Physics 610
1
Homework 4 Solutions
Asymptotic series
Consider the baby or toy version of scalar 4 theory, where it is just a single integral;
2 4
+
2
24
Z=
d exp
.
(1)
This is what the path integral for scalar 4 theory would look like if there were only o
Physics 610, Problem Set 3
due: Friday, September 30 at 3:30pm
Please place your completed problem sets in the Physics 610 box in the physics department mailroom
(Rutherford 103b) before the due date. Please do not leave them in my mailbox. You are encour
Physics 610, Problem Set 1
due: Friday, September 16 at 3:30pm
Please place your completed problem sets in the Physics 610 box in the physics department
mailroom (Rutherford 103b) before the due date. Please do not leave them in my mailbox. You
are encour
Physics 610, Problem Set 4
due: Friday, October 14 at 3:30pm
Please place your completed problem sets in the Physics 610 box in the physics department mailroom
(Rutherford 103b) before the due date. Please do not leave them in my mailbox. You are encourag
Physics 610
1
Homework 10 solutions
Renormalization
Consider scalar 4 theory, with one real scalar eld and Lagrangian
m2 2 0 4
1
.
(1)
L =
2
2
24
We have seen many times that the lowest-order matrix element for the scattering process
is M = 0 . Here we
Physics 610
1
Homework 7 Solutions
Spinors
In class we found explicit expressions for the spinors up,s and vp,s for the special case that
1
p = (E, 0, 0, pz ) (E 2 = p2 + m2 as usual) and s = 2 along the z axis.
z
Verify that these explicit expressions ob
Physics 610
1
Homework 8 Solutions
Complete Set of Grassmann States
For i , i , i , i each independent n-member sets of Grassmann variables, and using the
summation convention i i i i i , prove the identity
di di e e e .
e =
(1)
i
Hint: First show that it
Physics 610
1
Homework 9 Solutions
Spinor-Scalar Scattering in Yukawa Theory
Consider Yukawa theory, with one Dirac fermion and one real scalar eld , with Lagrangian
M2 2
1
/
4 (y i 5 y ) .
(1)
L = (i m) ( )( )
2
2
24
Assume that y and y are small but c
Physics 610
1
Homework 6 solutions
Fermionic harmonic oscillator
Free scalar eld theories reduce to a product of harmonic oscillators. Fermionic free theories
reduce to a product of fermionic harmonic oscillators. These are actually simpler than the
SHO,
Physics 610
1
Homework 5 Solutions
High-order Feynman diagrams
Consider the theory of one real scalar eld with Lagrangian density
m2 2
1
4 .
L[, ] =
2
2
24
1.1
(1)
6 external legs
In class we considered the scattering of two scalars into two scalars , w
Physics 610
Homework 1
Due Wed. 19 September 2012
1
4-vector notation and Maxwell equations
1.1
Problem
The purpose of this problem is to get you used to index notation and in particular to 4-vector
notation.
Recall that the electric and magnetic elds can
Physics 610
Homework 2
Due Thurs 27 September 2012
1
Commutation relations
In class we found that
(x) , (y) = i 3 (x y)
(1)
and then dened
(pm ) L3/2
d3 xeipm x (x)
(2)
and likewise for (pm ). Show that it really follows from these denitions that
(pn ) ,
Physics 610
1
Homework 3 Solutions
Projection operators
Consider the free eld theory of one scalar of mass m. Dene the state
|p = a |0 .
p
(1)
(Recall that ak , a = 2p (2 )3 3 (k p) and that ak |0 = 0.) Explain that the object
p
Frange
prange
d3 p
| p p|
Physics 610, Problem Set 2
due: Friday, September 23 at 3:30pm
Please place your completed problem sets in the Physics 610 box in the physics department
mailroom (Rutherford 103b) before the due date. Please do not leave them in my mailbox. You
are encour