AdS Space And Thermal Correlatorspinakib/presentation.pdf · 1 Introduction Motivation The AdS/CFT correspondence Brief review of AdS space 2 Thermal Correlators in QFT Minkowski

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




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





Outline

1 IntroductionMotivation

The AdS/CFT correspondenceBrief review of AdS space









Outline

1 IntroductionMotivationThe AdS/CFT correspondence

Brief review of AdS space









Outline

1 IntroductionMotivationThe AdS/CFT correspondenceBrief review of AdS space









Outline


2 Thermal Correlators in QFT

Minkowski spaceSample calculations for (0+1)d QFT








Outline


2 Thermal Correlators in QFTMinkowski space

Sample calculations for (0+1)d QFT








Outline










Outline



3 Thermal Correlators in AdS space

Euclidean spaceDifficulties in Minkowski space







Outline



3 Thermal Correlators in AdS spaceEuclidean space

Difficulties in Minkowski space







Outline










Outline










Outline










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.





















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.








⟨exp

(∫Sd φ

i0O)⟩

CFT= ZQG (φi0) (1)











⟨exp

(∫Sd φ

i0O)⟩

CFT= ZQG (φi0) (1)











⟨exp

(∫Sd φ

i0O)⟩

CFT= ZQG (φi0) (1)











⟨exp

(∫Sd φ

i0O)⟩

CFT= ZQG (φi0) (1)



Boundary conditions:

φi goes to φi0 on the boundary.








⟨exp

(∫Sd φ

i0O)⟩

CFT= ZQG (φi0) (1)








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)







Some Quadric surfaces :

Sphere :

d+1∑i=1

X 2i = R2 (2)

Hyperboloid :

d∑i=1

X 2i − U2 = ±R2 (3)








d+1∑i=1

X 2i = R2 (2)

Hyperboloid :

d∑i=1

X 2i − U2 = ±R2 (3)








d+1∑i=1

X 2i = R2 (2)

Hyperboloid :

d∑i=1

X 2i − U2 = ±R2 (3)






Some Quadric surfaces :Hyperbolic and de Sitter space :

ds2 =d∑

i=1

dX 2i − dU2 (4)

d∑i=1

X 2i − U2 = ∓R2 (5)






Some Quadric surfaces :Hyperbolic and de Sitter space :

ds2 =d∑

i=1

dX 2i − dU2 (4)

d∑i=1

X 2i − U2 = ∓R2 (5)






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







d−1∑i=1

X 2i − U2 − V 2 = −R2 (6)

ds2 =d−1∑i=1

dX 2i − dU2 − dV 2 (7)








d−1∑i=1

X 2i − U2 − V 2 = −R2 (6)

ds2 =d−1∑i=1

dX 2i − dU2 − dV 2 (7)








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







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







d−1∑i=1

X 2i − U2 − V 2 = −R2 (6)

ds2 =d−1∑i=1

dX 2i − dU2 − dV 2 (7)







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)












ds2d =

R2


2

d−2] (9)












ds2d =

R2


2

d−2] (9)












ds2d =

R2


2

d−2] (9)












ds2d =

R2


2

d−2] (9)






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 .








ds2 =R2

z2dz2 + (dx)2 − dt2 (10)









ds2 =R2

z2dz2 + (dx)2 − dt2 (10)






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)







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)







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)







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)





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)






T = 0:

GF (ω) =

[1

ω2 − ω20 + iε

](15)

GR,A(ω) =1


(16)

T 6= 0:

GF (ω) =1

(1− e−βω0)

1

(ω2 − ω20 + iε)

+e−βω0

(ω2 − ω20 − iε)

(17)






T = 0:

GF (ω) =

[1

ω2 − ω20 + iε

](15)

GR,A(ω) =1


(16)

T 6= 0:

GF (ω) =1

(1− e−βω0)

1

(ω2 − ω20 + iε)

+e−βω0

(ω2 − ω20 − iε)

(17)






T = 0:

GF (ω) =

[1

ω2 − ω20 + iε

](15)

GR,A(ω) =1


(16)

T 6= 0:

GF (ω) =1

(1− e−βω0)

1

(ω2 − ω20 + iε)

+e−βω0

(ω2 − ω20 − iε)

(17)






T = 0:

GF (ω) =

[1

ω2 − ω20 + iε

](15)

GR,A(ω) =1


(16)

T 6= 0:

GF (ω) =1

(1− e−βω0)

1

(ω2 − ω20 + iε)

+e−βω0

(ω2 − ω20 − iε)

(17)





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 ]







ds2 =R2

z2(dτ2 + dx2 + dz2) + R2d~Ω5

2(18)

⟨e∫∂M φ0O

⟩= e−Scl [φ], (19)


ds2 =R2

z2

(f (z)dτ2 + dx2 +

dz2

f (z)

)+ R2d~Ω5

2(20)








ds2 =R2

z2(dτ2 + dx2 + dz2) + R2d~Ω5

2(18)

⟨e∫∂M φ0O

⟩= e−Scl [φ], (19)


ds2 =R2

z2

(f (z)dτ2 + dx2 +

dz2

f (z)

)+ R2d~Ω5

2(20)






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)







∫∂M φ0O

⟩= e iScl [φ] (21)


S =

∫ √−gdd+1x

[DµφDµφ+ m2φ2)

](22)

S = K

∫ √−gd4x


Aφ+ m2φ2)]

(23)







∫∂M φ0O

⟩= e iScl [φ] (21)


S =

∫ √−gdd+1x

[DµφDµφ+ m2φ2)

](22)

S = K

∫ √−gd4x


Aφ+ m2φ2)]

(23)






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)






S = K

∫ √−gd4x

∫dz [−φ(−m2)φ︸︷︷︸

SEOM

] + K

∫ √−gd4x


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)






S = K

∫ √−gd4x

∫dz [−φ(−m2)φ︸︷︷︸

SEOM

] + K

∫ √−gd4x


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)






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) .







