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

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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

  1. 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

  2. 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

  3. 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 � X 2 i = R 2 (2) i =1 Pinaki Banerjee AdS Space And Thermal Correlators

  4. 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 � X 2 i = R 2 (2) i =1 Hyperboloid : d � i − U 2 = ± R 2 X 2 (3) i =1 Pinaki Banerjee AdS Space And Thermal Correlators

  5. 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 : d � ds 2 = dX 2 i − dU 2 (4) i =1 Pinaki Banerjee AdS Space And Thermal Correlators

  6. 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 : d � ds 2 = dX 2 i − dU 2 (4) i =1 d � i − U 2 = ∓ R 2 X 2 (5) i =1 Pinaki Banerjee AdS Space And Thermal Correlators

  7. 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

  8. 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 − U 2 − V 2 = − R 2 X 2 (6) i =1 Pinaki Banerjee AdS Space And Thermal Correlators

  9. 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 − U 2 − V 2 = − R 2 X 2 (6) i =1 d − 1 � ds 2 = i − dU 2 − dV 2 dX 2 (7) i =1 Pinaki Banerjee AdS Space And Thermal Correlators

  10. 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 − U 2 − V 2 = − R 2 X 2 (6) i =1 d − 1 � ds 2 = i − dU 2 − dV 2 dX 2 (7) i =1 The symmetry group : SO(2,d-1) Pinaki Banerjee AdS Space And Thermal Correlators

  11. 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 − U 2 − V 2 = − R 2 X 2 (6) i =1 d − 1 � ds 2 = i − dU 2 − dV 2 dX 2 (7) i =1 The symmetry group : SO(2,d-1) Allows closed time-like curve Pinaki Banerjee AdS Space And Thermal Correlators

  12. 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 − U 2 − V 2 = − R 2 X 2 (6) i =1 d − 1 � ds 2 = i − dU 2 − dV 2 dX 2 (7) i =1 The symmetry group : SO(2,d-1) Allows closed time-like curve Topology : AdS d → R d − 1 ⊗ S 1 ; dS d → S d − 1 ⊗ R 1 Pinaki Banerjee AdS Space And Thermal Correlators

  13. 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

  14. 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

  15. 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 X 1 = R sinh ρ cos φ ; X 2 = R sinh ρ sin φ Pinaki Banerjee AdS Space And Thermal Correlators

  16. 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 X 1 = R sinh ρ cos φ ; X 2 = R sinh ρ sin φ ds 2 = R 2 [ − cosh 2 d τ 2 + d ρ 2 + sinh 2 ρ d φ 2 ] (8) Pinaki Banerjee AdS Space And Thermal Correlators

  17. 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 X 1 = R sinh ρ cos φ ; X 2 = R sinh ρ sin φ ds 2 = R 2 [ − cosh 2 d τ 2 + d ρ 2 + sinh 2 ρ d φ 2 ] (8) The change of co-ordinate , tan θ = sinh ρ R 2 2 cos 2 θ [ − d τ 2 + d θ 2 + sin 2 θ d � ds 2 d = Ω d − 2 ] (9) Pinaki Banerjee AdS Space And Thermal Correlators

  18. 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

  19. 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 ds 2 = R 2 z 2 { dz 2 + ( d ¯ x ) 2 − dt 2 } (10) Pinaki Banerjee AdS Space And Thermal Correlators

  20. 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 ds 2 = R 2 z 2 { dz 2 + ( d ¯ x ) 2 − dt 2 } (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

  21. 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

  22. 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 . � ˜ d 4 xe − ik . x θ ( t ) � [ ˆ O ( x ) , ˆ G R ( k ) = − i O (0)] � (11) � ˜ d 4 xe − ik . x θ ( − t ) � [ ˆ O ( x ) , ˆ G A ( k ) = i O (0)] � (12) Pinaki Banerjee AdS Space And Thermal Correlators

  23. 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 . � ˜ d 4 xe − ik . x θ ( t ) � [ ˆ O ( x ) , ˆ G R ( k ) = − i O (0)] � (11) � ˜ d 4 xe − ik . x θ ( − t ) � [ ˆ O ( x ) , ˆ G A ( k ) = i O (0)] � (12) � ˜ d 4 xe − ik . x �| T { ˆ O ( x ) ˆ G F ( k ) = − i O (0) }|� (13) Pinaki Banerjee AdS Space And Thermal Correlators

