AdS Space And Thermal Correlators Pinaki Banerjee The Institute of - - PowerPoint PPT Presentation

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski 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

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Minkowski space

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Minkowski space Sample calculations for (0+1)d QFT

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Minkowski space Sample calculations for (0+1)d QFT

3 Thermal Correlators in AdS space

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Minkowski space Sample calculations for (0+1)d QFT

3 Thermal Correlators in AdS space

Euclidean space

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Minkowski space Sample calculations for (0+1)d QFT

3 Thermal Correlators in AdS space

Euclidean space Difficulties in Minkowski space

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Minkowski space Sample calculations for (0+1)d QFT

3 Thermal Correlators in AdS space

Euclidean space Difficulties in Minkowski space

4 Minkowski Space Correlators : prescription and sample calculations

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Outline

1 Introduction

Motivation The AdS/CFT correspondence Brief review of AdS space

2 Thermal Correlators in QFT

Minkowski space Sample calculations for (0+1)d QFT

3 Thermal Correlators in AdS space

Euclidean space Difficulties in Minkowski space

4 Minkowski Space Correlators : prescription and sample calculations 5 Conclusion and frontiers

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Motivation

The idea of gauge/gravity duality presents the most beautiful link between string theory and our observable world.

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Motivation

The idea of gauge/gravity duality presents the most beautiful link between string theory and our observable world. Historically it came out of string theory. But in the past few years this duality has proven its independent existence as an effective description of strongly-interacting quantum systems.

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Motivation

The idea of gauge/gravity duality presents the most beautiful link between string theory and our observable world. Historically it came out of string theory. But in the past few years this duality has proven its independent existence as an effective description of strongly-interacting quantum systems. The AdS/CFT correspondence is becoming the most promising toolkit of condense matter physicists to understand some strongly coupled systems such as real-time, finite temperature behavior of strongly interacting quantum systems, especially those near quantum critical points.

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

The AdS/CFT correspondence

Partition function of Gravity ≡ Partition function of QFT

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

The AdS/CFT correspondence

Partition function of Gravity ≡ Partition function of QFT The statement of the duality is following :

• exp
• Sd φi

0O

• CFT = ZQG(φi

0)

(1)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

The AdS/CFT correspondence

Partition function of Gravity ≡ Partition function of QFT The statement of the duality is following :

• exp
• Sd φi

0O

• CFT = ZQG(φi

0)

(1) This is in Euclidean signature.

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

The AdS/CFT correspondence

Partition function of Gravity ≡ Partition function of QFT The statement of the duality is following :

• exp
• Sd φi

0O

• CFT = ZQG(φi

0)

(1) This is in Euclidean signature. ZQG(φi

0) is the partition function of Quantum Gravity.

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

The AdS/CFT correspondence

Partition function of Gravity ≡ Partition function of QFT The statement of the duality is following :

• exp
• Sd φi

0O

• CFT = ZQG(φi

0)

(1) This is in Euclidean signature. ZQG(φi

0) is the partition function of Quantum Gravity.

Boundary conditions:

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

The AdS/CFT correspondence

Partition function of Gravity ≡ Partition function of QFT The statement of the duality is following :

• exp
• Sd φi

0O

• CFT = ZQG(φi

0)

(1) This is in Euclidean signature. ZQG(φi

0) is the partition function of Quantum Gravity.

Boundary conditions: φi goes to φi

0 on the boundary.

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Anti de Sitter space is a maximally symmetric space of Lorentzian signature (−, +, +, ..., +), but of constant negative curvature .

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Anti de Sitter space is a maximally symmetric space of Lorentzian signature (−, +, +, ..., +), but of constant negative curvature . Some Quadric surfaces :

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Anti de Sitter space is a maximally symmetric space of Lorentzian signature (−, +, +, ..., +), but of constant negative curvature . Some Quadric surfaces :

Sphere :

d+1

• i=1

X 2

i = R2

(2)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Anti de Sitter space is a maximally symmetric space of Lorentzian signature (−, +, +, ..., +), but of constant negative curvature . Some Quadric surfaces :

