Anti-de Sitter, scalar fields and boundary conditions

Anti-de Sitter, scalar fields and boundary conditions

Hugo Ferreira

Joint work with Claudio Dappiaggi

INFN / Università degli Studi di Pavia

14 September 2016, Lisboa

Spanish-Portuguese Relativity Meeting

0. Introduction

We study systematically the classical and quantum theory of a massive scalar fieldon anti-de Sitter (AdS) on a more mathematical precise fashion, extending the workof Avis, Isham, Storey (1978), Allen & Jacobson (1986) and others.

We consider all suitable boundary conditions at infinity, by treating the system as aSturm-Liouville problem, complementing the work of Wald & Ishibashi (2004).

We obtain the two-point functions for states obeying these boundary conditions andverify that they satisfy a natural generalisation of the global Hadamard condition fora spacetime with a timelike boundary.

We use this system as an example to study QFT on manifolds with boundaries andas the starting point to study generic asymptotically AdS spacetimes.

2 papers on arXiv (hopefully!) in the next few weeks.

1 AdS & Poincaré domain

2 Scalar field equation and boundary conditions

3 Two-point function and Hadamard condition

4 Conclusions

Outline




4 Conclusions

1. AdS & Poincaré domain

Anti-de Sitter AdSd+1 (d ≥ 2) is the maximally symmetric solution to Einstein’sequations with a negative cosmological constant. It is defined by the relation

−X20 −X2

1 +∑d+1

i=2 X2i = −`2 , ` > 0 ,

where (X0, . . . , Xd+1) are Cartesian coordinates of M2,d.

Poincaré patch (t, z, xi), t ∈ R, z ∈ R>0 and xi ∈ R, i = 1, . . . , d− 1,

ds2 =`2

z2(−dt2 + dz2 + δijdxidxj

).

The region covered by this chart is the Poincaré fundamental domain, PAdSd+1.

1. AdS & Poincaré domain

PAdSd+1 can be mapped to H̊d+1 .= R>0 × Rd ⊂M1,d via a conformal rescaling

ds2 7→ z2

`2ds2 = −dt2 + dz2 + δijdxidxj .

We can attach a conformal boundary as the locus z = 0 and obtainHd+1 .

= R≥0 × Rd, the upper half-plane.

Outline




4 Conclusions

2. Scalar field equation and boundary conditions

2.1. Field equation as a Sturm-Liouville equation

Poincaré domain (PAdSd+1, g), scalar field φ : PAdSd+1 → R,

Pφ =(�−m2

0 − ξR)φ = 0 .

Upper half-plane (Hd+1, η), scalar field Φ = ( z` )1−d2 φ : H̊d+1 → R,

PHΦ =

(�H −

m2

z2

)Φ = 0 ,

with m2 .= m2

0 − (ξ − d−14d )R.

From now on, ` = 1.



Fourier expansion. Fourier representation of Φ:

Φ =

∫Rd

ddk eik·x Φ̂k , x.= (t, x1, . . . , xd) , k

.= (ω, k1, . . . , kd) ,

where Φ̂k are solutions of the ODE

(− d2

dz2+m2

z2

)Φ̂k(z) = λ Φ̂k(z) , λ

.= ω2 −

d−1∑i=1

k2i

This is a Sturm-Liouville problem on z ∈ [0,+∞) with parameter λ.

General solution. The general solution in L2(0,∞) may be written as

Φ̂k(z) = c1(k)√z Jν

(z√λ)

+ c2(k)√z Yν

(z√λ),

ν.=

1

2

√1 + 4m2 .

The constants ci(k) depend on the boundary conditions.



Fourier expansion. Fourier representation of Φ:

Φ =

∫Rd

ddk eik·x Φ̂k , x.= (t, x1, . . . , xd) , k

.= (ω, k1, . . . , kd) ,

where Φ̂k are solutions of the ODE

(− d2

dz2+m2

z2

)Φ̂k(z) = λ Φ̂k(z) , λ

.= ω2 −

d−1∑i=1

k2i

This is a Sturm-Liouville problem on z ∈ [0,+∞) with parameter λ.

General solution. The general solution in L2(0,∞) may be written as

Φ̂k(z) = c1(k)√z Jν

(z√λ)

+ c2(k)√z Yν

(z√λ),

ν.=

1

2

√1 + 4m2 .

The constants ci(k) depend on the boundary conditions.



(− d2

dz2+m2

z2

)Φ̂k(z) = λ Φ̂k(z) , z ∈ [0,+∞)

Endpoint classification. The endpoint +∞ is singular (S). The endpoint 0 is:

1 regular (R) if the “potential” z 7→ m2

z2∈ L(0, c) for some c ∈ (0,+∞), i.e. if m2 = 0;

it is singular (S) otherwise, i.e. for m2 6= 0.