  24. 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 . � ˜ d 4 xe − ik . x θ ( t ) � [ ˆ O ( x ) , ˆ G R ( k ) = − i O (0)] � (11) � ˜ d 4 xe − ik . x θ ( − t ) � [ ˆ O ( x ) , ˆ G A ( k ) = i O (0)] � (12) � ˜ d 4 xe − ik . x �| T { ˆ O ( x ) ˆ G F ( k ) = − i O (0) }|� (13) � G ( k ) = 1 ˜ d 4 xe − ik . x � ˆ O ( x ) ˆ O (0) + ˆ O (0) ˆ O ( x ) � (14) 2 Pinaki Banerjee AdS Space And Thermal Correlators

  25. 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

  26. 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: � � 1 ˜ G F ( ω ) = (15) ω 2 − ω 2 0 + i ǫ Pinaki Banerjee AdS Space And Thermal Correlators

  27. 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: � � 1 ˜ G F ( ω ) = (15) ω 2 − ω 2 0 + i ǫ 1 ˜ G R , A ( ω ) = (16) ω 2 − ω 2 0 ∓ sgn ( ω ) i ǫ Pinaki Banerjee AdS Space And Thermal Correlators

  28. 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: � � 1 ˜ G F ( ω ) = (15) ω 2 − ω 2 0 + i ǫ 1 ˜ G R , A ( ω ) = (16) ω 2 − ω 2 0 ∓ sgn ( ω ) i ǫ T � = 0: Pinaki Banerjee AdS Space And Thermal Correlators

  29. 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: � � 1 ˜ G F ( ω ) = (15) ω 2 − ω 2 0 + i ǫ 1 ˜ G R , A ( ω ) = (16) ω 2 − ω 2 0 ∓ sgn ( ω ) i ǫ T � = 0: � � e − βω 0 1 1 ˜ G F ( ω ) = 0 + i ǫ ) + (17) ( ω 2 − ω 2 ( ω 2 − ω 2 (1 − e − βω 0 ) 0 − i ǫ ) Pinaki Banerjee AdS Space And Thermal Correlators

  30. 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 AdS 5 × S 5 . ds 2 = R 2 2 z 2 ( d τ 2 + d x 2 + dz 2 ) + R 2 d � Ω 5 (18) � ∂ M φ 0 O � � = e − S cl [ φ ] , (19) e Pinaki Banerjee AdS Space And Thermal Correlators

  31. 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 AdS 5 × S 5 . ds 2 = R 2 2 z 2 ( d τ 2 + d x 2 + dz 2 ) + R 2 d � Ω 5 (18) � ∂ M φ 0 O � � = e − S cl [ φ ] , (19) e To study thermal field theory metric will be a non-extremal one , � � ds 2 = R 2 f ( z ) d τ 2 + d x 2 + dz 2 2 + R 2 d � Ω 5 (20) z 2 f ( z ) Pinaki Banerjee AdS Space And Thermal Correlators

  32. 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 AdS 5 × S 5 . ds 2 = R 2 2 z 2 ( d τ 2 + d x 2 + dz 2 ) + R 2 d � Ω 5 (18) � ∂ M φ 0 O � � = e − S cl [ φ ] , (19) e To study thermal field theory metric will be a non-extremal one , � � ds 2 = R 2 f ( z ) d τ 2 + d x 2 + dz 2 2 + R 2 d � Ω 5 (20) z 2 f ( z ) H ; z H = ( π T ) − 1 ; τ ∼ τ + T − 1 & z = [0, z H ] f ( z ) = 1 − z 4 / z 4 Pinaki Banerjee AdS Space And Thermal Correlators

  33. 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 � ∂ M φ 0 O � � = e iS cl [ φ ] e i (21) Pinaki Banerjee AdS Space And Thermal Correlators

  34. 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 � ∂ M φ 0 O � � = e iS cl [ φ ] e i (21) For any curved (d+1) dimension the action of scalar field reads : � √− gd d +1 x � � D µ φ D µ φ + m 2 φ 2 ) S = (22) Pinaki Banerjee AdS Space And Thermal Correlators

