AdS Space And Thermal Correlatorspinakib/presentation.pdf · 1 Introduction Motivation The AdS/CFT...
Transcript of AdS Space And Thermal Correlatorspinakib/presentation.pdf · 1 Introduction Motivation The AdS/CFT...
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
AdS Space And Thermal Correlators
Pinaki Banerjee
The Institute of Mathematical Sciences
July 3, 2012
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 Introduction
MotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivation
The AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondence
Brief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFT
Minkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski space
Sample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS space
Euclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean space
Difficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Outline
1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space
2 Thermal Correlators in QFTMinkowski spaceSample calculations for (0+1)d QFT
3 Thermal Correlators in AdS spaceEuclidean spaceDifficulties in Minkowski space
4 Minkowski Space Correlators : prescription and sample calculations
5 Conclusion and frontiers
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionMotivation
The idea of gauge/gravity duality presents the most beautiful linkbetween string theory and our observable world.
Historically it came out of string theory. But in the past few years thisduality has proven its independent existence as an effectivedescription of strongly-interacting quantum systems.
The AdS/CFT correspondence is becoming the most promisingtoolkit of condense matter physicists to understand some stronglycoupled systems such as real-time, finite temperature behavior ofstrongly interacting quantum systems, especially those near quantumcritical points.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionMotivation
The idea of gauge/gravity duality presents the most beautiful linkbetween string theory and our observable world.
Historically it came out of string theory. But in the past few years thisduality has proven its independent existence as an effectivedescription of strongly-interacting quantum systems.
The AdS/CFT correspondence is becoming the most promisingtoolkit of condense matter physicists to understand some stronglycoupled systems such as real-time, finite temperature behavior ofstrongly interacting quantum systems, especially those near quantumcritical points.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionMotivation
The idea of gauge/gravity duality presents the most beautiful linkbetween string theory and our observable world.
Historically it came out of string theory. But in the past few years thisduality has proven its independent existence as an effectivedescription of strongly-interacting quantum systems.
The AdS/CFT correspondence is becoming the most promisingtoolkit of condense matter physicists to understand some stronglycoupled systems such as real-time, finite temperature behavior ofstrongly interacting quantum systems, especially those near quantumcritical points.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionThe AdS/CFT correspondence
Partition function of Gravity ≡ Partition function of QFT
The statement of the duality is following :
⟨exp
(∫Sd φ
i0O)⟩
CFT= ZQG (φi0) (1)
This is in Euclidean signature.
ZQG (φi0) is the partition function of Quantum Gravity.
Boundary conditions: φi goes to φi0 on the boundary.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionThe AdS/CFT correspondence
Partition function of Gravity ≡ Partition function of QFT
The statement of the duality is following :
⟨exp
(∫Sd φ
i0O)⟩
CFT= ZQG (φi0) (1)
This is in Euclidean signature.
ZQG (φi0) is the partition function of Quantum Gravity.
Boundary conditions: φi goes to φi0 on the boundary.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionThe AdS/CFT correspondence
Partition function of Gravity ≡ Partition function of QFT
The statement of the duality is following :
⟨exp
(∫Sd φ
i0O)⟩
CFT= ZQG (φi0) (1)
This is in Euclidean signature.
ZQG (φi0) is the partition function of Quantum Gravity.
Boundary conditions: φi goes to φi0 on the boundary.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionThe AdS/CFT correspondence
Partition function of Gravity ≡ Partition function of QFT
The statement of the duality is following :
⟨exp
(∫Sd φ
i0O)⟩
CFT= ZQG (φi0) (1)
This is in Euclidean signature.
ZQG (φi0) is the partition function of Quantum Gravity.
Boundary conditions: φi goes to φi0 on the boundary.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionThe AdS/CFT correspondence
Partition function of Gravity ≡ Partition function of QFT
The statement of the duality is following :
⟨exp
(∫Sd φ
i0O)⟩
CFT= ZQG (φi0) (1)
This is in Euclidean signature.
ZQG (φi0) is the partition function of Quantum Gravity.
Boundary conditions:
φi goes to φi0 on the boundary.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionThe AdS/CFT correspondence
Partition function of Gravity ≡ Partition function of QFT
The statement of the duality is following :
⟨exp
(∫Sd φ
i0O)⟩
CFT= ZQG (φi0) (1)
This is in Euclidean signature.
ZQG (φi0) is the partition function of Quantum Gravity.
Boundary conditions: φi goes to φi0 on the boundary.
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti de Sitter space is a maximally symmetric space of Lorentziansignature (−,+,+, ...,+), but of constant negative curvature .
Some Quadric surfaces :Sphere :
d+1∑i=1
X 2i = R2 (2)
Hyperboloid :
d∑i=1
X 2i − U2 = ±R2 (3)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti de Sitter space is a maximally symmetric space of Lorentziansignature (−,+,+, ...,+), but of constant negative curvature .
