Methods of QFT in CMT
WiSe 2012/13
Lecturer Dr. Oleg Yevtushenko, Tutor Dennis Schimmel
(dennis.schimmel@physik.uni-muenchen.de)
http://homepages.physik.uni-muenchen.de/~oleg.yevtushenko/qft-cm-WiSe-12-13/
OutLine-QFT-CM-WiSe-12-13.htm
Problem Set 14
“Field Theory on a closed time contour”
Exercise 1. Continuous versus Discrete Representation
In the lecture you saw the partition function for a system on a Keldysh-contour both in the
discrete and the continous setting. It was claimed that although the partition functions did
not agree, its cumulants did. Explicitely check this for the cumulants up to the third power.
That is, consider the generating functional in the discrete case,
Zd [V ] =
1 − ρ(ω0 )
R
1 − ρ(ω0 ) exp(−i C dtV (t))
3
δ ln Zd
2
3
and compute hhn3 (t)ii = (i/2)3 δ[V
q (t)]3 |V =0 (and similar for hhn (t)ii). Compute hhf (t)ii =
3
δ ln Zc
|
for the continuous partition function
i3 δ[V
(t)]3 V =0
Zc = exp(−T r(ln(1 − GV ))),
with G and V defined as in the lecture.
Exercise 2. Tunneling in a time-dependent potential
The aim of this exercise is to calculate the tunneling amplitude through a potential which
varies in time in a one-dimensional setup. Consider the potential
x2
1
V (x, t) = − ω02 x2 + 2 0 2 cos(Ωt),
2
x + x0
where x labels the position, t labels time and is a small positive constant. Later on we will
be interested in corrections to the tunneling probability of order . We will consider negative
energies (because we are interested in tunneling).
(a) As a first step, consider the static potential ( = 0). Look at the semiclassical limit. Then
the tunneling can described using complex times (instead of just real times). What is
the contour of the semiclassical tunneling trajectory? As a consistency check, recover the
WKB approximation.
(b) Find a tunneling solution for the unperturbed system.
(c) Use this to compute the change to the tunneling probability at order .
1
Exercise 3. Quantum particle coupled to an oscillator bath
Consider a quantum particle given by the action
Z
1
Sp = dt ((∂t q)2 + ω02 q 2 ).
2
Introduce a bath of bosonic oscillators given by the action
Z
dω X
ψ̄k,ω G−1
Sb =
k,ω ψk,ω ,
2π k
where we define
G−1
k,ω
=
ω − iδ − ωk
.
2iδ coth(ω/2T )
0
ω + iδ − ωk
(a) Rewrite the quantum particle in terms of the qc and qq .
(b) Introduce a coupling of the form
Z
dω X
Spb =
(γk qωT σ1 ψk,−ω + h.c.),
2π k
where γ are complex coupling constants and q T = (qc , qq ). Integrate out the bath, assuming
an ohmic dissipation |γ| = λ|k|1/2 , with λ > 0.
(c) Perform the sum over the k-modes.
(d) Now, let us have a look at the classical limit. This can be recovered by reintroducing ~
1
such that S → ~S
, coth(ω/2T ) → coth(ω~/2T ) → 2T
and qq → ~qq . What is the classical
~ω
equation of motion for qc ?
2