  35. 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 � ∂ M φ 0 O � � = e iS cl [ φ ] e i (21) For any curved (d+1) dimension the action of scalar field reads : � √− gd d +1 x � � D µ φ D µ φ + m 2 φ 2 ) S = (22) � √− gd 4 x � � � D A ( φ D A φ ) − φ D A D A φ + m 2 φ 2 ) S = K dz (23) Pinaki Banerjee AdS Space And Thermal Correlators

  36. 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 � √− gd 4 x � √− gd 4 x � � dz [ − φ ( � − m 2 ) φ dz [ D A ( φ D A φ )] S = K ] + K � �� � � �� � S EOM S Boundary Pinaki Banerjee AdS Space And Thermal Correlators

  37. 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 � √− gd 4 x � √− gd 4 x � � dz [ − φ ( � − m 2 ) φ dz [ D A ( φ D A φ )] S = K ] + K � �� � � �� � S EOM S Boundary √− g ∂ z ( √− gg zz ∂ z φ ) + g µν ∂ µ ∂ ν φ ) − m 2 φ = 0 1 (24) Pinaki Banerjee AdS Space And Thermal Correlators

  38. 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 � √− gd 4 x � √− gd 4 x � � dz [ − φ ( � − m 2 ) φ dz [ D A ( φ D A φ )] S = K ] + K � �� � � �� � S EOM S Boundary √− g ∂ z ( √− gg zz ∂ z φ ) + g µν ∂ µ ∂ ν φ ) − m 2 φ = 0 1 (24) � d 4 k (2 π ) 4 e ik . x f k ( z ) φ 0 ( k ) , φ ( z , x ) = (25) Pinaki Banerjee AdS Space And Thermal Correlators

  39. 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 φ 0 ( k ) is determined by the boundary condition , � d 4 k (2 π ) 4 e ik . x φ 0 ( k ) φ ( z B , x ) = ; f k ( z B ) = 1 . (26) Pinaki Banerjee AdS Space And Thermal Correlators

  40. 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 φ 0 ( k ) is determined by the boundary condition , � d 4 k (2 π ) 4 e ik . x φ 0 ( k ) φ ( z B , x ) = ; f k ( z B ) = 1 . (26) Now substituting it into the EOM we get , √− g ∂ z ( √− gg zz ∂ z f k ) − ( g µν k µ k ν + m 2 ) f k = 0 1 (27) Pinaki Banerjee AdS Space And Thermal Correlators

  41. 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 φ 0 ( k ) is determined by the boundary condition , � d 4 k (2 π ) 4 e ik . x φ 0 ( k ) φ ( z B , x ) = ; f k ( z B ) = 1 . (26) Now substituting it into the EOM we get , √− g ∂ z ( √− gg zz ∂ z f k ) − ( g µν k µ k ν + m 2 ) f k = 0 1 (27) Boundary condition on f k : 1 f k ( z B )=1 , and Pinaki Banerjee AdS Space And Thermal Correlators

  42. 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 φ 0 ( k ) is determined by the boundary condition , � d 4 k (2 π ) 4 e ik . x φ 0 ( k ) φ ( z B , x ) = ; f k ( z B ) = 1 . (26) Now substituting it into the EOM we get , √− g ∂ z ( √− gg zz ∂ z f k ) − ( g µν k µ k ν + m 2 ) f k = 0 1 (27) Boundary condition on f k : 1 f k ( z B )=1 , and 2 Satisfies the incoming wave boundary condition at horizon ( z = z H ) . Pinaki Banerjee AdS Space And Thermal Correlators

  43. 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 � √− gd 4 x � dz [ D A ( φ D A φ )] S Boundary = K � √− g d 4 x { φ g zz ( ∂ z φ ) } � z H � = K � � z B Pinaki Banerjee AdS Space And Thermal Correlators

