General Relativity is Not a Field General Relativity is Not a Field TheoryTheory

Cauchy-Kowalevskaya – Field Theory Has Cauchy-Kowalevskaya – Field Theory Has Finite # DOF per Space PointFinite # DOF per Space Point

Arnowitt Deser Misner: Space is Dynamical in Arnowitt Deser Misner: Space is Dynamical in GRGR

Hawking-Penrose: Generic Initial Conditions Hawking-Penrose: Generic Initial Conditions Lead to SingularitiesLead to Singularities

No Global Existence TheoremsNo Global Existence Theorems Cosmic Censorship Conjecture in Various Cosmic Censorship Conjecture in Various

Space-times With Fixed Asymptotic Structure Space-times With Fixed Asymptotic Structure (e.g. Asymptotically Flat):(e.g. Asymptotically Flat):

All Singularities are Black Holes.All Singularities are Black Holes.

Geometry of a Black HoleGeometry of a Black Hole

Asymptotically Stationary Outside RAsymptotically Stationary Outside RSS Shrinking d-2 Area and Infinite Length in time Shrinking d-2 Area and Infinite Length in time

~ R~ RS S In Region Inside R In Region Inside RSS Interior and Exterior Become Causally Interior and Exterior Become Causally

Disconnected Disconnected For Large Black Holes, Relation Between RFor Large Black Holes, Relation Between RS S

and M Depends on R :and M Depends on R : dsds22 = - f(r) dt = - f(r) dt22 + dr + dr22/ f(r) + r/ f(r) + r22 d dWW22

(1 – 2c(1 – 2cddM/rM/rd-3d-3 +/- (r/R) +/- (r/R)2 2 ) = f(r)) = f(r) 3R3R -2 -2 = -/+ = -/+ LL L LPP

2 2 LLP P = 10 = 10-33 -33 cm h/2cm h/2pp = c = 1 = c = 1

In a Black Hole, Space Stretches In a Black Hole, Space Stretches and Squeezes Faster Than Lightand Squeezes Faster Than Light

Particle Scattering at b < 2 EParticle Scattering at b < 2 Ecm cm G = R G = RS S (E) (E)

Leads to Black Hole Formation (Penrose, Leads to Black Hole Formation (Penrose, Amati-Veneziano, Matschull, TB Fischler, Amati-Veneziano, Matschull, TB Fischler, Giddings-Eardley etc.)Giddings-Eardley etc.)

Suggests that probes of distances smaller Suggests that probes of distances smaller than Lthan LP P ~ 10~ 10-33 -33 cm. fail, and instead create cm. fail, and instead create

larger and larger black holes – the UV/IR larger and larger black holes – the UV/IR correspondencecorrespondence

Black Hole Entropy Formula 4S LBlack Hole Entropy Formula 4S LPPd-2 d-2 = A of = A of

Bekenstein and Hawking Leads to Association of Bekenstein and Hawking Leads to Association of DOF With Boundary d – 2 Surfaces: Thorn, ‘t DOF With Boundary d – 2 Surfaces: Thorn, ‘t Hooft, Susskind, Fischler, BoussoHooft, Susskind, Fischler, Bousso

Feynman & Wilson: Definition of a Quantum Feynman & Wilson: Definition of a Quantum Theory Comes From High Energy Theory Comes From High Energy

Asymptotic Darkness: Black Holes Dominate Asymptotic Darkness: Black Holes Dominate High Energy Spectrum and Definition of High Energy Spectrum and Definition of HamiltonianHamiltonian

Vacua: Different IR Superselection Sectors of Vacua: Different IR Superselection Sectors of Single Local Field TheorySingle Local Field Theory

This Concept is Not Applicable in Quantum This Concept is Not Applicable in Quantum Theories of GravityTheories of Gravity

Trying to Create Other VacuaTrying to Create Other Vacua Guth Farhi: Create Local Region of Meta-stable Guth Farhi: Create Local Region of Meta-stable

dS space, which can Inflate, in Asymptotically dS space, which can Inflate, in Asymptotically flat background. Instead find Black Hole: flat background. Instead find Black Hole: inflating region separated from exterior by BH inflating region separated from exterior by BH singularity.singularity.

