. Further, under what limit does one recover the result obtained for
nonrelativistic Coulomb scattering? Show that in this
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509

Fall 2011
Princeton University
Physics 509
Quantum Field Theory
Problem Set 4
14 Particle creation by a classical source
Do problem 4.1 in Peskin and Schroeder.
15 Twobody decay
Suppose that particle 1 decays into several other particles 1 2 + 3 + 4 + + n . The
de
Relativistic Quantum Theory (Introduction to Quantum Field Theory)
PHY 509

Fall 2011
Physics 509: Relativistic Quantum Field Theory
Problem Set 2
Due Friday, 07 October 2011
7. Convergence of Perturbation Theory in a 0Dimensional QFT
Much insight into QFT can be gained by studying the integral
1
2
dx e 2 x
Z (j ) =
4! x4 +jx
,
(1)
where
Ph509 Problem Set 2
Wentao Fan
November 15, 2016
1.
(a)
In terms of creation and annihilation operators we can write the position operator and the conjugate momentum operator as
1 (a
2m
respectively, with
(0) =
p
+ a ) and (0) = i m
2 (a a )
[H, a] = m[a