View
4
Download
0
Category
Preview:
Citation preview
Computing the Wave Function
of the Universe
Penn State, 9 May 2012
Alex Maloney, McGill University
Castro, A. M., to appearCastro, Lashkari & A. M.
Overview
The Problem
Quantum cosmology is confusing:
I How is Unitarity consistent with singularities, inflation, . . . ?
I Is quantum mechanics modified in cosmological settings?
I What are the appropriate observables for eternal inflation?
I What is the meaning and origin of the entropy of acosmological horizon?
Similar questions are answered in the context of black hole physicsby AdS/CFT.
Let us be bold and apply the same techniques to cosmology.
The Wave-function of the Universe
One lesson of AdS/CFT is that the ”wave function of theuniverse” |ψ〉 exists and is computable.
The Hartle-Hawking state
〈h|ψ〉 ∼∫
g |∂M=h
Dg e−S
includes contributions from all geometries. It is the natural“vacuum state” of quantum gravity.
In AdS the radial wave function is a CFT partition function.
Goal: Compute 〈h|ψ〉 in de Sitter space.
de Sitter Space
For three dimensional general relativity with a positivecosmological constant
S [g ] =1
G
∫M
√−g
(R − 2
`2
)the partition function
Z =
∫Dg e−S[g ]
can be computed exactly.
We will be inspired by AdS/CFT but we will not use it.
The Idea
The saddle point approximation is
Z =
∫Dg e−S[g ] =
∑g0
e−kS0+S1+ 1kS2+...
where k = `/G is the coupling. The approximation becomes exactif we can
I Find all classical saddles
I Compute all perturbative corrections around each saddle
We will do both.
The Result
We will compute the wave function of the universe as a function ofthe topology and (conformal) geometry of I+.
This is a version of the Hartle-Hawking computation, where oneanalytically continues from dS to Euclidean AdS space.
The resulting wave function 〈h|ψ〉 is non-normalizable, and peakedwhen the geometry h is singular.
I This comes from a non-perturbative effect
Thus, for pure 2+1 dimensional gravity, de Sitter space is unstable.
The Plan for Today:
• The Wave Function
• The Computation
• The Result
The Wave Function of the Universe
We wish to compute the wave function of the universe 〈h|ψ〉.
Near I+, h→∞ and one can take the WKB approximation of theWheeler-de Witt equation.
Up to local counterterms, the Hartle-Hawking wave functiondepends only on the conformal structure of h
〈h|ψ〉 → e iSct(h)ψ(h)
The dS/CFT conjecture identifies ψ(h) with the partition functionof a Euclidean conformal field theory (Strominger, Maldacena, ...).
We will not use dS/CFT. Instead we will compute ψ(h) directly.
Hartle-Hawking WavefunctionHartle & Hawking propose that 〈h|ψ〉 should be computed byintegrating over smooth, compact Euclidean geometries:
This is difficult to compute.
If we take `→ i`, z → iz the metric of Lorentzian dS becomes thatof Euclidean AdS:
ds2 = `2−dz2 + dx2
z2→ `2
dz2 + dx2
z2
Future infinity of dS becomes the asymptotic boundary of EAdS.
Asymptotics
The Bunch-Davies boundary conditions (ψ ∼ e−iωz → eωz atz → −∞) tell us that wave functions are smooth in the interior ofEAdS.
The Computation
Thus the wave function can be computed by a path integral oversmooth field configurations in Euclidean AdS.
The wave function ψ(h) can be interpreted as the partition of aEuclidean CFT with imaginary central charge, since the analyticcontinuation takes
k =`
G→ i
`
G
Such CFTs are confusing. They do not obey standard unitarity(reflection positivity) conditions.
So instead of trying to identify the CFT we will perform a directbulk computation.
Aside:
This notion of Euclidean continuation is natural if we think ofgravity as a gauge theory
I Analytically continue signature of spacetime withoutcontinuing the local Lorentz group
I Very natural in 3D gravity, Vasiliev theory, . . .
This explains why the AdS/CFT duality between the O(N) modeland Vasiliev theory can be turned into a dS/CFT.
For theories with more complicated degrees of freedom (e.g. RRfields) this is less useful.