Some Quadric surfaces :
Sphere :
d+1∑i=1
X 2i = R2 (2)
Hyperboloid :
d∑i=1
X 2i − U2 = ±R2 (3)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti de Sitter space is a maximally symmetric space of Lorentziansignature (−,+,+, ...,+), but of constant negative curvature .
Some Quadric surfaces :Sphere :
d+1∑i=1
X 2i = R2 (2)
Hyperboloid :
d∑i=1
X 2i − U2 = ±R2 (3)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti de Sitter space is a maximally symmetric space of Lorentziansignature (−,+,+, ...,+), but of constant negative curvature .
Some Quadric surfaces :Sphere :
d+1∑i=1
X 2i = R2 (2)
Hyperboloid :
d∑i=1
X 2i − U2 = ±R2 (3)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Hyperbolic and de Sitter space :
ds2 =d∑
i=1
dX 2i − dU2 (4)
d∑i=1
X 2i − U2 = ∓R2 (5)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Hyperbolic and de Sitter space :
ds2 =d∑
i=1
dX 2i − dU2 (4)
d∑i=1
X 2i − U2 = ∓R2 (5)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Anti-de Sitter space :
d−1∑i=1
X 2i − U2 − V 2 = −R2 (6)
ds2 =d−1∑i=1
dX 2i − dU2 − dV 2 (7)
The symmetry group : SO(2,d-1)Allows closed time-like curveTopology : AdSd → Rd−1 ⊗ S1 ; dSd → Sd−1 ⊗ R1
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Anti-de Sitter space :
d−1∑i=1
X 2i − U2 − V 2 = −R2 (6)
ds2 =d−1∑i=1
dX 2i − dU2 − dV 2 (7)
The symmetry group : SO(2,d-1)Allows closed time-like curveTopology : AdSd → Rd−1 ⊗ S1 ; dSd → Sd−1 ⊗ R1
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Anti-de Sitter space :
d−1∑i=1
X 2i − U2 − V 2 = −R2 (6)
ds2 =d−1∑i=1
dX 2i − dU2 − dV 2 (7)
The symmetry group : SO(2,d-1)Allows closed time-like curveTopology : AdSd → Rd−1 ⊗ S1 ; dSd → Sd−1 ⊗ R1
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Anti-de Sitter space :
d−1∑i=1
X 2i − U2 − V 2 = −R2 (6)
ds2 =d−1∑i=1
dX 2i − dU2 − dV 2 (7)
The symmetry group : SO(2,d-1)
Allows closed time-like curveTopology : AdSd → Rd−1 ⊗ S1 ; dSd → Sd−1 ⊗ R1
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Anti-de Sitter space :
d−1∑i=1
X 2i − U2 − V 2 = −R2 (6)
ds2 =d−1∑i=1
dX 2i − dU2 − dV 2 (7)
The symmetry group : SO(2,d-1)Allows closed time-like curve
Topology : AdSd → Rd−1 ⊗ S1 ; dSd → Sd−1 ⊗ R1
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Some Quadric surfaces :Anti-de Sitter space :
d−1∑i=1
X 2i − U2 − V 2 = −R2 (6)
ds2 =d−1∑i=1
dX 2i − dU2 − dV 2 (7)
The symmetry group : SO(2,d-1)Allows closed time-like curveTopology : AdSd → Rd−1 ⊗ S1 ; dSd → Sd−1 ⊗ R1
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti-de Sitter space in different co-ordinates :
Global co-ordinates :
U = R cosh ρ sin τ ; V = R cosh ρ cos τ
X1 = R sinh ρ cosφ ; X2 = R sinh ρ sinφ
ds2 = R2[− cosh2 dτ2 + dρ2 + sinh2 ρdφ2] (8)
The change of co-ordinate , tan θ = sinh ρ
ds2d =
R2
cos2 θ[−dτ2 + dθ2 + sin2 θd~Ω
2
d−2] (9)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti-de Sitter space in different co-ordinates :
Global co-ordinates :
U = R cosh ρ sin τ ; V = R cosh ρ cos τ
X1 = R sinh ρ cosφ ; X2 = R sinh ρ sinφ
ds2 = R2[− cosh2 dτ2 + dρ2 + sinh2 ρdφ2] (8)
The change of co-ordinate , tan θ = sinh ρ
ds2d =
R2
cos2 θ[−dτ2 + dθ2 + sin2 θd~Ω
2
d−2] (9)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti-de Sitter space in different co-ordinates :
Global co-ordinates :
U = R cosh ρ sin τ ; V = R cosh ρ cos τ
X1 = R sinh ρ cosφ ; X2 = R sinh ρ sinφ
ds2 = R2[− cosh2 dτ2 + dρ2 + sinh2 ρdφ2] (8)
The change of co-ordinate , tan θ = sinh ρ
ds2d =
R2
cos2 θ[−dτ2 + dθ2 + sin2 θd~Ω
2
d−2] (9)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti-de Sitter space in different co-ordinates :
Global co-ordinates :
U = R cosh ρ sin τ ; V = R cosh ρ cos τ
X1 = R sinh ρ cosφ ; X2 = R sinh ρ sinφ
ds2 = R2[− cosh2 dτ2 + dρ2 + sinh2 ρdφ2] (8)
The change of co-ordinate , tan θ = sinh ρ
ds2d =
R2
cos2 θ[−dτ2 + dθ2 + sin2 θd~Ω
2
d−2] (9)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Anti-de Sitter space in different co-ordinates :
Global co-ordinates :
U = R cosh ρ sin τ ; V = R cosh ρ cos τ
X1 = R sinh ρ cosφ ; X2 = R sinh ρ sinφ
ds2 = R2[− cosh2 dτ2 + dρ2 + sinh2 ρdφ2] (8)
The change of co-ordinate , tan θ = sinh ρ
ds2d =
R2
cos2 θ[−dτ2 + dθ2 + sin2 θd~Ω
2
d−2] (9)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Poincare Co-ordinates :
In this coordinates AdS metric takes the form
ds2 =R2
z2dz2 + (dx)2 − dt2 (10)
Here , z behaves as radial coordinate and the AdS space in two regions ,depending on whether z > 0 or z < 0 . These are known as Poincarecharts .
