# Computing the Wave Function of the media/ppcc/talks/ ¢ \vacuum state" of quantum...

date post

06-Aug-2021Category

## Documents

view

0download

0

Embed Size (px)

### Transcript of Computing the Wave Function of the media/ppcc/talks/ ¢ \vacuum state" of quantum...

Alex Maloney, McGill University

Overview

I What are the appropriate observables for eternal inflation?

I What is the meaning and origin of the entropy of a cosmological horizon?

Similar questions are answered in the context of black hole physics by 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 the universe” |ψ exists and is computable.

The Hartle-Hawking state

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 positive cosmological constant

S [g ] = 1

∫ Dg e−S[g ]

can be computed exactly.

We will be inspired by AdS/CFT but we will not use it.

The Idea

Z =

e−kS0+S1+ 1 k S2+...

where k = `/G is the coupling. The approximation becomes exact if 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 of the topology and (conformal) geometry of I+.

This is a version of the Hartle-Hawking computation, where one analytically continues from dS to Euclidean AdS space.

The resulting wave function h|ψ is non-normalizable, and peaked when 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 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 the Wheeler-de Witt equation.

Up to local counterterms, the Hartle-Hawking wave function depends only on the conformal structure of h

h|ψ → e iSct(h)ψ(h)

The dS/CFT conjecture identifies ψ(h) with the partition function of a Euclidean conformal field theory (Strominger, Maldacena, ...).

We will not use dS/CFT. Instead we will compute ψ(h) directly.

Hartle-Hawking Wavefunction Hartle & Hawking propose that h|ψ should be computed by integrating over smooth, compact Euclidean geometries:

This is difficult to compute.

If we take `→ i`, z → iz the metric of Lorentzian dS becomes that of Euclidean AdS:

ds2 = `2 −dz2 + dx2

Future infinity of dS becomes the asymptotic boundary of EAdS.

Asymptotics

The Bunch-Davies boundary conditions (ψ ∼ e−iωz → eωz at z → −∞) tell us that wave functions are smooth in the interior of EAdS.

The Computation

Thus the wave function can be computed by a path integral over smooth field configurations in Euclidean AdS.

The wave function ψ(h) can be interpreted as the partition of a Euclidean CFT with imaginary central charge, since the analytic continuation takes

k = `

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 direct bulk computation.

Aside:

This notion of Euclidean continuation is natural if we think of gravity as a gauge theory

I Analytically continue signature of spacetime without continuing the local Lorentz group

I Very natural in 3D gravity, Vasiliev theory, . . .

This explains why the AdS/CFT duality between the O(N) model and Vasiliev theory can be turned into a dS/CFT.

For theories with more complicated degrees of freedom (e.g. RR fields) 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 quotients dS3/Γ.

I They can have arbitrary topology and conformal structure at future infinity.

At early times these universes have Milne-type big bang singularities.

We wish to compute the topology and conformal moduli dependence 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 conformal structure 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 conformal structure τ . This includes a sum over topologies

ψ(τ) = ∑ M

ψM(τ)

In the semiclassical (~→ 0) limit only M which admit a classical solution contribute

ψ = ∑ g0

e−k S(0) + S(1) + k−1S(2)+...

For pure gravity we can find all classical saddles and compute all perturbative corrections.

The Classical Solutions:

The smooth classical solutions are solid tori Mγ , labelled by

elements γ = (

) ∈ SL(2,Z)

T

X

The Mγ are quotients EAdS3/Z. They are related by modular transformations

ψγ(τ) = ψ0(γτ), γτ = aτ + b

cτ + d

Classical Action:

The wave function is a sum over the modular group

ψ(τ) = ∑

ψ0(γτ)

where ψ0 is the contribution from geometries continuously connected 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:

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-trivial diffeomorphisms, a la Brown & Henneaux.

ψ0 = |q|−k ∏ n

The Results

Modular Invariance When we sum over SL(2,Z), the resulting ψ(τ) is invariant under τ → aτ+b

cτ+d . It is a function on H2/SL2(Z).

The Result

ψ(τ) ∼ e−πiτ/12

575 000

580 000

585 000

590 000

=τ2 |ψ(τ)|2 =∞

Peaked at the infinitely stretched torus with τ = i∞.

Thus de Sitter space is unstable; the spatial geometry wants to be infinity distorted.

I Effect invisible at all orders in perturbation theory around the standard saddle

I Comes from loop terms around the non-perturbative saddles

Conclusions

The wave function of the universe at I+ is computable as a Euclidean gravity path integral.

The computation is more easily done in Euclidean AdS, rather than Euclidean dS.

