Heidelberg University
WS 2016/2017
Quantum Gravity and the Renormalization Group
Assignment 10
This assignment will be discussed in the tutorials on Jan. 15, 2017.
Problem 2: Wetterich Equation for Quantum Gravity
As it was discussed in the lecture and on assignment 3, the quantization of gravity
requires the introduction of Faddeev-Popov ghost fields C̄µ , Cµ . Moreover, we need
to introduce a background field ḡµν with gµν = ḡµν + hµν . Therefore, the Wetterich
equation in quantum gravity
−1
1 (2)
∂t Rk ,
(1)
∂t Γk [ḡ, h, C̄, C] = Tr Γk + Rk
2
depends on the super-field Φ = hµν , C̄µ , Cµ (note that the introduction of the superfield has nothing to do with supersymmetry; it’s just for notational convenience).
The super-field Φ is a three-component vector and the two-point function as well as
the regulator are three-by-three matrices,
(2)
Γk,ij =
δ 2 Γk
.
δΦi δΦj
(2)
The trace in the flow equation is then also a trace in the corresponding vector-space
of fields. We write a truncation of the effective action as
Γk [ḡ; h, C̄, C] = Γk,grav [ḡ; g] + Sghost [ḡ; h, C̄, C] ,
(3)
where Sghost is given by the ghost action derived on assignemt 3. Argue why the
cutoff-operator 4Sk that is added in the path integral should be chosen of the form
Z
Z
1
µνκλ
µν
hµν Rk,hh hκλ + C̄µ Rk,
4Sk =
C .
(4)
C̄C ν
2
Show that in that case the flow equation takes the form
−1
−1
1 (2)
(2)
∂t Rk,C̄C ,
∂t Γk [ḡ; h, C̄, C] = tr Γk,hh + Rk,hh
∂t Rk,hh − tr Γk,C̄C + Rk,C̄C
2
(5)
where tr means that the trace in field space is already carried out.
Problem 2: The Heat Kernel
The operators in the traces in the flow equation are in general functions of the
covariant Laplace operator ∆, covariant derivatives ∇ and the curvature R. A very
general and powerful method to evaluate these traces are the so-called heat-kernel
techniques. Let us assume that we are dealing with functions f (∆) of the Laplace
operator only.
(a) The heat-kernel
H(x, y, s) := e−s∆ δ(x − y) ,
(6)
owes its name to the the heat equation
∂s u(x, s) = −∆u(x, s) .
(7)
Show that the formal solution to the heat-equation with initial condition u0 is
given by
Z
u(x, s) =
H(x, y, s)u0 (y) .
(8)
y
(b) Show that the heat-kernel in flat space is given by
H(x, y, s) =
(x−y)2
1
− 4s
,
e
(4πs)d/2
(9)
where d is the dimension of spacetime.
(b) The trace over an operator is of course always defined with respect to some
space of functions on which this operator acts. Argue that the trace tr in the
flow equation over some operator A is given by
Z Z X
tr A =
ϕ̄n (x)A(x, y)ϕn (y) ,
(10)
x
y
n
where ϕn is a complete of normalized functions that span the space of fields
we integrate over in the path integral. In flat space, use a plane wave-basis in
order to derive the integral representation
Z Z ∞
z (d−2)/2
tr f (∆) =
dz
f (z) ,
(11)
(4π)d/2 Γ(d/2)
x 0
with the Gamma-function Γ.
(c) We now introduce the Laplace-transform f˜ for a function f : R+ −→ R,
given by
Z ∞
f (z) =
dsf˜(s)e−sz .
(12)
0
The Laplace transform has the property
Z ∞
Z ∞
1
−x ˜
ds s f (s) =
dz z x−1 f (z)
Γ(x)
0
0
(13)
Writing the trace as a coincidence limit and using the Laplace-transform, show
that
Z Z ∞
trf (∆) =
dsf˜(s)H(x, x, s) .
(14)
x
0
Taking the flat-space limit, re-derive the result of (b).