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Poincare Co-ordinates :
In this coordinates AdS metric takes the form
ds2 =R2
z2dz2 + (dx)2 − dt2 (10)
Here , z behaves as radial coordinate and the AdS space in two regions ,depending on whether z > 0 or z < 0 . These are known as Poincarecharts .
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
IntroductionBrief review of AdS space
Poincare Co-ordinates :
In this coordinates AdS metric takes the form
ds2 =R2
z2dz2 + (dx)2 − dt2 (10)
Here , z behaves as radial coordinate and the AdS space in two regions ,depending on whether z > 0 or z < 0 . These are known as Poincarecharts .
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Thermal Correlators in QFTMikowski space
O → local, Bosonic operator in a finite temperature QFT .
GR(k) = −i∫
d4xe−ik.xθ(t)〈[O(x), O(0)]〉 (11)
GA(k) = i
∫d4xe−ik.xθ(−t)〈[O(x), O(0)]〉 (12)
GF (k) = −i∫
d4xe−ik.x〈|TO(x)O(0)|〉 (13)
G (k) =1
2
∫d4xe−ik.x〈O(x)O(0) + O(0)O(x)〉 (14)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Thermal Correlators in QFTMikowski space
O → local, Bosonic operator in a finite temperature QFT .
GR(k) = −i∫
d4xe−ik.xθ(t)〈[O(x), O(0)]〉 (11)
GA(k) = i
∫d4xe−ik.xθ(−t)〈[O(x), O(0)]〉 (12)
GF (k) = −i∫
d4xe−ik.x〈|TO(x)O(0)|〉 (13)
G (k) =1
2
∫d4xe−ik.x〈O(x)O(0) + O(0)O(x)〉 (14)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Thermal Correlators in QFTMikowski space
O → local, Bosonic operator in a finite temperature QFT .
GR(k) = −i∫
d4xe−ik.xθ(t)〈[O(x), O(0)]〉 (11)
GA(k) = i
∫d4xe−ik.xθ(−t)〈[O(x), O(0)]〉 (12)
GF (k) = −i∫
d4xe−ik.x〈|TO(x)O(0)|〉 (13)
G (k) =1
2
∫d4xe−ik.x〈O(x)O(0) + O(0)O(x)〉 (14)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Thermal Correlators in QFTMikowski space
O → local, Bosonic operator in a finite temperature QFT .
GR(k) = −i∫
d4xe−ik.xθ(t)〈[O(x), O(0)]〉 (11)
GA(k) = i
∫d4xe−ik.xθ(−t)〈[O(x), O(0)]〉 (12)
GF (k) = −i∫
d4xe−ik.x〈|TO(x)O(0)|〉 (13)
G (k) =1
2
∫d4xe−ik.x〈O(x)O(0) + O(0)O(x)〉 (14)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Thermal Correlators in QFTSample calculations for (0+1)d QFT
T = 0:
GF (ω) =
[1
ω2 − ω20 + iε
](15)
GR,A(ω) =1
ω2 − ω20 ∓ sgn(ω)iε
(16)
T 6= 0:
GF (ω) =1
(1− e−βω0)
1
(ω2 − ω20 + iε)
+e−βω0
(ω2 − ω20 − iε)
(17)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Thermal Correlators in QFTSample calculations for (0+1)d QFT
T = 0:
GF (ω) =
[1
ω2 − ω20 + iε
](15)
GR,A(ω) =1
ω2 − ω20 ∓ sgn(ω)iε
(16)
T 6= 0:
GF (ω) =1
(1− e−βω0)
1
(ω2 − ω20 + iε)
+e−βω0
(ω2 − ω20 − iε)
(17)
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Thermal Correlators in QFTSample calculations for (0+1)d QFT
T = 0:
GF (ω) =
[1
ω2 − ω20 + iε
](15)
GR,A(ω) =1
ω2 − ω20 ∓ sgn(ω)iε
(16)
T 6= 0:
GF (ω) =1
(1− e−βω0)
1
(ω2 − ω20 + iε)
+e−βω0
(ω2 − ω20 − iε)
(17)
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Conclusion and frontiers
Thermal Correlators in QFTSample calculations for (0+1)d QFT
T = 0:
GF (ω) =
[1
ω2 − ω20 + iε
](15)
GR,A(ω) =1
ω2 − ω20 ∓ sgn(ω)iε
(16)
T 6= 0:
GF (ω) =1
(1− e−βω0)
1
(ω2 − ω20 + iε)
+e−βω0
(ω2 − ω20 − iε)
(17)
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Thermal Correlators in QFTSample calculations for (0+1)d QFT
T = 0:
GF (ω) =
[1
ω2 − ω20 + iε
](15)
GR,A(ω) =1
ω2 − ω20 ∓ sgn(ω)iε
(16)
T 6= 0:
GF (ω) =1
(1− e−βω0)
1
(ω2 − ω20 + iε)
+e−βω0
(ω2 − ω20 − iε)
(17)
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Thermal Correlators in AdS spaceEuclidean space
N =4 SYM theory and classical gravity (SUGRA) on AdS5 × S5 .