φ(zB , x) =

∫d4k

(2π)4e ik.xφ0(k) ; fk(zB) = 1. (26)


1√−g ∂z(










φ(zB , x) =

∫d4k

(2π)4e ik.xφ0(k) ; fk(zB) = 1. (26)


1√−g ∂z(



1 fk(zB)=1 , and

2 Satisfies the incoming wave boundary condition at horizon (z = zH) .







φ(zB , x) =

∫d4k

(2π)4e ik.xφ0(k) ; fk(zB) = 1. (26)


1√−g ∂z(









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)






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)






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.





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)






GR(k) = −2F (k, z)

∣∣∣∣zB

(31)









GR(k) = −2F (k, z)

∣∣∣∣zB

(31)


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 .







GR(k) = −2F (k, z)

∣∣∣∣zB

(31)


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 .







GR(k) = −2F (k, z)

∣∣∣∣zB

(31)









GR(k) = −2F (k, z)

∣∣∣∣zB

(31)








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)







〈e∫∂M φ0O〉 = e−SE [φ] (32)


ds25 =

R2

z2(dz2 + dx2) (33)


SE = K

∫d4x

zH=∞∫zB=ε


](34)







〈e∫∂M φ0O〉 = e−SE [φ] (32)


ds25 =

R2

z2(dz2 + dx2) (33)


SE = K

∫d4x

zH=∞∫zB=ε


](34)







〈e∫∂M φ0O〉 = e−SE [φ] (32)


ds25 =

R2

z2(dz2 + dx2) (33)


SE = K

∫d4x

zH=∞∫zB=ε


](34)






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)






SE =π3R8

4κ210

∫dz

∫d4xz−3

((∂zφ)2 +

z2

R2(∂iφ)2 +

R2m2

z2φ2

)(35)

SE ∼∫

dz

∫d4k

(2π)4

1


R2m2


f ′′k (z)− 3z f′k(z)−

(k2 + m2R2

z2

)fk(z) = 0 (36)






SE =π3R8

4κ210

∫dz

∫d4xz−3

((∂zφ)2 +

z2

R2(∂iφ)2 +

R2m2

z2φ2

)(35)

SE ∼∫

dz

∫d4k

(2π)4

1


R2m2


f ′′k (z)− 3z f′k(z)−

(k2 + m2R2

z2

)fk(z) = 0 (36)






SE =π3R8

4κ210

∫dz

∫d4xz−3

((∂zφ)2 +

z2

R2(∂iφ)2 +

R2m2

z2φ2

)(35)

SE ∼∫

dz

∫d4k

(2π)4

1


R2m2


f ′′k (z)− 3z f′k(z)−

(k2 + m2R2

z2

)fk(z) = 0 (36)






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)









fk(z) =z2Kν(kz)

ε2Kν(kε)(38)


SE =π3R8

4κ210

∫d4kd4k ′

(2π)8φ0(k)φ0(k ′)F (z , k , k ′)

∣∣∣∣∞ε

(39)









fk(z) =z2Kν(kz)

ε2Kν(kε)(38)


SE =π3R8

4κ210

∫d4kd4k ′

(2π)8φ0(k)φ0(k ′)F (z , k , k ′)

∣∣∣∣∞ε

(39)






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)






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





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)






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)






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)






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)






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)






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)






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)






More generally ,

GR(k) =N2K 4

64π2


)(49)


GF (k) =N2K 4

64π2

[ln |k2| − iπθ(−k2)

](50)


GE (kE ) = −N2K 4

E

64π2ln k2

E (51)






More generally ,

GR(k) =N2K 4

64π2


)(49)


GF (k) =N2K 4

64π2

[ln |k2| − iπθ(−k2)

](50)


GE (kE ) = −N2K 4

E

64π2ln k2

E (51)






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 !






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



GF (ω) =

1ω2−ω2

0+iε+ −i2π


1−e−βω0δ(ω2 −m2)

2πie−βω0/2

1−e−βω0δ(ω2 −m2) −1

ω2−ω20−iε

+ −i2πeβω0−1

δ(ω2 −m2)










GF (ω) =

1ω2−ω2

0+iε+ −i2π


1−e−βω0δ(ω2 −m2)

2πie−βω0/2

1−e−βω0δ(ω2 −m2) −1

ω2−ω20−iε

+ −i2πeβω0−1

δ(ω2 −m2)










GF (ω) =

1ω2−ω2

0+iε+ −i2π


1−e−βω0δ(ω2 −m2)

2πie−βω0/2

1−e−βω0δ(ω2 −m2) −1

ω2−ω20−iε

+ −i2πeβω0−1

δ(ω2 −m2)










GF (ω) =

1ω2−ω2

0+iε+ −i2π


1−e−βω0δ(ω2 −m2)

2πie−βω0/2

1−e−βω0δ(ω2 −m2) −1

ω2−ω20−iε

+ −i2πeβω0−1

δ(ω2 −m2)










GF (ω) =

1ω2−ω2

0+iε+ −i2π


1−e−βω0δ(ω2 −m2)

2πie−βω0/2

1−e−βω0δ(ω2 −m2) −1

ω2−ω20−iε

+ −i2πeβω0−1

δ(ω2 −m2)






thank you!


Documents

AdS Space And Thermal Correlatorspinakib/presentation.pdf · 1 Introduction Motivation The AdS/CFT correspondence Brief review of AdS space 2 Thermal Correlators in QFT Minkowski