The resulting wave function is peaked when the spatial geometry is (infinitely) inhomogeneous.

I de Sitter space is non-perturbatively unstable.

Overview

I What are the appropriate observables for eternal inflation?

I What is the meaning and origin of the entropy of a cosmological horizon?

Similar questions are answered in the context of black hole physics by 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 the universe” |ψ exists and is computable.

The Hartle-Hawking state

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 positive cosmological constant

S [g ] = 1

∫ Dg e−S[g ]

can be computed exactly.

We will be inspired by AdS/CFT but we will not use it.

The Idea

Z =

e−kS0+S1+ 1 k S2+...

where k = `/G is the coupling. The approximation becomes exact if 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 of the topology and (conformal) geometry of I+.

This is a version of the Hartle-Hawking computation, where one analytically continues from dS to Euclidean AdS space.

The resulting wave function h|ψ is non-normalizable, and peaked when 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 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 the Wheeler-de Witt equation.

Up to local counterterms, the Hartle-Hawking wave function depends only on the conformal structure of h

h|ψ → e iSct(h)ψ(h)

The dS/CFT conjecture identifies ψ(h) with the partition function of a Euclidean conformal field theory (Strominger, Maldacena, ...).

We will not use dS/CFT. Instead we will compute ψ(h) directly.

Hartle-Hawking Wavefunction Hartle & Hawking propose that h|ψ should be computed by integrating over smooth, compact Euclidean geometries:

This is difficult to compute.

If we take `→ i`, z → iz the metric of Lorentzian dS becomes that of Euclidean AdS:

ds2 = `2 −dz2 + dx2

Future infinity of dS becomes the asymptotic boundary of EAdS.

Asymptotics

The Bunch-Davies boundary conditions (ψ ∼ e−iωz → eωz at z → −∞) tell us that wave functions are smooth in the interior of EAdS.

The Computation

Thus the wave function can be computed by a path integral over smooth field configurations in Euclidean AdS.

The wave function ψ(h) can be interpreted as the partition of a Euclidean CFT with imaginary central charge, since the analytic continuation takes

k = `

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 direct bulk computation.

Aside:

This notion of Euclidean continuation is natural if we think of gravity as a gauge theory

I Analytically continue signature of spacetime without continuing the local Lorentz group

I Very natural in 3D gravity, Vasiliev theory, . . .

This explains why the AdS/CFT duality between the O(N) model and Vasiliev theory can be turned into a dS/CFT.

For theories with more complicated degrees of freedom (e.g. RR fields) 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 quotients dS3/Γ.

I They can have arbitrary topology and conformal structure at future infinity.

At early times these universes have Milne-type big bang singularities.

We wish to compute the topology and conformal moduli dependence 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 conformal structure 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 conformal structure τ . This includes a sum over topologies

ψ(τ) = ∑ M

ψM(τ)

In the semiclassical (~→ 0) limit only M which admit a classical solution contribute

ψ = ∑ g0

e−k S(0) + S(1) + k−1S(2)+...

For pure gravity we can find all classical saddles and compute all perturbative corrections.

The Classical Solutions:

The smooth classical solutions are solid tori Mγ , labelled by

elements γ = (

) ∈ SL(2,Z)

T

X

The Mγ are quotients EAdS3/Z. They are related by modular transformations

ψγ(τ) = ψ0(γτ), γτ = aτ + b

cτ + d

Classical Action:

The wave function is a sum over the modular group

ψ(τ) = ∑

ψ0(γτ)

where ψ0 is the contribution from geometries continuously connected 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:

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-trivial diffeomorphisms, a la Brown & Henneaux.

ψ0 = |q|−k ∏ n

The Results

Modular Invariance When we sum over SL(2,Z), the resulting ψ(τ) is invariant under τ → aτ+b

cτ+d . It is a function on H2/SL2(Z).

The Result

ψ(τ) ∼ e−πiτ/12

575 000

580 000

585 000

590 000

=τ2 |ψ(τ)|2 =∞

Peaked at the infinitely stretched torus with τ = i∞.

Thus de Sitter space is unstable; the spatial geometry wants to be infinity distorted.

I Effect invisible at all orders in perturbation theory around the standard saddle

I Comes from loop terms around the non-perturbative saddles

Conclusions

The wave function of the universe at I+ is computable as a Euclidean gravity path integral.

The computation is more easily done in Euclidean AdS, rather than Euclidean dS.

The resulting wave function is peaked when the spatial geometry is (infinitely) inhomogeneous.

I de Sitter space is non-perturbatively unstable.

*View more*