Physics 735: Homework 2
Due Thursday, October 2, 2008 (Give to Hiren Patel)
1. Using the explicit forms of the 2x2 Pauli matrices, compute their commutation
and anti-commutation relationships.
2. Using the Klein-Gordon equation together with the replacement
∂
µ
→ ∂
µ
+iqA
µ
,
find the form of the potential,
ˆ
VKG
, in the corresponding equation,
+m2
( )
φ
=−ˆ
VKG
φ
in terms of A.
3. In Majorana representation γ matrices are:
γ
0=0i
σ
1
−i
σ
10
,
γ
1=
iI 0
0−iI
,
γ
2=0
σ
2
−
σ
20
,
γ
3=0iI
iI 0
,
γ
5=0i
σ
3
−i
σ
30
a. Show that this is a valid representation, i.e.,
γµ
,
γν
{ }
=2g
µν
I,
γµ
†=
γ
0
γµγ
0,
γ
5,
γν
{ }
=0,
γ
5
( )
2=I
b. Show that
γµ
*=−
γµ
c. Show that if ψ satisfies Dirac equation, then its complex conjugate also
does. What can you conclude about the particle represented by ψ?
4. Problem 5.4 from Aitchison & Hey
5. Problem 5.6 from Aitchison & Hey
6. Problem 5.7 from Aitchison & Hey
7. Problem 5.8 from Aitchison & Hey
8. Problem 6.6 from Aitchison & Hey
9. Problem 6.7 from Aitchison & Hey
10. Problem 6.9 from Aitchison & Hey
Aitchison & Hey is available on reserve in the physics library.
A scanned copy of chapter 5-7 are also available online.
http://www.hep.wisc.edu/~dasu/classes/physics735