TB: Same true for large regions of zero c.c. TB: Same true for large regions of zero c.c. vacuum separated from another by potential vacuum separated from another by potential barrier. Rbarrier. RS S = M = T R = M = T R2 2 >> R Also true for >> R Also true for regions on moduli space separated by regions on moduli space separated by f > f > mmPP

Matrix Theory and AdS/CFT Confirm Idea that Matrix Theory and AdS/CFT Confirm Idea that Change of Vacuum Corresponds to Change of Change of Vacuum Corresponds to Change of Parameters in the HamiltonianParameters in the Hamiltonian

AdS/CFTAdS/CFT

Quantum Theory of AdSQuantum Theory of AdSd d X Y is CFT on X Y is CFT on Conformal Boundary R X SConformal Boundary R X Sd-2 d-2 : :

dsds2 2 = - (1 + r= - (1 + r22 / R / R22 ) dt ) dt22 + dr + dr22/(1 + r/(1 + r22 / R / R22 ) ) + r+ r22 d dWW22

BH Entropy Formula fits CFT Entropy S = BH Entropy Formula fits CFT Entropy S = c (TR)c (TR)d-2d-2

If c = (RMIf c = (RMPP))k k k = (dk = (d22 - 3d – 6)/(d – 1) - 3d – 6)/(d – 1)C.C. counts # of Degrees of Freedom in C.C. counts # of Degrees of Freedom in

CFTCFT

Large RMLarge RMP P Insufficient for Low Insufficient for Low

Curvature Space-timeCurvature Space-time Scalar field in Euclidean AdSScalar field in Euclidean AdSdd:: f ~ f ~ r r - - DD

D = D = (1/2) [(d – 1) +/- ((d – 1)(1/2) [(d – 1) +/- ((d – 1)2 2 + (m R)+ (m R)22))1/21/2]] Zero mass, marginal op. ; negative mZero mass, marginal op. ; negative m2 2 , real , real D, D,

BBreitenlohner Freedman allowed tachyon.reitenlohner Freedman allowed tachyon. Normal CFT spectrum of primaries: Normal CFT spectrum of primaries:

exponentially growing number of bulk fields. exponentially growing number of bulk fields. “R is string scale” (generally no weakly “R is string scale” (generally no weakly coupled string interpretation). coupled string interpretation).

Only known examples with gap in dimension Only known examples with gap in dimension spectrum are exactly supersymmetric CFTs spectrum are exactly supersymmetric CFTs

Holographic Renormalization Holographic Renormalization GroupGroup

Asymptotic values of moduli are lines of fixed pointsAsymptotic values of moduli are lines of fixed points d dimensional SUGRA (and gauged SUGRA) with scalars has d dimensional SUGRA (and gauged SUGRA) with scalars has

potential with multiple AdS minima. Static domain wall solutions potential with multiple AdS minima. Static domain wall solutions interpolating between minima (Poincare patch) are mapped by interpolating between minima (Poincare patch) are mapped by AdS/CFT to RG flows between Super-conformal field theories. Two AdS/CFT to RG flows between Super-conformal field theories. Two minima have different “c” (c thm becomes area thm). Domain walls minima have different “c” (c thm becomes area thm). Domain walls interpolate between theories, not vacua.interpolate between theories, not vacua.

No examples where lower fixed point is stable SUSY violating CFT.No examples where lower fixed point is stable SUSY violating CFT. Non Susic Orbifolds of e.g. SU(N) MSYM are not conformal beyond Non Susic Orbifolds of e.g. SU(N) MSYM are not conformal beyond

planar order (cf. Scherk-Schwarz in “flat space”)planar order (cf. Scherk-Schwarz in “flat space”) Horowitz-Hertog: Coleman de Lucia instanton for “decay of SUSic Horowitz-Hertog: Coleman de Lucia instanton for “decay of SUSic

AdS” corresponds to perturbation of SCFT by SUSY violating AdS” corresponds to perturbation of SCFT by SUSY violating marginal operator which is unbounded from below (subtle details of marginal operator which is unbounded from below (subtle details of IR b.c.)IR b.c.)

The Holographic PrincipleThe Holographic Principle

In asymptotically flat and AdS (including In asymptotically flat and AdS (including approximately AdS) space-times, theory is approximately AdS) space-times, theory is defined on conformal boundary.defined on conformal boundary.

Boundary correlators in AdS, S-matrix in Boundary correlators in AdS, S-matrix in AF only gauge invariant objects. AF only gauge invariant objects.

