de Sitter space and Holography
de Sitter space and Holography de Sitter space and Holography
For example• In static coordinates:l 2 P(X,X ′ ) = −√(r 2 − l 2 )(r ′2 − l 2 ) cosh t − t′l+ rr ′ cos Θwhere Θ is the geodesic distance of two points on the unitS d−2 .• In planer coordinates:l 2 P(X,X ′ ) = −l 2 cosh t − t′l+ 1 2 e−t−t′ l δ ij (x i − y i )(x j − y j )Consider a scalar field in dS spaceS = − 1 ∫d d x √ −g[(∇φ) 2 + m 2 φ 2 ]2The Green function G(X,Y ) = 〈0|φ(X)φ(Y )|0〉 obeys(∇ 2 − m 2 )G(X,Y ) = 0Since dS space is maximally symmetric, the Green functiondepends on X and Y only through P(X,Y ).26
For any function f(P) one can seel 2 (∇ 2 − m 2 )f(P) = (1 − P 2 ) d2 f df− PddP2dP − m2 l 2 fTherefore one has[(1 − P 2 ) d2dP 2 − Pd ddP − m2 l 2 ]G(P(X,Y )) = 0Which as solution in terms of hypergeometric functionsG = const.F(h + , h − , d 2 , 1 + P2)whereh ± = 1 [(d − 1) ± √ ](d − 1)22 − 4m 2 l 2Since the above equation is symmetric under P → −P thereis another solutionG = const.F(h + , h − , d 2 , 1 − P2)One parameter family of dS invariant Green functioncorresponding to a linear combination of these solutions.G α (X,Y ) = 〈α|φ(X)φ(Y )|α〉27
- Page 1 and 2: de Sitter space and HolographyThird
- Page 3: 1. Historical review and motivation
- Page 7 and 8: 5. Technologies:There are a number
- Page 9 and 10: Let us know study different coordin
- Page 11 and 12: Conformal coordinates: T, θ i , i
- Page 13 and 14: Planer (Inflationary) coordinates:
- Page 15 and 16: The corresponding metric isdS 2 =
- Page 17 and 18: It is easy to see thatX 0 + X d =
- Page 20 and 21: 1.2 Causal structure (Penrose diagr
- Page 22 and 23: IINorth PoleOSouth PoleNorth PoleOS
- Page 24 and 25: The entire region of the dS space i
- Page 28 and 29: An observer moving along a timelike
- Page 30 and 31: 3. dS/CFT correspondenceFrom what w
- Page 32 and 33: In d dimensions the Einstein equati
- Page 34 and 35: One can generalize it for a general
- Page 36 and 37: • One can always decompose the me
- Page 38 and 39: Consider the Brown-York stress tens
- Page 40 and 41: Using the definition of Brown-York
- Page 42 and 43: Using this diffeomorphism one may s
- Page 44 and 45: lim dtdtr→∞∫I ′ dφdφ ′
- Page 46 and 47: dS static patch (spatial)dg =0 g =1
- Page 48 and 49: • Lower dimensional Plank mass is
- Page 50 and 51: One can use AdS/CFT correspondence
- Page 52 and 53: Under this conformal map one has:
- Page 54 and 55: Conformal AnomalyAdS/CFT instructs
- Page 56 and 57: • Naively one would think that gr
- Page 58: This talk is based on the following
For example• In static coordinates:l 2 P(X,X ′ ) = −√(r 2 − l 2 )(r ′2 − l 2 ) cosh t − t′l+ rr ′ cos Θwhere Θ is the geo<strong>de</strong>sic distance of two points on the unitS d−2 .• In planer coordinates:l 2 P(X,X ′ ) = −l 2 cosh t − t′l+ 1 2 e−t−t′ l δ ij (x i − y i )(x j − y j )Consi<strong>de</strong>r a scalar field in dS <strong>space</strong>S = − 1 ∫d d x √ −g[(∇φ) 2 + m 2 φ 2 ]2The Green function G(X,Y ) = 〈0|φ(X)φ(Y )|0〉 obeys(∇ 2 − m 2 )G(X,Y ) = 0Since dS <strong>space</strong> is maximally symmetric, the Green function<strong>de</strong>pends on X <strong>and</strong> Y only through P(X,Y ).26
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