ds2 =R2
z2(dτ2 + dx2 + dz2) + R2d~Ω5
2(18)
⟨e∫∂M φ0O
⟩= e−Scl [φ], (19)
To study thermal field theory metric will be a non-extremal one ,
ds2 =R2
z2
(f (z)dτ2 + dx2 +
dz2
f (z)
)+ R2d~Ω5
2(20)
f (z) = 1− z4/z4H ; zH = (πT )−1 ; τ ∼ τ + T−1 & z = [0, zH ]
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Thermal Correlators in AdS spaceEuclidean space
N =4 SYM theory and classical gravity (SUGRA) on AdS5 × S5 .
ds2 =R2
z2(dτ2 + dx2 + dz2) + R2d~Ω5
2(18)
⟨e∫∂M φ0O
⟩= e−Scl [φ], (19)
To study thermal field theory metric will be a non-extremal one ,
ds2 =R2
z2
(f (z)dτ2 + dx2 +
dz2
f (z)
)+ R2d~Ω5
2(20)
f (z) = 1− z4/z4H ; zH = (πT )−1 ; τ ∼ τ + T−1 & z = [0, zH ]
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Thermal Correlators in AdS spaceEuclidean space
N =4 SYM theory and classical gravity (SUGRA) on AdS5 × S5 .
ds2 =R2
z2(dτ2 + dx2 + dz2) + R2d~Ω5
2(18)
⟨e∫∂M φ0O
⟩= e−Scl [φ], (19)
To study thermal field theory metric will be a non-extremal one ,
ds2 =R2
z2
(f (z)dτ2 + dx2 +
dz2
f (z)
)+ R2d~Ω5
2(20)
f (z) = 1− z4/z4H ; zH = (πT )−1 ; τ ∼ τ + T−1 & z = [0, zH ]
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
The Minkowski analog of the AdS/CFT Correspondence is⟨e i
∫∂M φ0O
⟩= e iScl [φ] (21)
For any curved (d+1) dimension the action of scalar field reads :
S =
∫ √−gdd+1x
[DµφDµφ+ m2φ2)
](22)
S = K
∫ √−gd4x
∫dz[DA(φDAφ)− φDAD
Aφ+ m2φ2)]
(23)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
The Minkowski analog of the AdS/CFT Correspondence is⟨e i
∫∂M φ0O
⟩= e iScl [φ] (21)
For any curved (d+1) dimension the action of scalar field reads :
S =
∫ √−gdd+1x
[DµφDµφ+ m2φ2)
](22)
S = K
∫ √−gd4x
∫dz[DA(φDAφ)− φDAD
Aφ+ m2φ2)]
(23)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
The Minkowski analog of the AdS/CFT Correspondence is⟨e i
∫∂M φ0O
⟩= e iScl [φ] (21)
For any curved (d+1) dimension the action of scalar field reads :
S =
∫ √−gdd+1x
[DµφDµφ+ m2φ2)
](22)
S = K
∫ √−gd4x
∫dz[DA(φDAφ)− φDAD
Aφ+ m2φ2)]
(23)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
S = K
∫ √−gd4x
∫dz [−φ(−m2)φ︸ ︷︷ ︸
SEOM
] + K
∫ √−gd4x
∫dz [DA(φDAφ)]︸ ︷︷ ︸
SBoundary
1√−g ∂z(
√−gg zz∂zφ) + gµν∂µ∂νφ)−m2φ = 0 (24)
φ(z , x) =
∫d4k
(2π)4e ik.x fk(z)φ0(k), (25)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
S = K
∫ √−gd4x
∫dz [−φ(−m2)φ︸ ︷︷ ︸
SEOM
] + K
∫ √−gd4x
∫dz [DA(φDAφ)]︸ ︷︷ ︸
SBoundary
1√−g ∂z(
√−gg zz∂zφ) + gµν∂µ∂νφ)−m2φ = 0 (24)
φ(z , x) =
∫d4k
(2π)4e ik.x fk(z)φ0(k), (25)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
S = K
∫ √−gd4x
∫dz [−φ(−m2)φ︸ ︷︷ ︸
SEOM
] + K
∫ √−gd4x
∫dz [DA(φDAφ)]︸ ︷︷ ︸
SBoundary
1√−g ∂z(
√−gg zz∂zφ) + gµν∂µ∂νφ)−m2φ = 0 (24)
φ(z , x) =
∫d4k
(2π)4e ik.x fk(z)φ0(k), (25)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
φ0(k) is determined by the boundary condition ,
φ(zB , x) =
∫d4k
(2π)4e ik.xφ0(k) ; fk(zB) = 1. (26)
Now substituting it into the EOM we get ,
1√−g ∂z(
√−gg zz∂z fk)− (gµνkµkν + m2)fk = 0 (27)
Boundary condition on fk :
1 fk(zB)=1 , and2 Satisfies the incoming wave boundary condition at horizon (z = zH) .