  44. 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 � √− gd 4 x � dz [ D A ( φ D A φ )] S Boundary = K � √− g d 4 x { φ g zz ( ∂ z φ ) } � z H � = K � � z B Now substituting the expression for φ we get , � �� � z H d 4 k � S Boundary = φ 0 ( − k ) F ( k , z ) φ 0 ( k ) (28) � (2 π ) 4 � z B where F ( k , z ) = K √− gg zz f − k ( z ) ∂ z f k ( z ) . (29) Pinaki Banerjee AdS Space And Thermal Correlators

  45. 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 , � � z H z H � � ˜ G ( k ) = − F ( k , z ) − F ( − k , z ) (30) � � � � z B z B 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

  46. 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 � � ˜ G R ( k ) = − 2 F ( k , z ) (31) � � z B Pinaki Banerjee AdS Space And Thermal Correlators

  47. 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 � � ˜ G R ( k ) = − 2 F ( k , z ) (31) � � z B 1 Find a solution to the (27) with following properties : Pinaki Banerjee AdS Space And Thermal Correlators

  48. 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 � � ˜ G R ( k ) = − 2 F ( k , z ) (31) � � z B 1 Find a solution to the (27) with following properties : It equals to 1 at boundary z = z B ; Pinaki Banerjee AdS Space And Thermal Correlators

  49. 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 � � ˜ G R ( k ) = − 2 F ( k , z ) (31) � � z B 1 Find a solution to the (27) with following properties : It equals to 1 at boundary z = z B ; time-like momenta : It satisfies incoming wave boundary condition at horizon . Pinaki Banerjee AdS Space And Thermal Correlators

  50. 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 � � ˜ G R ( k ) = − 2 F ( k , z ) (31) � � z B 1 Find a solution to the (27) with following properties : It equals to 1 at boundary z = z B ; 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

  51. 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 � � ˜ G R ( k ) = − 2 F ( k , z ) (31) � � z B 1 Find a solution to the (27) with following properties : It equals to 1 at boundary z = z B ; 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 = − 2 F ∂ M . (at z = z B ) Pinaki Banerjee AdS Space And Thermal Correlators

  52. 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 , � ∂ M φ 0 O � = e − S E [ φ ] � e (32) Pinaki Banerjee AdS Space And Thermal Correlators

  53. 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 , � ∂ M φ 0 O � = e − S E [ φ ] � e (32) Euclidean AdS 5 metric is 5 = R 2 z 2 ( dz 2 + d x 2 ) ds 2 (33) Pinaki Banerjee AdS Space And Thermal Correlators

  54. 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 , � ∂ M φ 0 O � = e − S E [ φ ] � e (32) Euclidean AdS 5 metric is 5 = R 2 z 2 ( dz 2 + d x 2 ) ds 2 (33) The action of massive scalar field on this background is , z H = ∞ � � dz √ g � g zz ( ∂ z φ ) 2 + g µν ( ∂ µ φ )( ∂ ν φ ) + m 2 φ 2 � d 4 x S E = K Pinaki Banerjee AdS Space And Thermal Correlators z B = ǫ

  55. 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 , � ∂ M φ 0 O � = e − S E [ φ ] � e (32) Euclidean AdS 5 metric is 5 = R 2 z 2 ( dz 2 + d x 2 ) ds 2 (33) The action of massive scalar field on this background is , z H = ∞ � � dz √ g � g zz ( ∂ z φ ) 2 + g µν ( ∂ µ φ )( ∂ ν φ ) + m 2 φ 2 � d 4 x S E = K Pinaki Banerjee AdS Space And Thermal Correlators z B = ǫ

  56. 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 � � � � S E = π 3 R 8 ( ∂ z φ ) 2 + z 2 R 2 ( ∂ i φ ) 2 + R 2 m 2 d 4 xz − 3 φ 2 (35) dz 4 κ 2 z 2 10 Pinaki Banerjee AdS Space And Thermal Correlators

  57. 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 � � � � S E = π 3 R 8 ( ∂ z φ ) 2 + z 2 R 2 ( ∂ i φ ) 2 + R 2 m 2 d 4 xz − 3 φ 2 (35) dz 4 κ 2 z 2 10 � � d 4 k z 3 { ( ∂ z f k )( ∂ z f − k ) + k 2 f k f − k + R 2 m 2 1 S E ∼ dz f k f − k } φ 0 ( k ) φ 0 ( − k ) (2 π ) 4 z 2 Pinaki Banerjee AdS Space And Thermal Correlators