In rigorously established examples, small In rigorously established examples, small c.c. realized only with exact (AF) or c.c. realized only with exact (AF) or asymptotically exact (AAdS) SUSYasymptotically exact (AAdS) SUSY

An Approach to Local Description An Approach to Local Description of Quantum Gravity & Cosmologyof Quantum Gravity & Cosmology

Causal Diamond: Intersection of Interior of Causal Diamond: Intersection of Interior of Forward Light Cone of P and Backward Light-Forward Light Cone of P and Backward Light-cone of Q in Future of P. Region under cone of Q in Future of P. Region under experimental control of time-like observer experimental control of time-like observer travelling between P and Q.travelling between P and Q.

The Holographic Screen of the Causal Diamond The Holographic Screen of the Causal Diamond of a Local Observer Has Finite Area: Local of a Local Observer Has Finite Area: Local Physics Has Inherent Quantum Ambiguity – Physics Has Inherent Quantum Ambiguity – Quantum Origin of General CovarianceQuantum Origin of General Covariance

Holographic CosmologyHolographic Cosmology

HHn n ((xx) Hilbert Space of Observer n Time Steps ) Hilbert Space of Observer n Time Steps From Big BangFrom Big Bang

Dim [Dim [HHn n ((xx)] = Dim [)] = Dim [KK]]n n K K Irrep. Of Pixel Irrep. Of Pixel Algebra Defined BelowAlgebra Defined Below

Equal Area Step Time Slicing.Equal Area Step Time Slicing. Dynamics Takes Place in Dynamics Takes Place in HHnmax nmax ((xx) for Maximal ) for Maximal

Area Slice, ButArea Slice, But H(n,k,H(n,k,xx) = H) = Hinin (k,k,x) + H (k,k,x) + Houtout (n,k, (n,k,xx)) Enforces Concept of Particle Horizon: D.O.F. Enforces Concept of Particle Horizon: D.O.F.

Inside Horizon Do Not Interact With Those Inside Horizon Do Not Interact With Those Outside Until Horizon ExpandsOutside Until Horizon Expands

Degrees of Freedom of Quantum Degrees of Freedom of Quantum GravityGravity

SSa a (y) Real Components of d – 2 Spinor (y) Real Components of d – 2 Spinor Determines Orientation of Holoscreen at y via Determines Orientation of Holoscreen at y via SST T ggm1 … mkm1 … mk S 1<k<d -2 (Cartan – Penrose) S 1<k<d -2 (Cartan – Penrose)

SSaaII (m) S (m) Sbb

JJ (n ) + S (n ) + SbbJJ (n) S (n) Saa

II (m) = (m) = ddabab d dmn mn MMIJIJ

m,n pixelation of holoscreen. I,J refer to m,n pixelation of holoscreen. I,J refer to compact dimensionscompact dimensions

DOF of Supersymmetric Massless Particles DOF of Supersymmetric Massless Particles Penetrating Pixels of HoloscreenPenetrating Pixels of Holoscreen

16 16 Real Components per pixel implies Real Components per pixel implies graviton in spectrumgraviton in spectrum

The Dense Black Hole FluidThe Dense Black Hole Fluid A full holographic cosmology introduces a spatial lattice A full holographic cosmology introduces a spatial lattice

of observers with the topology of d-1 Euclidean spaceof observers with the topology of d-1 Euclidean space Nearest neighbor observers have overlap Hilbert space Nearest neighbor observers have overlap Hilbert space

of dimension (dimof dimension (dim K) K)n-1n-1 at Time Step n. Dynamics must at Time Step n. Dynamics must agree on overlap. Only known solution of these difficult agree on overlap. Only known solution of these difficult conditions. Hconditions. Hn n ((xx) is the same random Hamiltonian for ) is the same random Hamiltonian for each each x. x. Chosen from a distribution with free fermion Chosen from a distribution with free fermion spectrum for large n.spectrum for large n.

Gives rise to Emergent Space-Time Geometry : Flat Gives rise to Emergent Space-Time Geometry : Flat FRW with p = FRW with p = r. r.

Horizon Filling Black Hole for Every Observer at Every Horizon Filling Black Hole for Every Observer at Every TimeTime

Asymptotically Flat Space Super-Asymptotically Flat Space Super-Poincare Invariant?Poincare Invariant?