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
φ0(k) is determined by the boundary condition ,
φ(zB , x) =
∫d4k
(2π)4e ik.xφ0(k) ; fk(zB) = 1. (26)
Now substituting it into the EOM we get ,
1√−g ∂z(
√−gg zz∂z fk)− (gµνkµkν + m2)fk = 0 (27)
Boundary condition on fk :
1 fk(zB)=1 , and2 Satisfies the incoming wave boundary condition at horizon (z = zH) .
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
φ0(k) is determined by the boundary condition ,
φ(zB , x) =
∫d4k
(2π)4e ik.xφ0(k) ; fk(zB) = 1. (26)
Now substituting it into the EOM we get ,
1√−g ∂z(
√−gg zz∂z fk)− (gµνkµkν + m2)fk = 0 (27)
Boundary condition on fk :
1 fk(zB)=1 , and
2 Satisfies the incoming wave boundary condition at horizon (z = zH) .
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
φ0(k) is determined by the boundary condition ,
φ(zB , x) =
∫d4k
(2π)4e ik.xφ0(k) ; fk(zB) = 1. (26)
Now substituting it into the EOM we get ,
1√−g ∂z(
√−gg zz∂z fk)− (gµνkµkν + m2)fk = 0 (27)
Boundary condition on fk :
1 fk(zB)=1 , and2 Satisfies the incoming wave boundary condition at horizon (z = zH) .
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
SBoundary = K
∫ √−gd4x
∫dz [DA(φDAφ)]
= K
∫ √−g d4xφg zz(∂zφ)
∣∣∣∣zHzB
Now substituting the expression for φ we get ,
SBoundary =
∫d4k
(2π)4
φ0(−k)F (k , z)φ0(k)
∣∣∣∣zHzB
(28)
where
F (k , z) = K√−gg zz f−k(z)∂z fk(z). (29)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
SBoundary = K
∫ √−gd4x
∫dz [DA(φDAφ)]
= K
∫ √−g d4xφg zz(∂zφ)
∣∣∣∣zHzB
Now substituting the expression for φ we get ,
SBoundary =
∫d4k
(2π)4
φ0(−k)F (k , z)φ0(k)
∣∣∣∣zHzB
(28)
where
F (k , z) = K√−gg zz f−k(z)∂z fk(z). (29)
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Thermal Correlators in AdS spaceDifficulties in Minkowski space
The Green’s function is ,
G (k) = −F (k , z)
∣∣∣∣zHzB
−F (−k, z)
∣∣∣∣zHzB
(30)
The problem with this Green’s function is , it is completely real. Butretarded Green’s functions are complex in general.
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Prescription for Minkowski Space CorrelatorsRecipe
GR(k) = −2F (k, z)
∣∣∣∣zB
(31)
1 Find a solution to the (27) with following properties :
It equals to 1 at boundary z = zB ;time-like momenta : It satisfies incoming wave boundary condition athorizon .space-like momenta : The solution is regular at horizon .
2 The retarded Green’s function is given by G = −2F∂M . (at z = zB)
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Prescription for Minkowski Space CorrelatorsRecipe
GR(k) = −2F (k, z)
∣∣∣∣zB
(31)
1 Find a solution to the (27) with following properties :
It equals to 1 at boundary z = zB ;time-like momenta : It satisfies incoming wave boundary condition athorizon .space-like momenta : The solution is regular at horizon .
2 The retarded Green’s function is given by G = −2F∂M . (at z = zB)
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Prescription for Minkowski Space CorrelatorsRecipe
GR(k) = −2F (k, z)
∣∣∣∣zB
(31)
1 Find a solution to the (27) with following properties :
It equals to 1 at boundary z = zB ;
time-like momenta : It satisfies incoming wave boundary condition athorizon .space-like momenta : The solution is regular at horizon .
2 The retarded Green’s function is given by G = −2F∂M . (at z = zB)
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Prescription for Minkowski Space CorrelatorsRecipe
GR(k) = −2F (k, z)
∣∣∣∣zB
(31)
1 Find a solution to the (27) with following properties :
It equals to 1 at boundary z = zB ;time-like momenta : It satisfies incoming wave boundary condition athorizon .
space-like momenta : The solution is regular at horizon .
