Physics 230A In-Class Problems:
Interactions II
1. Consider the meson-nucleon theory with interaction
Lint = g .
(1)
Here is a real scalar eld of mass M
(x) =
(dp)M eipx (p) + h.c.
(2)
and is a complex scalar eld of mass m
(x) =
(dp)m eipx (p) + eipx (p)
Physics 230a
Homework 1 Solutions
1. (10 points)
(a) (2 points) At t = 0, our particle is in state (~, t = 0). To get the particles
p
state at a later time, we solve the time-dependent Schrdinger equation and
o
get:
(~, t) = e
p
(~, t = 0)
p
~
= e iEp t (
Physics 230a
Homework 2 Solutions
1. (10 points)
(a) (3 points)The smallest dimension term allowed by both shift symmetry and
Lorentz invariance is
( )2 .
Rearranging indices, we could also have the term
but this term is the same as the previous term wi
Physics 230a
Midterm Solutions
1. (10 points) Remembering our denitions for real scalar elds we have
(x) =
(dk) (k)eikx + (k)eikx
so then
(x) =
=
(dk) (k)(ik )eikx + (k)(ik )eikx
(dk)(ik ) (k)eikx + (k)eikx .
Our matrix element is only non zero when we h
Physics 230a
Homework 3 Solutions
1. (10 points) Our interaction Lagrangian is
Lint = g.
(a) (5 points) We want to compute the scattering amplitude for fermion-antifermion goes to fermionantifermion. Our initial and nal states are
|i = s1 (p1 )s2 (p2 ) |0
Physics 230A In-Class Problems:
Lorentz Transformations and Index Gymnastics
1. Write the following Lorentz transformations of 2-index tensors in index-free matrix
notation:
A = A
B
1
(1)
= ( ) B ,
(2)
C = (1 ) (1 ) C .
(3)
Show that the tensors
X = A C
Physics 230A In-Class Problems:
Relativistic Particle States
1. Consider the theory of a single relativistic particle dened in the lectures. Just as in
non-relativistic quantum mechanics, the states can be represented by wavefunctions,
that is, functions
Physics 230A In-Class Problems:
Symmetries in Quantum Mechanics
1. (a) Use the creation and annihilation operator formalism to show that for a simple
harmonic oscillator in Heisenberg picture
i
0|[q(t), q(t )]|0 =
sin[(t t )].
(1)
M
(b) Recall that we ca
Physics 230A In-Class Problems:
Interactions I
1. Consider a real scalar eld theory with interaction term
Lint = 4 .
(1)
4!
In this problem you will compute the O(0 ) and O() contribution to the S-matrix
element f |S|i , where |i and |f are 2-particle sta
Physics 230A In-Class Problems:
Continuous Groups and Representations
1. A group representation is dened by a linear transformation R(g) that acts on a
vector space of states, such that the group multiplication rule is satised:
R(g1 )R(g2 ) = R(g1 g2 ).
(
Physics 230A In-Class Problems:
Feynman Rules for Dirac Fermions
1. Consider the theory of a Dirac fermion coupled to a real scalar eld. The
interaction Lagrangian is
Lint = g.
(1)
(a) Compute the scattering amplitude for fermion-antifermion annihilation.
Physics 230A In-Class Problems:
Feynman Rules for Dirac Fermions II
Suppose that the electron is coupled to a new kind of scalar by a Yukawa coupling
Lint = y .
(1)
Compute the spin-averaged squared amplitude for e e . You may neglect the
mass of the elec
Physics 230A In-Class Problems:
Symmetry II
1. Consider a complex scalar with classical Lagrangian density
L = m2 ( )2 .
4
(1)
This Lagrangian is invariant under the U (1) symmetry
ei ,
(2)
where is a real phase.
(a) Find the Noether current associated w
Physics 230A Assignment 6
Due May 19
Quantum field theory is also useful in cases where special relativity is not important,
and particles are neither created nor destroyed. Non-relativistic quantum field theory
has extensive applications in condensed mat