  58. 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 � � � � S E = π 3 R 8 ( ∂ z φ ) 2 + z 2 R 2 ( ∂ i φ ) 2 + R 2 m 2 d 4 xz − 3 φ 2 (35) dz 4 κ 2 z 2 10 � � d 4 k z 3 { ( ∂ z f k )( ∂ z f − k ) + k 2 f k f − k + R 2 m 2 1 S E ∼ dz f k f − k } φ 0 ( k ) φ 0 ( − k ) (2 π ) 4 z 2 � � k 2 + m 2 R 2 k ( z ) − 3 f ′′ z f ′ k ( z ) − f k ( z ) = 0 (36) z 2 Pinaki Banerjee AdS Space And Thermal Correlators

  59. 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 � � � � S E = π 3 R 8 ( ∂ z φ ) 2 + z 2 R 2 ( ∂ i φ ) 2 + R 2 m 2 d 4 xz − 3 φ 2 (35) dz 4 κ 2 z 2 10 � � d 4 k z 3 { ( ∂ z f k )( ∂ z f − k ) + k 2 f k f − k + R 2 m 2 1 S E ∼ dz f k f − k } φ 0 ( k ) φ 0 ( − k ) (2 π ) 4 z 2 � � k 2 + m 2 R 2 k ( z ) − 3 f ′′ z f ′ k ( z ) − f k ( z ) = 0 (36) z 2 Pinaki Banerjee AdS Space And Thermal Correlators

  60. 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 , φ k ( z ) = Az 2 I ν ( kz ) + Bz 2 I − ν ( kz ) (37) Pinaki Banerjee AdS Space And Thermal Correlators

  61. 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 , φ k ( z ) = Az 2 I ν ( kz ) + Bz 2 I − ν ( kz ) (37) The solution is regular at z = ∞ and equals to 1 at z = ǫ , therefore , f k ( z ) = z 2 K ν ( kz ) (38) ǫ 2 K ν ( k ǫ ) Pinaki Banerjee AdS Space And Thermal Correlators

  62. 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 , φ k ( z ) = Az 2 I ν ( kz ) + Bz 2 I − ν ( kz ) (37) The solution is regular at z = ∞ and equals to 1 at z = ǫ , therefore , f k ( z ) = z 2 K ν ( kz ) (38) ǫ 2 K ν ( k ǫ ) On shell , the action reduces to the boundary term � d 4 kd 4 k ′ � S E = π 3 R 8 ∞ � (2 π ) 8 φ 0 ( k ) φ 0 ( k ′ ) F ( z , k , k ′ ) (39) � 4 κ 2 � 10 ǫ Pinaki Banerjee AdS Space And Thermal Correlators

  63. 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 � δ 2 Z [ φ 0 ] � � O ( k ) O ( k ′ ) � = Z − 1 (40) � δφ 0 ( k ) δφ 0 ( k ′ ) � φ 0 =0 � ∞ � = − 2 F ( z , k , k ′ ) � � ǫ � = − (2 π ) 4 δ 4 ( k + k ′ ) π 3 R 8 ∞ f k ′ ( z ) ∂ z f k ( z ) � � 2 κ 2 z 3 � 10 ǫ Pinaki Banerjee AdS Space And Thermal Correlators

  64. 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 � δ 2 Z [ φ 0 ] � � O ( k ) O ( k ′ ) � = Z − 1 (40) � δφ 0 ( k ) δφ 0 ( k ′ ) � φ 0 =0 � ∞ � = − 2 F ( z , k , k ′ ) � � ǫ � = − (2 π ) 4 δ 4 ( k + k ′ ) π 3 R 8 ∞ f k ′ ( z ) ∂ z f k ( z ) � � 2 κ 2 z 3 � 10 ǫ Putting the value of f k ( z ) we get , � O ( k ) O ( k ′ ) � = − π 3 R 8 ǫ 2(∆ − d ) (2 π ) 4 δ 4 ( k + k ′ ) k 2 ν 2 1 − 2 ν Γ(1 − ν ) + ... 2 κ 2 Γ( ν ) 10 (41) Pinaki Banerjee AdS Space And Thermal Correlators