2 The retarded Green’s function is given by G = −2F∂M . (at z = zB)
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Prescription for Minkowski Space CorrelatorsRecipe
GR(k) = −2F (k, z)
∣∣∣∣zB
(31)
1 Find a solution to the (27) with following properties :
It equals to 1 at boundary z = zB ;time-like momenta : It satisfies incoming wave boundary condition athorizon .space-like momenta : The solution is regular at horizon .
2 The retarded Green’s function is given by G = −2F∂M . (at z = zB)
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Prescription for Minkowski Space CorrelatorsRecipe
GR(k) = −2F (k, z)
∣∣∣∣zB
(31)
1 Find a solution to the (27) with following properties :
It equals to 1 at boundary z = zB ;time-like momenta : It satisfies incoming wave boundary condition athorizon .space-like momenta : The solution is regular at horizon .
2 The retarded Green’s function is given by G = −2F∂M . (at z = zB)
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Prescription for Minkowski Space CorrelatorsSample calculation
For Euclidean correlator of a CFT operator O ,
〈e∫∂M φ0O〉 = e−SE [φ] (32)
Euclidean AdS5 metric is
ds25 =
R2
z2(dz2 + dx2) (33)
The action of massive scalar field on this background is ,
SE = K
∫d4x
zH=∞∫zB=ε
dz√g[g zz(∂zφ)2 + gµν(∂µφ)(∂νφ) + m2φ2
](34)
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Prescription for Minkowski Space CorrelatorsSample calculation
For Euclidean correlator of a CFT operator O ,
〈e∫∂M φ0O〉 = e−SE [φ] (32)
Euclidean AdS5 metric is
ds25 =
R2
z2(dz2 + dx2) (33)
The action of massive scalar field on this background is ,
SE = K
∫d4x
zH=∞∫zB=ε
dz√g[g zz(∂zφ)2 + gµν(∂µφ)(∂νφ) + m2φ2
](34)
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Prescription for Minkowski Space CorrelatorsSample calculation
For Euclidean correlator of a CFT operator O ,
〈e∫∂M φ0O〉 = e−SE [φ] (32)
Euclidean AdS5 metric is
ds25 =
R2
z2(dz2 + dx2) (33)
The action of massive scalar field on this background is ,
SE = K
∫d4x
zH=∞∫zB=ε
dz√g[g zz(∂zφ)2 + gµν(∂µφ)(∂νφ) + m2φ2
](34)
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Prescription for Minkowski Space CorrelatorsSample calculation
For Euclidean correlator of a CFT operator O ,
〈e∫∂M φ0O〉 = e−SE [φ] (32)
Euclidean AdS5 metric is
ds25 =
R2
z2(dz2 + dx2) (33)
The action of massive scalar field on this background is ,
SE = K
∫d4x
zH=∞∫zB=ε
dz√g[g zz(∂zφ)2 + gµν(∂µφ)(∂νφ) + m2φ2
](34)
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Prescription for Minkowski Space CorrelatorsSample calculation
SE =π3R8
4κ210
∫dz
∫d4xz−3
((∂zφ)2 +
z2
R2(∂iφ)2 +
R2m2
z2φ2
)(35)
SE ∼∫
dz
∫d4k
(2π)4
1
z3(∂z fk)(∂z f−k) + k2fk f−k +
R2m2
z2fk f−kφ0(k)φ0(−k)
f ′′k (z)− 3z f′k(z)−
(k2 + m2R2
z2
)fk(z) = 0 (36)
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Prescription for Minkowski Space CorrelatorsSample calculation
SE =π3R8
4κ210
∫dz
∫d4xz−3
((∂zφ)2 +
z2
R2(∂iφ)2 +
R2m2
z2φ2
)(35)
SE ∼∫
dz
∫d4k
(2π)4
1
z3(∂z fk)(∂z f−k) + k2fk f−k +
R2m2
z2fk f−kφ0(k)φ0(−k)
f ′′k (z)− 3z f′k(z)−
(k2 + m2R2
z2
)fk(z) = 0 (36)
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Prescription for Minkowski Space CorrelatorsSample calculation
SE =π3R8
4κ210
∫dz
∫d4xz−3
((∂zφ)2 +
z2
R2(∂iφ)2 +
R2m2
z2φ2
)(35)
SE ∼∫
dz
∫d4k
(2π)4
1
z3(∂z fk)(∂z f−k) + k2fk f−k +
R2m2
z2fk f−kφ0(k)φ0(−k)