Sphere :

d+1

• i=1

X 2

i = R2

(2) Hyperboloid :

d

• i=1

X 2

i − U2 = ±R2

(3)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Hyperbolic and de Sitter space : ds2 =

d

• i=1

dX 2

i − dU2

(4)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Hyperbolic and de Sitter space : ds2 =

d

• i=1

dX 2

i − dU2

(4)

d

• i=1

X 2

i − U2 = ∓R2

(5)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Anti-de Sitter space :

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Anti-de Sitter space :

d−1

• i=1

X 2

i − U2 − V 2 = −R2

(6)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Anti-de Sitter space :

d−1

• i=1

X 2

i − U2 − V 2 = −R2

(6) ds2 =

d−1

• i=1

dX 2

i − dU2 − dV 2

(7)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Anti-de Sitter space :

d−1

• i=1

X 2

i − U2 − V 2 = −R2

(6) ds2 =

d−1

• i=1

dX 2

i − dU2 − dV 2

(7) The symmetry group : SO(2,d-1)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Anti-de Sitter space :

d−1

• i=1

X 2

i − U2 − V 2 = −R2

(6) ds2 =

d−1

• i=1

dX 2

i − dU2 − dV 2

(7) The symmetry group : SO(2,d-1) Allows closed time-like curve

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Some Quadric surfaces :

Anti-de Sitter space :

d−1

• i=1

X 2

i − U2 − V 2 = −R2

(6) ds2 =

d−1

• i=1

dX 2

i − 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

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Anti-de Sitter space in different co-ordinates :

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Anti-de Sitter space in different co-ordinates :

Global co-ordinates :

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief 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 cos φ ; X2 = R sinh ρ sin φ

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief 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 cos φ ; X2 = R sinh ρ sin φ ds2 = R2[− cosh2 dτ 2 + dρ2 + sinh2 ρdφ2] (8)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief 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 cos φ ; X2 = R sinh ρ sin φ ds2 = R2[− cosh2 dτ 2 + dρ2 + sinh2 ρdφ2] (8) The change of co-ordinate , tan θ = sinh ρ ds2

d =

R2 cos2 θ[−dτ 2 + dθ2 + sin2 θd Ω

2 d−2]

(9)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Poincare Co-ordinates :

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Poincare Co-ordinates : In this coordinates AdS metric takes the form ds2 = R2 z2 {dz2 + (d¯ x)2 − dt2} (10)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Introduction

Brief review of AdS space

Poincare Co-ordinates : In this coordinates AdS metric takes the form ds2 = R2 z2 {dz2 + (d¯ x)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 Poincare charts .

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Mikowski space

ˆ O → local, Bosonic operator in a finite temperature QFT .

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Mikowski space

ˆ O → local, Bosonic operator in a finite temperature QFT . ˜ GR(k) = −i

• d4xe−ik.xθ(t)[ ˆ

O(x), ˆ O(0)] (11) ˜ GA(k) = i

• d4xe−ik.xθ(−t)[ ˆ

O(x), ˆ O(0)] (12)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Mikowski space

ˆ O → local, Bosonic operator in a finite temperature QFT . ˜ GR(k) = −i

• d4xe−ik.xθ(t)[ ˆ

O(x), ˆ O(0)] (11) ˜ GA(k) = i

• d4xe−ik.xθ(−t)[ ˆ

O(x), ˆ O(0)] (12) ˜ GF(k) = −i

• d4xe−ik.x|T{ ˆ

O(x) ˆ O(0)}| (13)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Mikowski space

ˆ O → local, Bosonic operator in a finite temperature QFT . ˜ GR(k) = −i

• d4xe−ik.xθ(t)[ ˆ

O(x), ˆ O(0)] (11) ˜ GA(k) = i

• d4xe−ik.xθ(−t)[ ˆ

O(x), ˆ O(0)] (12) ˜ GF(k) = −i

• d4xe−ik.x|T{ ˆ

O(x) ˆ O(0)}| (13) ˜ G(k) = 1 2

• d4xe−ik.x ˆ

O(x) ˆ O(0) + ˆ O(0) ˆ O(x) (14)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Sample calculations for (0+1)d QFT

T = 0:

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Sample calculations for (0+1)d QFT

T = 0: ˜ GF(ω) =

• 1

ω2 − ω2

0 + iǫ

• (15)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Sample calculations for (0+1)d QFT

T = 0: ˜ GF(ω) =

• 1

ω2 − ω2

0 + iǫ

• (15)