The Computation
Wave Function on dS3
The wave function ψ(h) is a function of conformal structure of I+.
A sphere (I+ of global de Sitter) has a unique conformal structure,so ψ is just a number.
But I+ can have more complicated topology:
I Solutions to the classical equations of motion are quotientsdS3/Γ.
I They can have arbitrary topology and conformal structure atfuture infinity.
At early times these universes have Milne-type big bangsingularities.
We wish to compute the topology and conformal modulidependence of ψ(h).
Wave Function on T 2
Today, focus on the case where I+ is a torus.
The quotient dS3/Z
ds2 = `2(−dt2 + cosh2 tdφ2 + sinh2 tdθ2
)with θ ∼ θ + 2π, φ ∼ φ+ 2π has a Milne singularity at t = 0.
This is a de Sitter analogue of the BTZ black hole.
For this geometry, the torus at I+ is square.
More generally the wave function depends on the conformalstructure parameter τ of the torus at I+.
z ∼ z + 1 ∼ z + τ
The Path Integral:
The wave function ψ(τ) is computed by the path integral
ψ(τ) =
∫Dg e−S[g ]
over locally EAdS3 manifolds with T 2 boundary with conformalstructure τ . This includes a sum over topologies
ψ(τ) =∑M
ψM(τ)
In the semiclassical (~→ 0) limit only M which admit a classicalsolution contribute
ψ =∑g0
e−k S(0) + S(1) + k−1S(2)+...
For pure gravity we can find all classical saddles and compute allperturbative corrections.
The Classical Solutions:
The smooth classical solutions are solid tori Mγ , labelled by
elements γ =(
a bc d
)∈ SL(2,Z)
T
X
The Mγ are quotients EAdS3/Z. They are related by modulartransformations
ψγ(τ) = ψ0(γτ), γτ =aτ + b
cτ + d
where ψ0 is the contribution from thermal AdS.
Classical Action:
The wave function is a sum over the modular group
ψ(τ) =∑
γ∈SL(2,Z)
ψ0(γτ)
where ψ0 is the contribution from geometries continuouslyconnected to thermal AdS.
The classical action is obtained by computing the (regularized)volume of EAdS
ψ0(τ) = |q|−2k q = e2πiτ
This becomes a pure phase
ψ0(τ) = |q|2ik = e4πik=τ
upon continuation back to dS.
Quantum Effects:
The one loop term is not pure phase
ψ0(τ) = |q|−2k det∇(1)
√det∇(0) det∇(2)
It is non-trivial even though there are no local degrees of freedom.
This comes from quantizing the space of non-trivialdiffeomorphisms, a la Brown & Henneaux.
ψ0 = |q|−k∏n
1
|1− qn|2
The result is one-loop exact.
The Results
Modular InvarianceWhen we sum over SL(2,Z), the resulting ψ(τ) is invariant underτ → aτ+b
cτ+d . It is a function on H2/SL2(Z).
The Result
ψ(τ) can be written as a convergent series expansion
ψ(τ) ∼ e−πiτ/12
(−6 +
(π6 − 6π)(11 + 24k)
9ζ(3)=τ+ . . .
)
1.000 1.005 1.010 1.015 1.020ImHΤL570 000
575 000
580 000
585 000
590 000
595 000 ΨHΤL¤
Divergence
The divergence at τ = i∞ is non-normalizable∫d2τ
=τ2|ψ(τ)|2 =∞
Peaked at the infinitely stretched torus with τ = i∞.
Thus de Sitter space is unstable; the spatial geometry wants to beinfinity distorted.
I Effect invisible at all orders in perturbation theory around thestandard saddle
I Comes from loop terms around the non-perturbative saddles
Conclusions
The wave function of the universe at I+ is computable as aEuclidean gravity path integral.
The computation is more easily done in Euclidean AdS, rather thanEuclidean dS.
The resulting wave function is peaked when the spatial geometry is(infinitely) inhomogeneous.
I de Sitter space is non-perturbatively unstable.
All known constructions of de Sitter space in string theory arenon-perturbatively unstable.
I We have found the three dimensional version of this instability.
Recommended