Gauge/Gravity Duality Foundations and Applications Solutions to Exercises Martin Ammon Johanna Erdmenger

Gauge/Gravity Duality
Foundations and Applications
Solutions to Exercises
Martin Ammon
Johanna Erdmenger

1
t
References to equations involving Arabic chapter numbering refer to the equations given in the book. The equations in this set of solutions are labelled by
a Roman number counting the chapters, followed by an arabic equation number
(eg. (V.3)).
1 Elements of field theory
Exercise 1.1.1
The equation of motion for a free real scalar field reads (✷ − m2
)φ(x) = 0 and is
linear. It is therefore natural to expect that it is solved by a plane wave ansatz
φ(x) = 1
(2π)
d
Z
d
d
k a(k)e
−ikx
, (I.1)
where k
µ = (ω, k), x
µ = (t, ~x) and kx = kµx
µ = −ωt + ~k~x. Inserting this ansatz
into the equation of motion, we obtain
(✷ − m2
)φ(x) = 1
(2π)
d
Z
d
d
k (−k
2 − m2
) a(k) e
−ikx
. (I.2)
Thus the ansatz is only a solution if k
2+m2 = 0. Modifying the ansatz to implement
this condition, we have
φ(x) = 1
(2π)
d
Z
d
d
k 2π δ(k
2 + m2
)Θ(k
0
)

a(k)e
−ikx + c.c.


, (I.3)
with Θ the step function. This ansatz satisfies the equation of motion. Note that
φ(x) is real, which we achieved by adding the complex conjugated terms to the
expression. We perform the k
0
-integration using the identity
δ(f(k
0
)) = X
i,single zeros
δ(k
0 − k
0
i
)
f
0(k
0
i
)
, (I.4)
where k
0
i
are the single zeros of the function. For k
2 + m2 = −(k
0
)
2 +~k
2 + m2
, the
zeros are located at k
0 = ±ωk and we obtain
φ(x) = 1
(2π)
d−1
Z
d
d−1~k
2ωk

a(k)e
−ikx + a(k)
∗
e
ikx

 
 
 
k0=ωk
. (I.5)
Exercise 1.1.2
The Lagrangian reads
L = −
1
2
η
µν∂µφ(x)∂νφ(x) −
1
2
m2φ(x)
2 −
g
4!φ(x)
4
. (I.6)
The associated Euler-Lagrange equation
∂µ

∂L
∂(∂µφ(x))
=
∂L
∂φ(x)
(I.