˜ GR,A(ω) = 1 ω2 − ω2

0 ∓ sgn(ω)iǫ

(16)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Sample calculations for (0+1)d QFT

T = 0: ˜ GF(ω) =

• 1

ω2 − ω2

0 + iǫ

• (15)

˜ GR,A(ω) = 1 ω2 − ω2

0 ∓ sgn(ω)iǫ

(16) T = 0:

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in QFT

Sample calculations for (0+1)d QFT

T = 0: ˜ GF(ω) =

• 1

ω2 − ω2

0 + iǫ

• (15)

˜ GR,A(ω) = 1 ω2 − ω2

0 ∓ sgn(ω)iǫ

(16) T = 0: ˜ GF(ω) = 1 (1 − e−βω0)

• 1

(ω2 − ω2

0 + iǫ) +

e−βω0 (ω2 − ω2

0 − iǫ)

• (17)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Euclidean space

N =4 SYM theory and classical gravity (SUGRA) on AdS5 × S5 . ds2 = R2 z2 (dτ 2 + dx2 + dz2) + R2d Ω5

2

(18)

• e
• ∂M φ0O

= e−Scl[φ], (19)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Euclidean space

N =4 SYM theory and classical gravity (SUGRA) on AdS5 × S5 . ds2 = R2 z2 (dτ 2 + 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)dτ 2 + dx2 + dz2

f (z)

• + R2d

Ω5

2

(20)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Euclidean space

N =4 SYM theory and classical gravity (SUGRA) on AdS5 × S5 . ds2 = R2 z2 (dτ 2 + 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)dτ 2 + dx2 + dz2

f (z)

• + R2d

Ω5

2

(20) f (z) = 1 − z4/z4

H ; zH = (πT)−1 ; τ ∼ τ + T −1 & z = [0, zH]

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

The Minkowski analog of the AdS/CFT Correspondence is

• ei
• ∂M φ0O

= eiScl[φ] (21)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

The Minkowski analog of the AdS/CFT Correspondence is

• ei
• ∂M φ0O

= eiScl[φ] (21) For any curved (d+1) dimension the action of scalar field reads : S = √−gdd+1x

• DµφDµφ + m2φ2)
• (22)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

The Minkowski analog of the AdS/CFT Correspondence is

• ei
• ∂M φ0O

= eiScl[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) − φDADAφ + m2φ2)
• (23)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

S = K √−gd4x

• dz[−φ( − m2)φ
• SEOM

] + K √−gd4x

• dz[DA(φDAφ)]
• SBoundary

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

S = K √−gd4x

• dz[−φ( − m2)φ
• SEOM

] + K √−gd4x

• dz[DA(φDAφ)]
• SBoundary

1 √−g ∂z(√−ggzz∂zφ) + gµν∂µ∂νφ) − m2φ = 0

(24)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

S = K √−gd4x

• dz[−φ( − m2)φ
• SEOM

] + K √−gd4x

• dz[DA(φDAφ)]
• SBoundary

1 √−g ∂z(√−ggzz∂zφ) + gµν∂µ∂νφ) − m2φ = 0

(24) φ(z, x) =

• d4k

(2π)4 eik.xfk(z)φ0(k), (25)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

φ0(k) is determined by the boundary condition , φ(zB, x) =

• d4k

(2π)4 eik.xφ0(k) ; fk(zB) = 1. (26)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

φ0(k) is determined by the boundary condition , φ(zB, x) =

• d4k

(2π)4 eik.xφ0(k) ; fk(zB) = 1. (26) Now substituting it into the EOM we get ,

1 √−g ∂z(√−ggzz∂zfk) − (gµνkµkν + m2)fk = 0

(27)

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Thermal Correlators in AdS space

Difficulties in Minkowski space

φ0(k) is determined by the boundary condition , φ(zB, x) =

• d4k

(2π)4 eik.xφ0(k) ; fk(zB) = 1. (26) Now substituting it into the EOM we get ,

1 √−g ∂z(√−ggzz∂zfk) − (gµνkµkν + m2)fk = 0

(27) Boundary condition on fk :