2 limit-circle (LC) if all solutions of the equation are in L2(0, c) for some c ∈ (0,+∞);it is limit-point (LP) otherwise.

m2 ν.= 1

2

√1 + 4m2 Classification

m2 = 0 ν = 12 Regular (R)

m2 ∈ [−14 ,

34), m2 6= 0 ν ∈ [0, 1), ν 6= 1

2 Limit-circle (LC)m2 ∈ [34 ,∞) ν ∈ [1,∞) Limit-point (LP)

N.B. z 7→√z Yν

(z√λ)∼z→0 z

−ν+ 12 is in L2(0, c) only if ν ∈ [0, 1).

The endpoint classification restricts the type of boundary conditions that may be appliedto the field solution at those endpoints.



(− d2

dz2+m2

z2

)Φ̂k(z) = λ Φ̂k(z) , z ∈ [0,+∞)

Endpoint classification. The endpoint +∞ is singular (S). The endpoint 0 is:

1 regular (R) if the “potential” z 7→ m2

z2∈ L(0, c) for some c ∈ (0,+∞), i.e. if m2 = 0;

it is singular (S) otherwise, i.e. for m2 6= 0.

2 limit-circle (LC) if all solutions of the equation are in L2(0, c) for some c ∈ (0,+∞);it is limit-point (LP) otherwise.

m2 ν.= 1

2

√1 + 4m2 Classification

m2 = 0 ν = 12 Regular (R)

m2 ∈ [−14 ,

34), m2 6= 0 ν ∈ [0, 1), ν 6= 1

2 Limit-circle (LC)m2 ∈ [34 ,∞) ν ∈ [1,∞) Limit-point (LP)

N.B. z 7→√z Yν

(z√λ)∼z→0 z

−ν+ 12 is in L2(0, c) only if ν ∈ [0, 1).

The endpoint classification restricts the type of boundary conditions that may be appliedto the field solution at those endpoints.


2.2. Boundary conditions

Singular problem. For a boundary-value problem with one or two singular endpoints,regular boundary conditions are no longer valid and need to be generalised.

Wronskian. Given two differentiable functions u, v, the Wronskian is

W [u, v](z).= u(z)v̄′(z)− v̄(z)u′(z) .

The Wronskian has a finite limit at each endpoint, even if singular.

Robin boundary condition. At an endpoint a, for a solution Φ, when1 a is regular (R), takes the form

Φ(a) +AΦ′(a) = 0 , A ∈ R .

2 a is limit-circle (LC), takes the form

W [Φ,Ψ1](a) +AW [Φ,Ψ2](a) = 0 , A ∈ R ,

where {Ψ1,Ψ2} is a linearly independent basis of solutions.

3 a is limit-point (LP), is not required.



Singular problem. For a boundary-value problem with one or two singular endpoints,regular boundary conditions are no longer valid and need to be generalised.

Wronskian. Given two differentiable functions u, v, the Wronskian is

W [u, v](z).= u(z)v̄′(z)− v̄(z)u′(z) .

The Wronskian has a finite limit at each endpoint, even if singular.

Robin boundary condition. At an endpoint a, for a solution Φ, when1 a is regular (R), takes the form

Φ(a) +AΦ′(a) = 0 , A ∈ R .

2 a is limit-circle (LC), takes the form

W [Φ,Ψ1](a) +AW [Φ,Ψ2](a) = 0 , A ∈ R ,

where {Ψ1,Ψ2} is a linearly independent basis of solutions.

3 a is limit-point (LP), is not required.



Basis of linearly independent solutions: {Ψ1,Ψ2}

Ψ1(z) =√z Jν

(z√λ), Ψ2(z) =

√z Yν

(z√λ).

ν.= 1

2

√1 + 4m2 Classification Boundary condition at z = 0

ν = 12 Regular (R) Φ̂k(0) +A Φ̂′k(0) = 0

ν ∈ [0, 1), ν 6= 12 Limit-circle (LC) W

[Φ̂k,Ψ1

](0) +BW

[Φ̂k,Ψ2

](0) = 0

ν ∈ [1,∞) Limit-point (LP) Not required

In general, we can write the solution as

Φ̂k(z) = N[√

z Jν(z√λ)

+A√z Yν

(z√λ)],

{A ∈ R , ν ∈ [0, 1) ,

A = 0 , ν ∈ [1,∞) .

N.B. For these boundary conditions (when needed), the operator − d2

dz2+ m2

z2is

essentially self-adjoint on C∞0 (0,∞). The study of the existence of the self-adjointextensions to this operator was done by Wald & Ishibashi (2004).

