Problem Set 4
Phys 7240
Due: Feb 24
1 Correlation Functions: Dirac Fermions
The action for the massless Dirac fermions reads
S=Zd2ξhψ†
R¯
∂ψR+ψ†
L∂ψLi.(1.1)
Here ξ0is imaginary time, ξ1is space, ∂=∂
∂z , and z=ξ0+iξ1.
Find the correlation functions Dψ†
R(ξ)ψR(0) E,Dψ†
L(ξ)ψL(0) Eand Dψ†
R(ξ)ψL(0) E.
2 Correlation Functions: Bosons
Free bosons have the following action
S=1
8πZd2ξ(∂µϕ)2.(2.1)
1. Find the correlation function hϕ(ξ)ϕ(0) i.
2. Find the correlation function Deiϕ(ξ)e−iϕ(0) E. This can be calculated by using the
definition
Deiϕ(ξ)e−iϕ(0) E=1
RDϕ e−SZDϕ eiϕ(ξ)−iϕ(0)−S,(2.2)
and noticing that this is simply a Gaussian functional integral.
3. Compare the correlation functions in this problem and in problem 1. The reason why
some of the correlation functions coincide is due to bosonization, the equivalence between
free bosons and Dirac fermions.