1 fk(zB)=1 , and Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

φ0(k) is determined by the boundary condition , φ(zB, x) =

• d4k

(2π)4 eik.xφ0(k) ; fk(zB) = 1. (26) Now substituting it into the EOM we get ,

1 √−g ∂z(√−ggzz∂zfk) − (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) . Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

SBoundary = K √−gd4x

• dz[DA(φDAφ)]

= K √−g d4x{φgzz(∂zφ)}

• zH

zB

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

SBoundary = K √−gd4x

• dz[DA(φDAφ)]

= K √−g d4x{φgzz(∂zφ)}

• zH

zB

Now substituting the expression for φ we get , SBoundary =

• d4k

(2π)4

• φ0(−k)F(k, z)φ0(k)
• zH

zB

(28) where F(k, z) = K√−ggzzf−k(z)∂zfk(z). (29)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Thermal Correlators in AdS space

Difficulties in Minkowski space

The Green’s function is , ˜ G(k) = −F(k, z)

• zH

zB

− F(−k, z)

• zH

zB

(30) The problem with this Green’s function is , it is completely real. But retarded Green’s functions are complex in general.

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Recipe

˜ GR(k) = −2F(k, z)

• zB

(31)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Recipe

˜ GR(k) = −2F(k, z)

• zB

(31)

1 Find a solution to the (27) with following properties : Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Recipe

˜ 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 ;

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Recipe

˜ 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 at horizon .

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Recipe

˜ 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 at horizon . space-like momenta : The solution is regular at horizon .

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Recipe

˜ 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 at horizon . space-like momenta : The solution is regular at horizon .

2 The retarded Green’s function is given by G = −2F∂M. (at z = zB) Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For Euclidean correlator of a CFT operator O , e

• ∂M φ0O = e−SE [φ]

(32)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For Euclidean correlator of a CFT operator O , e

• ∂M φ0O = e−SE [φ]

(32) Euclidean AdS5 metric is ds2

5 = R2

z2 (dz2 + dx2) (33)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For Euclidean correlator of a CFT operator O , e

• ∂M φ0O = e−SE [φ]

(32) Euclidean AdS5 metric is ds2

5 = R2

z2 (dz2 + dx2) (33) The action of massive scalar field on this background is , SE = K

• d4x

zH=∞

• zB=ǫ

dz√g

• gzz(∂zφ)2 + gµν(∂µφ)(∂νφ) + m2φ2

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For Euclidean correlator of a CFT operator O , e

• ∂M φ0O = e−SE [φ]

(32) Euclidean AdS5 metric is ds2

5 = R2

z2 (dz2 + dx2) (33) The action of massive scalar field on this background is , SE = K

• d4x

zH=∞

• zB=ǫ

dz√g

• gzz(∂zφ)2 + gµν(∂µφ)(∂νφ) + m2φ2

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

SE = π3R8 4κ2

10

• dz
• d4xz−3
• (∂zφ)2 + z2

R2 (∂iφ)2 + R2m2 z2 φ2

• (35)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

SE = π3R8 4κ2

10

• dz
• d4xz−3
• (∂zφ)2 + z2

R2 (∂iφ)2 + R2m2 z2 φ2

• (35)

SE ∼

• dz
• d4k

(2π)4 1 z3 {(∂zfk)(∂zf−k) + k2fkf−k + R2m2 z2 fkf−k}φ0(k)φ0(−k)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

SE = π3R8 4κ2

10

• dz
• d4xz−3
• (∂zφ)2 + z2

R2 (∂iφ)2 + R2m2 z2 φ2

• (35)

SE ∼

• dz
• d4k

(2π)4 1 z3 {(∂zfk)(∂zf−k) + k2fkf−k + R2m2 z2 fkf−k}φ0(k)φ0(−k) f ′′

k (z) − 3 z f ′ k(z) −

• k2 + m2R2

z2

• fk(z) = 0

(36)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

SE = π3R8 4κ2

10

• dz
• d4xz−3
• (∂zφ)2 + z2

R2 (∂iφ)2 + R2m2 z2 φ2

• (35)