f ′′k (z)− 3z f′k(z)−
(k2 + m2R2
z2
)fk(z) = 0 (36)
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SE =π3R8
4κ210
∫dz
∫d4xz−3
((∂zφ)2 +
z2
R2(∂iφ)2 +
R2m2
z2φ2
)(35)
SE ∼∫
dz
∫d4k
(2π)4
1
z3(∂z fk)(∂z f−k) + k2fk f−k +
R2m2
z2fk f−kφ0(k)φ0(−k)
f ′′k (z)− 3z f′k(z)−
(k2 + m2R2
z2
)fk(z) = 0 (36)
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Prescription for Minkowski Space CorrelatorsSample calculation
Its general solution is ,
φk(z) = Az2Iν(kz) + Bz2I−ν(kz) (37)
The solution is regular at z =∞ and equals to 1 at z = ε , therefore ,
fk(z) =z2Kν(kz)
ε2Kν(kε)(38)
On shell , the action reduces to the boundary term
SE =π3R8
4κ210
∫d4kd4k ′
(2π)8φ0(k)φ0(k ′)F (z , k , k ′)
∣∣∣∣∞ε
(39)
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Its general solution is ,
φk(z) = Az2Iν(kz) + Bz2I−ν(kz) (37)
The solution is regular at z =∞ and equals to 1 at z = ε , therefore ,
fk(z) =z2Kν(kz)
ε2Kν(kε)(38)
On shell , the action reduces to the boundary term
SE =π3R8
4κ210
∫d4kd4k ′
(2π)8φ0(k)φ0(k ′)F (z , k , k ′)
∣∣∣∣∞ε
(39)
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Prescription for Minkowski Space CorrelatorsSample calculation
Its general solution is ,
φk(z) = Az2Iν(kz) + Bz2I−ν(kz) (37)
The solution is regular at z =∞ and equals to 1 at z = ε , therefore ,
fk(z) =z2Kν(kz)
ε2Kν(kε)(38)
On shell , the action reduces to the boundary term
SE =π3R8
4κ210
∫d4kd4k ′
(2π)8φ0(k)φ0(k ′)F (z , k , k ′)
∣∣∣∣∞ε
(39)
Pinaki Banerjee AdS Space And Thermal Correlators
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Prescription for Minkowski Space CorrelatorsSample calculation
The two point function is given by
〈O(k)O(k ′)〉 =Z−1 δ2Z [φ0]
δφ0(k)δφ0(k ′)
∣∣∣∣φ0=0
(40)
=− 2F (z , k , k ′)
∣∣∣∣∞ε
=− (2π)4δ4(k + k ′)π3R8
2κ210
fk ′(z)∂z fk(z)
z3
∣∣∣∣∞ε
Putting the value of fk(z) we get ,
〈O(k)O(k ′)〉 = −π3R8
2κ210
ε2(∆−d)(2π)4δ4(k + k ′)k2ν21−2ν Γ(1− ν)
Γ(ν)+ ...
(41)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
The two point function is given by
〈O(k)O(k ′)〉 =Z−1 δ2Z [φ0]
δφ0(k)δφ0(k ′)
∣∣∣∣φ0=0
(40)
=− 2F (z , k , k ′)
∣∣∣∣∞ε
=− (2π)4δ4(k + k ′)π3R8
2κ210
fk ′(z)∂z fk(z)
z3
∣∣∣∣∞ε
Putting the value of fk(z) we get ,
〈O(k)O(k ′)〉 = −π3R8
2κ210
ε2(∆−d)(2π)4δ4(k + k ′)k2ν21−2ν Γ(1− ν)
Γ(ν)+ ...
(41)Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
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Prescription for Minkowski Space CorrelatorsSample calculation
For integer ∆ , the propagator will be ,
〈O(k)O(k ′)〉 = − (−1)∆
(∆− 3)!
N2
8π2(2π)4δ4(k + k ′)
k2∆−4
22∆−5ln k2 (42)
For massless case (∆ = 4) , we have
〈O(k)O(k ′)〉 = − N2
64π4(2π)4δ4(k + k ′)k4 ln k2 (43)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
For integer ∆ , the propagator will be ,
〈O(k)O(k ′)〉 = − (−1)∆
(∆− 3)!
N2
8π2(2π)4δ4(k + k ′)
k2∆−4
22∆−5ln k2 (42)
For massless case (∆ = 4) , we have
〈O(k)O(k ′)〉 = − N2
64π4(2π)4δ4(k + k ′)k4 ln k2 (43)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
The EOM :
f ′′k (z)− 3
zf ′k(z)− k2fk(z) = 0 (44)
For spacelike momenta , k2 > 0 , we can follow the steps identical tothe Euclidean case.
GR(k) = +N2k4
64π2ln k2 ; k2 > 0 (45)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
The EOM :
f ′′k (z)− 3
zf ′k(z)− k2fk(z) = 0 (44)
For spacelike momenta , k2 > 0 , we can follow the steps identical tothe Euclidean case.