  65. 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) ∆ 8 π 2 (2 π ) 4 δ 4 ( k + k ′ ) k 2∆ − 4 N 2 2 2∆ − 5 ln k 2 (42) (∆ − 3)! Pinaki Banerjee AdS Space And Thermal Correlators

  66. 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) ∆ 8 π 2 (2 π ) 4 δ 4 ( k + k ′ ) k 2∆ − 4 N 2 2 2∆ − 5 ln k 2 (42) (∆ − 3)! For massless case (∆ = 4) , we have � O ( k ) O ( k ′ ) � = − N 2 64 π 4 (2 π ) 4 δ 4 ( k + k ′ ) k 4 ln k 2 (43) Pinaki Banerjee AdS Space And Thermal Correlators

  67. 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 : k ( z ) − 3 k ( z ) − k 2 f k ( z ) = 0 f ′′ z f ′ (44) Pinaki Banerjee AdS Space And Thermal Correlators

  68. 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 : k ( z ) − 3 k ( z ) − k 2 f k ( z ) = 0 f ′′ z f ′ (44) For spacelike momenta , k 2 > 0 , we can follow the steps identical to the Euclidean case. G R ( k ) = + N 2 k 4 ; k 2 > 0 ˜ 64 π 2 ln k 2 (45) Pinaki Banerjee AdS Space And Thermal Correlators

  69. 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 √ − k 2 . For timelike momenta , we introduce q = f k ( z ) = z 2 H (1) 2 ( qz ) if ω > 0 (46) ǫ 2 H (1)( q ǫ ) ν = z 2 H (2) 2 ( qz ) if ω < 0 (47) ǫ 2 H (2)( q ǫ ) 2 Pinaki Banerjee AdS Space And Thermal Correlators

  70. 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 √ − k 2 . For timelike momenta , we introduce q = f k ( z ) = z 2 H (1) 2 ( qz ) if ω > 0 (46) ǫ 2 H (1)( q ǫ ) ν = z 2 H (2) 2 ( qz ) if ω < 0 (47) ǫ 2 H (2)( q ǫ ) 2 G R ( k ) = N 2 k 4 64 π 2 (ln k 2 − i π sgn ω ) ˜ (48) Pinaki Banerjee AdS Space And Thermal Correlators

  71. 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 , G R ( k ) = N 2 K 4 � � ˜ ln | k 2 | − i πθ ( − k 2 ) sgn ω (49) 64 π 2 Pinaki Banerjee AdS Space And Thermal Correlators

  72. 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 , G R ( k ) = N 2 K 4 � � ˜ ln | k 2 | − i πθ ( − k 2 ) sgn ω (49) 64 π 2 We can get the Feynman propagator , G F ( k ) = N 2 K 4 � � ˜ ln | k 2 | − i πθ ( − k 2 ) (50) 64 π 2 Pinaki Banerjee AdS Space And Thermal Correlators

  73. 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 , G R ( k ) = N 2 K 4 � � ˜ ln | k 2 | − i πθ ( − k 2 ) sgn ω (49) 64 π 2 We can get the Feynman propagator , G F ( k ) = N 2 K 4 � � ˜ ln | k 2 | − i πθ ( − k 2 ) (50) 64 π 2 we can also get it by Wick rotating the Euclidean correlator , G E ( k E ) = − N 2 K 4 ˜ 64 π 2 ln k 2 E (51) E Pinaki Banerjee AdS Space And Thermal Correlators

  74. 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

  75. 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

  76. 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

  77. 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

  78. 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  e βω 0 − 1 δ ( ω 2 − m 2 ) 1 − e − βω 0 δ ( ω 2 − m 2 ) 2 π ie − βω 0 / 2 1 − i 2 π 0 + i ǫ + ω 2 − ω 2 ˜ G F ( ω ) =  1 − e − βω 0 δ ( ω 2 − m 2 ) e βω 0 − 1 δ ( ω 2 − m 2 2 π ie − βω 0 / 2 − 1 − i 2 π 0 − i ǫ + ω 2 − ω 2 Pinaki Banerjee AdS Space And Thermal Correlators

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