STOCHASTIC QUANTIZATION AND HOLOGRAPHY

Transcription

1 STOCHASTIC QUANTIZATION AND HOLOGRAPHY WORK WITH D.MANSI & A. MAURI: TO APPEAR TASSOS PETKOU UNIVERSITY OF CRETE

2 OUTLINE CONFORMAL HOLOGRAPHY STOCHASTIC QUANTIZATION STOCHASTIC QUANTIZATION VS HOLOGRAPHY IS HOLOGRAPHY STOCHASTIC?

3 ASYMPTOTIC BEHAVIOUR f(r, x) φ 0 ( x) + rφ 1 ( x) + THE ON-SHELL ACTION YIELDS THE (HOLOGRAPHIC) EFFECTIVE ACTION FOR THE DUAL BOUNDARY THEORY I.E. THE THEORY WITH AN OPERATOR OF DIMENSION=1. I S[φ 0 ] Γ[φ 0 ] ( ) 1 i φ 0 i φ 0 + κφ φ 0 φ 0 ϕ 2 Γ[φ 0 ] Γ[ϕ] ( 1 2 iϕ i ϕ + κ ) 8 ϕ6 + CONFORMAL HOLOGRAPHY APPEARS TO RELATE A BULK ACTION IN 4D TO A BOUNDARY ACTION IN 3D - CONTRAST WITH STANDARD HOLOGRAPHY

4 CONFORMAL HOLOGRAPHY T. P., S. DE HARO, I. PAPADIMITRIOU (06) THE HOLOGRAPHY OF CONFORMALLY COUPLED SCALAR FIELDS (THE MOST RELEVANT CASE FOR ADS4/CFT3) REDUCES(ESSENTIALLY) TO MASSLESS THEORIES IN FLAT SPACE WITH A BOUNDARY: EXAMPLE I = d 4 x g ( 1 2 gµν µ φ ν φ m2 φ 2 + λ ) 4 φ4 ds 2 = L2 r 2 ( dr 2 + d x 2) m 2 L 2 = 2 g µν = Ω(x) 2 η µν, φ = Ω(x)f, Ω(x) = r L I = = 0 dr ( 1 2 ηµν µ f ν f + λ ) 4 f r f 2 0 SUBTRACT-RENORMALIZATION

5 A SIMPLE WAY TO REPRODUCE THE RESULT: HAMILTONIAN ANALYSIS I = 0 dr [ ] fπ H, H = 1 2 (π 2 i f i f λ2 f 4 ) THE 2ND ORDER E.O.M. HAVE A GENERAL INSTANTON SOLUTION λ < 0 f + i i f = λf 3 ˆf(r, x) = 8 λ b b 2 + r 2 + x 2 GENERICALLY INSTANTONS MAY NOT SIT AT THE ORIGIN SUCH THAT THE ON-SHELL HAMILTONIAN DENSITY IS Ĥ = 1 ( ) ˆπ 2 λ i ˆf i ˆf 2 2 ˆf 4 = i i ˆf λ ˆπ = ± 2 ˆf λ ˆf ˆf 2 4 i i ˆf 3λ ˆf i i ˆf 2 4

6 CHOOSING THE + SIGN WE FIND THE ON-SHELL ACTION IN A LARGE- LAMBDA EXPANSION, AS: I Γ[φ 0 ] = 1 18λ ( λφ ) i φ 0 i φ 0 + φ 0 SETTING φ = ϕ 2 WE OBTAIN THE STANDARD ϕ 6 THEORY THIS IS REMARKABLE! ADS/CFT GIVES THE EFFECTIVE ACTION OF COMPOSITE BOUNDARY OPERATORS. IT RATHER LOOKS LIKE A TOPOLOGICAL RELATION: da da = M M A da

7 STOCHASTIC QUANTIZATION (PARISI-WU (81)) φ( x)... = 1 Z (Dφ)e S cl[φ] φ( x)... CONSIDER THE EUCLIDEAN PATH INTEGRAL IN D-DIMENSIONS AS THE STATIONARY DISTRIBUTION OF A STOCHASTIC PROCESS IN D+1- DIMENSIONS; INTRODUCE STOCHASTIC TIME. LANGEVIN EQ. φ( x) φ(t, x), φ(t, x) t + k δs cl δφ(t, x) = η(t, x) WHITE NOISE S cl [φ] = d d x CONSTANT KERNEL ( ) 1 2 iφ i φ + V (λ, φ) GENERICALLY, FOR SMALL-LAMBDA THE LANGEVIN GIVES A NON- RELATIVISTIC DISPERSION RELATION. HENCE, THE D+1 DIMENSIONAL STOCHASTIC PROCESS IS NON-RELATIVISTIC CONSIDER THE EUCLIDEAN PATH INTEGRAL IN IN D-DIMENSIONS AS AS

8 UPSHOT: E.G. P. DAMGAARD & H. HUEFFEL PHYS. REP. (87) Z = = = (Dη)e 1 4k R dtd d x η 2 (t, x) (Dφ)P [φ, 0]e R dt R» d d x (Dφ 0 )e 1 2 R d d xs cl [φ 0 ] 1 φ2 4k +k δs cl 2 k δφ 2 δ 2 S cl δφ 2 (Dφ)e R dtd d 1 x φ 2 4k + k 4 ( δs δφ ) 2 k 2 δ 2 S δφ 2 FOKKER-PLANCK LAGRANGIAN P [φ, 0] : φ(t t M, x) φ 0 ( x) DIRICHLET B.C. det δη δφ = e k 2 R dt R d d x δ2 S cl δφ 2 I CAN USE Z TO CALCULATE CORRELATION FUNCTIONS OF φ(t, x) WHICH FOR LARGE TIMES RELAX TO THE EUCLIDEAN CORRELATION FUNCTIONS OF THE D-DIMENSIONAL THEORY.