GR(k) = +N2k4
64π2ln k2 ; k2 > 0 (45)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
For timelike momenta , we introduce q =√−k2 .
fk(z) =z2H
(1)2 (qz)
ε2H(1)(qε)ν
if ω > 0 (46)
=z2H
(2)2 (qz)
ε2H(2)(qε)2
if ω < 0 (47)
GR(k) =N2k4
64π2(ln k2 − iπ sgn ω) (48)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
For timelike momenta , we introduce q =√−k2 .
fk(z) =z2H
(1)2 (qz)
ε2H(1)(qε)ν
if ω > 0 (46)
=z2H
(2)2 (qz)
ε2H(2)(qε)2
if ω < 0 (47)
GR(k) =N2k4
64π2(ln k2 − iπ sgn ω) (48)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
More generally ,
GR(k) =N2K 4
64π2
(ln |k2| − iπθ(−k2) sgn ω
)(49)
We can get the Feynman propagator ,
GF (k) =N2K 4
64π2
[ln |k2| − iπθ(−k2)
](50)
we can also get it by Wick rotating the Euclidean correlator ,
GE (kE ) = −N2K 4
E
64π2ln k2
E (51)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
More generally ,
GR(k) =N2K 4
64π2
(ln |k2| − iπθ(−k2) sgn ω
)(49)
We can get the Feynman propagator ,
GF (k) =N2K 4
64π2
[ln |k2| − iπθ(−k2)
](50)
we can also get it by Wick rotating the Euclidean correlator ,
GE (kE ) = −N2K 4
E
64π2ln k2
E (51)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Prescription for Minkowski Space CorrelatorsSample calculation
More generally ,
GR(k) =N2K 4
64π2
(ln |k2| − iπθ(−k2) sgn ω
)(49)
We can get the Feynman propagator ,
GF (k) =N2K 4
64π2
[ln |k2| − iπθ(−k2)
](50)
we can also get it by Wick rotating the Euclidean correlator ,
GE (kE ) = −N2K 4
E
64π2ln k2
E (51)
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Conclusion and frontiers
Previous correlators of SHO are useful...
but ambiguous !
Use better techniques :
Schwinger-Keldysh formalism
GF (ω) =
1ω2−ω2
0+iε+ −i2π
eβω0−1δ(ω2 −m2) 2πie−βω0/2
1−e−βω0δ(ω2 −m2)
2πie−βω0/2
1−e−βω0δ(ω2 −m2) −1
ω2−ω20−iε
+ −i2πeβω0−1
δ(ω2 −m2)
Thermo-field Dynamics !
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Conclusion and frontiers
Previous correlators of SHO are useful... but ambiguous !
Use better techniques :
Schwinger-Keldysh formalism
GF (ω) =
1ω2−ω2
0+iε+ −i2π
eβω0−1δ(ω2 −m2) 2πie−βω0/2
1−e−βω0δ(ω2 −m2)
2πie−βω0/2
1−e−βω0δ(ω2 −m2) −1
ω2−ω20−iε
+ −i2πeβω0−1
δ(ω2 −m2)
Thermo-field Dynamics !
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Conclusion and frontiers
Previous correlators of SHO are useful... but ambiguous !
Use better techniques :
Schwinger-Keldysh formalism
GF (ω) =
1ω2−ω2
0+iε+ −i2π
eβω0−1δ(ω2 −m2) 2πie−βω0/2
1−e−βω0δ(ω2 −m2)
2πie−βω0/2
1−e−βω0δ(ω2 −m2) −1
ω2−ω20−iε
+ −i2πeβω0−1
δ(ω2 −m2)
Thermo-field Dynamics !
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Conclusion and frontiers
Previous correlators of SHO are useful... but ambiguous !
Use better techniques :
Schwinger-Keldysh formalism
GF (ω) =
1ω2−ω2
0+iε+ −i2π
eβω0−1δ(ω2 −m2) 2πie−βω0/2
1−e−βω0δ(ω2 −m2)
2πie−βω0/2
1−e−βω0δ(ω2 −m2) −1
ω2−ω20−iε
+ −i2πeβω0−1
δ(ω2 −m2)
Thermo-field Dynamics !
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Conclusion and frontiers
Previous correlators of SHO are useful... but ambiguous !
Use better techniques :
Schwinger-Keldysh formalism
GF (ω) =
1ω2−ω2
0+iε+ −i2π
eβω0−1δ(ω2 −m2) 2πie−βω0/2
1−e−βω0δ(ω2 −m2)
2πie−βω0/2
1−e−βω0δ(ω2 −m2) −1
ω2−ω20−iε
+ −i2πeβω0−1
δ(ω2 −m2)
Thermo-field Dynamics !
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
Conclusion and frontiers
Previous correlators of SHO are useful... but ambiguous !
Use better techniques :
Schwinger-Keldysh formalism
GF (ω) =
1ω2−ω2
0+iε+ −i2π
eβω0−1δ(ω2 −m2) 2πie−βω0/2
1−e−βω0δ(ω2 −m2)
2πie−βω0/2
1−e−βω0δ(ω2 −m2) −1
ω2−ω20−iε
+ −i2πeβω0−1
δ(ω2 −m2)
Thermo-field Dynamics !
Pinaki Banerjee AdS Space And Thermal Correlators
IntroductionThermal Correlators in QFT
Thermal Correlators in AdS spaceMinkowski Space Correlators : prescription and sample calculations
Conclusion and frontiers
thank you!
Pinaki Banerjee AdS Space And Thermal Correlators