anti-de sitter, scalar fields and boundary conditions
TRANSCRIPT
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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
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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.
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1 AdS & Poincaré domain
2 Scalar field equation and boundary conditions
3 Two-point function and Hadamard condition
4 Conclusions
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Outline
1 AdS & Poincaré domain
2 Scalar field equation and boundary conditions
3 Two-point function and Hadamard condition
4 Conclusions
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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.
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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.
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Outline
1 AdS & Poincaré domain
2 Scalar field equation and boundary conditions
3 Two-point function and Hadamard condition
4 Conclusions
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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.
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2. Scalar field equation and boundary conditions
2.1. Field equation as a Sturm-Liouville equation
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.
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2. Scalar field equation and boundary conditions
2.1. Field equation as a Sturm-Liouville equation
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.
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2. Scalar field equation and boundary conditions
2.1. Field equation as a Sturm-Liouville equation
(− 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.
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2. Scalar field equation and boundary conditions
2.1. Field equation as a Sturm-Liouville equation
(− 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.
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2. Scalar field equation and boundary conditions
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.
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2. Scalar field equation and boundary conditions
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.
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2. Scalar field equation and boundary conditions
2.2. Boundary conditions
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).
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Outline
1 AdS & Poincaré domain
2 Scalar field equation and boundary conditions
3 Two-point function and Hadamard condition
4 Conclusions
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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.
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3. Two-point function and Hadamard condition
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)).
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3. Two-point function and Hadamard condition
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.
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3. Two-point function and Hadamard condition
3.2. 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′
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3. Two-point function and Hadamard condition
3.2. Hadamard condition
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.
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Outline
1 AdS & Poincaré domain
2 Scalar field equation and boundary conditions
3 Two-point function and Hadamard condition
4 Conclusions
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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.
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