Outline




4 Conclusions

3. Two-point function and Hadamard condition

3.1. Two point-function

Quantum state. We investigate if one can define a quantum state which obeys theaforementioned boundary conditions and which satisfies a natural analogue of theHadamard condition for globally hyperbolic spacetimes.

Green’s function. Two linearly independent solutions to (�x −m2)G(x, x′) = 0:

G1(x, x′) =

F

(d2 + ν, 12 + ν; 1 + 2ν;

[cosh

(√2σ2

)]−2)[cosh

(√2σ2

)] d2+ν

,

G2(x, x′) =

F

(d2 − ν,

12 − ν; 1− 2ν;

[cosh

(√2σ2

)]−2)[cosh

(√2σ2

)] d2−ν

,

where σ is the half-squared geodesic distance in AdSd+1, ν > 0 and F is thehypergeometric function.


3.1. Two point-function

These solutions can also be obtained by integration over the field mode solutions:

G1(x, x′) =

F

(d2 + ν, 12 + ν; 1 + 2ν;

[cosh

(√2σ2

)]−2)[cosh

(√2σ2

)] d2+ν

∝∫Rd

ddk eik·(x−x′) (zz′)

d2 Jν

(z√λ)Jν(z′√λ),

which is the Dirichlet-type two-point function,

G2(x, x′) =

F

(d2 − ν,

12 − ν; 1− 2ν;

[cosh

(√2σ2

)]−2)[cosh

(√2σ2

)] d2−ν

∝∫Rd

ddk eik·(x−x′) (zz′)

d2 J−ν

(z√λ)J−ν

(z′√λ),

which is the Neumann-type two-point function (for ν ∈ (0, 1)).


3.2. Hadamard condition

Expansion around σ = 0. In the case of d+ 1 = 4 spacetime dimensions,

G1(x, x′) =

F

(32 + ν, 12 + ν; 1 + 2ν;

[cosh

(√2σ2

)]−2)[cosh

(√2σ2

)] 32+ν

∝ 1

σ+

1

2

(ν2 − 1

4

)log(σ) +O(σ0)

∝ H(x, x′) +O(σ0) ,

where H(x, x′) is the Hadamard parametrix.

The same can be shown to be true for G2 and for other spacetime dimensions.

Local Hadamard condition. The state satisfies the local Hadamard condition. SinceAdS is not globally hyperbolic, one cannot immediately conclude from this that it alsosatisfies the global Hadamard condition.



Expansion around σ(−) = 0. Further expanding,

G1(x, x′)−H(x, x′) ∝ 1

σ(−)+

1

2

(ν2 − 1

4

)log(σ(−)) +O

((σ(−))0

)∝ H(−)(x, x′) +O(σ0) ,

where σ(−)(x, z;x′, z′) .= σ(x,−z;x′, z′).

The two-point function is thus singular at σ = 0 and σ(−) = 0.

−z0

x(−)

z0

x

z

tx′



Wavefront set. The wavefront set of the two-point function is given by

WF (G) ={

(x, k;x′, k′) ∈ T ∗(R1,d × R1,d) \ {0} : (x, k) ∼± (x′, k′), k . 0}

∼±: ∃ null geodesics γ, γ(−) : [0, 1]→ R1,d with

γ(0) = x = (x, z), γ(−)(0) = x(−) = (x,−z) and γ(1) = x′;k = (kx, kz) (k(−) = (kx,−kz)) is coparallel to γ (γ(−)) at 0;

−k′ is the parallel transport of k (k(−)) along γ (γ(−)) at 1;

k . 0: k is future-directed.

−z0

x(−)

z0

x

z

t x′

kk(−)

−k′

Hadamard condition. We say that a quantum state on AdS for which the wavefrontset of its two-point function is as above is an Hadamard state.

Outline




4 Conclusions

4. Conclusions

We studied the classical and quantum field theory of a massive scalar field on AdSon a more mathematical rigorous way. We treated the classical dynamics as asingular Sturm-Liouville problem, determining all the suitable boundary conditionsat infinity, which only depend on the mass of the field.

We obtained the two-point functions for states obeying these boundary conditionsand showed that, besides the usual singularity at the coincidence limit, there existsonly one extra singularity given by the method of images, independently of the massof the field. We call such states Hadamard states.

Next steps:

extend this formalism to a larger class of spacetimes with boundaries;

relate the states constructed in AdS to states on an QFT at the boundary in order toconstruct Hadamard states in asymptotically AdS spacetimes.

THANK YOU FOR YOUR ATTENTION!

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Anti-de Sitter, scalar fields and boundary conditions