SE ∼

• dz
• d4k

(2π)4 1 z3 {(∂zfk)(∂zf−k) + k2fkf−k + R2m2 z2 fkf−k}φ0(k)φ0(−k) f ′′

k (z) − 3 z f ′ k(z) −

• k2 + m2R2

z2

• fk(z) = 0

(36)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

Its general solution is , φk(z) = Az2Iν(kz) + Bz2I−ν(kz) (37)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample 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 = ǫ , therefore , fk(z) = z2Kν(kz) ǫ2Kν(kǫ) (38)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample 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 = ǫ , therefore , fk(z) = z2Kν(kz) ǫ2Kν(kǫ) (38) On shell , the action reduces to the boundary term SE = π3R8 4κ2

10

d4kd4k′ (2π)8 φ0(k)φ0(k′)F(z, k, k′)

• ∞

ǫ

(39)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample 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′)

• ∞

ǫ

= − (2π)4δ4(k + k′)π3R8 2κ2

10

fk′(z)∂zfk(z) z3

• ∞

ǫ

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample 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′)

• ∞

ǫ

= − (2π)4δ4(k + k′)π3R8 2κ2

10

fk′(z)∂zfk(z) z3

• ∞

ǫ

Putting the value of fk(z) we get , O(k)O(k′) = −π3R8 2κ2

10

ǫ2(∆−d)(2π)4δ4(k + k′)k2ν21−2ν Γ(1 − ν) Γ(ν) + ... (41)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For integer ∆ , the propagator will be , O(k)O(k′) = − (−1)∆ (∆ − 3)! N2 8π2 (2π)4δ4(k + k′)k2∆−4 22∆−5 ln k2 (42)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For integer ∆ , the propagator will be , O(k)O(k′) = − (−1)∆ (∆ − 3)! N2 8π2 (2π)4δ4(k + k′)k2∆−4 22∆−5 ln 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

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

The EOM : f ′′

k (z) − 3

z f ′

k(z) − k2fk(z) = 0

(44)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

The EOM : f ′′

k (z) − 3

z f ′

k(z) − k2fk(z) = 0

(44) For spacelike momenta , k2 > 0 , we can follow the steps identical to the Euclidean case. ˜ GR(k) = +N2k4 64π2 ln k2 ; k2 > 0 (45)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For timelike momenta , we introduce q = √ −k2 . fk(z) =z2H(1)

2 (qz)

ǫ2H(1)(qǫ)

ν

if ω > 0 (46) =z2H(2)

2 (qz)

ǫ2H(2)(qǫ)

2

if ω < 0 (47)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

For timelike momenta , we introduce q = √ −k2 . fk(z) =z2H(1)

2 (qz)

ǫ2H(1)(qǫ)

ν

if ω > 0 (46) =z2H(2)

2 (qz)

ǫ2H(2)(qǫ)

2

if ω < 0 (47) ˜ GR(k) = N2k4 64π2 (ln k2 − iπ sgn ω) (48)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample calculation

More generally , ˜ GR(k) = N2K 4 64π2

• ln |k2| − iπθ(−k2) sgn ω
• (49)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample 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)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Prescription for Minkowski Space Correlators

Sample 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

64π2 ln k2

E

(51)

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Conclusion and frontiers

Previous correlators of SHO are useful...

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Conclusion and frontiers

Previous correlators of SHO are useful... but ambiguous !

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

Conclusion and frontiers

Previous correlators of SHO are useful... but ambiguous ! Use better techniques :

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski 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

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski 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) =  

1 ω2−ω2

0+iǫ +

−i2π eβω0−1δ(ω2 − m2) 2πie−βω0/2 1−e−βω0 δ(ω2 − m2) 2πie−βω0/2 1−e−βω0 δ(ω2 − m2) −1 ω2−ω2

0−iǫ +

−i2π eβω0−1δ(ω2 − m2

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski 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) =  

1 ω2−ω2

0+iǫ +

−i2π eβω0−1δ(ω2 − m2) 2πie−βω0/2 1−e−βω0 δ(ω2 − m2) 2πie−βω0/2 1−e−βω0 δ(ω2 − m2) −1 ω2−ω2

0−iǫ +

−i2π eβω0−1δ(ω2 − m2

Thermo-field Dynamics !

Pinaki Banerjee AdS Space And Thermal Correlators

Introduction Thermal Correlators in QFT Thermal Correlators in AdS space Minkowski Space Correlators : prescription and sample calculations Conclusion and frontiers

thank you!

Pinaki Banerjee AdS Space And Thermal Correlators