9 HOLOGRAPHY VS STOCHASTIC QUANTIZATION THE S.Q. PARTITION FUNCTION IS FORMALLY EQUIVALENT TO THE HOLOGRAPHIC FORMULA UPON INTEGRATION OVER THE BOUNDARY DATA (MODULO RENORMALIZATIONS ) 1, UP TO BULK LOOPS Z hol [φ 0 ] = Z = STOCHASTIC QUANTIZATION 1 S cl = 2 18λ CALCULATIONS.. S3+1 F.P. = dt FOR LARGE-LAMBDA! (Dφ) φ φ0 e S d+1[φ] e Γ d[φ 0 ] (Dφ 0 )e S d[φ 0 ]+Γ d [φ 0 ] FOKKER-PLANCK G. COMPERE & D. MAROLF (08) ( λφ ) φ iφ i φ ( 1 2 φ iφ i φ + λ ) 4 φ4 +

10 RECAPITULATE: I HAVE SHOWN THAT S cl ( λφ ) φ iφ i φ STOCHASTIC QUANTIZATION LARGE-LAMBDA INSTANTONS HOLOGRAPHY S 3+1 dt ( 1 2 φ iφ i φ + λ ) 4 φ4 CAN WE UNDERSTAND WHAT IS GOING ON?

11 IS HOLOGRAPHY STOCHASTIC? A GEOMETRIC PICTURE OF STOCHASTIC HOLOGRAPHY AND CONFORMAL HOLOGRAPHY: I EH = 1 16πG d 4 x g (R 2Λ) CONSIDER CONFORMALLY FLAT METRICS g µν (x) = ϕ 2 (x)η µν I EH = 3 4π d 4 x ( 1 2 ηµν µ φ ν φ λ ) 6 φ4, φ = ϕ G, λ = Λ G FOR POSITIVE C.C. THE INSTANTON SOLUTION IS JUST THE 4-SPHERE φ(x) = 12 λ R R 2 + x 2 ds2 = 4R 4 (R 2 + x 2 ) 2 dx2, Λ = 3 R 2

12 BOUNDARY S 3 Rφ(x 1, x) cos 2 θ S 4 x 1 0 x 1 cos θ 0 S 3 θ half S 4 BULK R 0 INSTANTONS SATISFY x 1 i log φ = 2 i log φ φ = e C(t) φ 2 LANGEVIN (NO WHITE-NOISE)

13 GEOMETRIC PICTURE: THE 4D EFFECTIVE E-H ACTION FOR CONFORMALLY FLAT METRICS, AS A PATH INTEGRAL MEASURE, RELAXES FOR LARGE TIMES TO A THREE- DIMENSIONAL E-H ACTION. 1 I EH = g (R 2Λ) 16πG g µν = ϕ 2 ( x)η µν I EH = 1 4π φ = f 2 I EH = 2 π ( ) 1 φ iφ i φ λφ 3 ( 1 2 if i f λ ) 8 f 6, φ = ϕ G, λ = Λ G 2 FOR λ > 0 (I.E. POSITIVE C.C.) THIS HAS INSTANTON SOLUTIONS (3-SPHERE) f( x) = ( ) 1/4 4 R 1/2 λ (R 2 + x 2 ) 1/2 THESE ARE THE BOUNDARY VALUES OF THE BULK INSTANTONS IF λ 4 = 3λ 3 Λ 3 Λ 4 = G 3 2G 4

14 TOWARDS A STOCHASTIC HOLOGRAPHY? I HAVE SHOWN AN EXPLICIT EXAMPLE OF 4D AND 3D THEORIES THAT ARE RELATED: 1)HOLOGRAPHICALLY, 2) VIA STOCHASTIC QUANTIZATION AND 3)GEOMETRICALLY. THE CRUCIAL POINT WAS THE RELATIVISTIC LANGEVIN EQ. SATISFIED BY THE INSTANTON CONFIGURATIONS. THIS IN NOT A STRONG/WEAK COUPLING CORRESPONDENCE. THE BULK AND BOUNDARY THEORIES ARE IN THE SAME REGIME (I.E. D- INSTANTONS/YM INSTANTONS IN ADS5/CFT4).

15 RELATED TO RICCI-FLOW?(1ST-ORDER EQUATIONS) INTIMATELY RELATED TO THE WELL-KNOWN STOCHASTIC QUANTIZATION RESULT: TOPOLOGIGAL YANG-MILLS/ CHERN-SIMONS. THIS ARISES FROM THE BRST GAUGE FIXING OF THE TOPOLOGICAL GAUGE INVARIANCE. IT IS A GENERALIZATION OF THE STOKES FORMULA. D. MANSI & A. MAURI: TO APPEAR

16 THE HOLOGRAPHIC WEB BULK (ADS)THEORY NON-RELATIVISTIC LIFSCHITZ LIMIT INSTANTONS PERTURBATIVE LIMIT HOLOGRAPHY - STOCHASTIC QUANTIZATION WEAK-COUPLING INSTANTONS BOUNDARY THEORY STRONG COUPLING BULK (ADS)THEORY

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