Superstring/M-theory Provides Ample Evidence Superstring/M-theory Provides Ample Evidence This is TrueThis is True

Multi-parameter Web of Supersymmetric Multi-parameter Web of Supersymmetric Theories in SpaceTime d = 4 … 11Theories in SpaceTime d = 4 … 11

Strange Dualities and Connections (11D Theory Strange Dualities and Connections (11D Theory Compactified on K3 4-folds = 10D Heterotic Compactified on K3 4-folds = 10D Heterotic String Compactified on 3-torus etc. ) String Compactified on 3-torus etc. ) Explanation of Origin of Gauge Theory and Explanation of Origin of Gauge Theory and ChiralityChirality

No Consistent AF Space-time w/o SUSYNo Consistent AF Space-time w/o SUSY

These are mathematical theories of These are mathematical theories of quantum gravity, but don’t describe the quantum gravity, but don’t describe the real worldreal world

Exact SUSY, Poincare Invariance, Exact SUSY, Poincare Invariance, Massless Spin Zero ParticlesMassless Spin Zero Particles

No CosmologyNo CosmologyAsymptotically Anti-deSitter (negative c.c.) Asymptotically Anti-deSitter (negative c.c.)

String Theories Lead to Similar String Theories Lead to Similar Conclusions.Conclusions.

AdS/CFT Gives Rigorous Evidence for AdS/CFT Gives Rigorous Evidence for UV/IR Connection Between Black Hole UV/IR Connection Between Black Hole Spectrum and c.c..Spectrum and c.c..

The Real World Has(?) Positive The Real World Has(?) Positive LL

Evidence From Distant Supernovae, Ages Evidence From Distant Supernovae, Ages of Globular Clusters/Universe, Large Scale of Globular Clusters/Universe, Large Scale Structure, Cosmic Microwave BackgroundStructure, Cosmic Microwave Background

If True: Holographic Principle Implies If True: Holographic Principle Implies Finite Number (ln N = 10Finite Number (ln N = 10120120) of Quantum ) of Quantum States (TB – Fischler) States (TB – Fischler)

(1 – 2c(1 – 2cdd M/r M/rd-3d-3 -(r/R) -(r/R)2 2 ) = 0) = 0No Exact Scattering Theory as in No Exact Scattering Theory as in

Conventional String TheoryConventional String Theory

The Sombrero Galaxy: See It The Sombrero Galaxy: See It Before It’s Too LateBefore It’s Too Late

The Quantum Theory of de Sitter The Quantum Theory of de Sitter (dS) Space(dS) Space

Holographic Principle Implies Finite Holographic Principle Implies Finite Number of States (TB Fischler)Number of States (TB Fischler)

Holoscreen Variables Holoscreen Variables cciiAA : :

[c[ciiAA , c* , c*BB

kk ] ]++ = d = diikk ddAA

B B 22N(N + 1) N(N + 1) States StatesSpinor Bundle Over Fuzzy 2 SphereSpinor Bundle Over Fuzzy 2 SphereN ~ RN ~ RdSdS

Static Hamiltonian H : Everything dS Static Hamiltonian H : Everything dS d(cays): Spectrum d(cays): Spectrum e [0, e [0, ccTT ] ] T = T = 1/21/2ppRR

So What Are Ordinary Energies? : PSo What Are Ordinary Energies? : P00

[P[P0 0 , H] ~ f(P, H] ~ f(P00/R) (bounded) : P/R) (bounded) : P0 0 Resolves Resolves Degeneracy of H. Low PDegeneracy of H. Low P0 0 Eigenstates Eigenstates Approximately Stable. Particle and Black Hole Approximately Stable. Particle and Black Hole Masses, etc.Masses, etc.

Semiclassical Result That Semiclassical Result That r r Is Thermal In PIs Thermal In P0 0 Is Is Reproduced If PReproduced If P0 0 Eigenvalue is Related To Eigenvalue is Related To Entropy Deficit of EigenspaceEntropy Deficit of Eigenspace

This Relation Valid For Small Black Holes S = This Relation Valid For Small Black Holes S = SSdS dS - 2- 2ppRPRP00

If H is Random Hamiltonian, then Ergodic Thm Implies Random Initial State Time Averages Are Thermal With Temperature T (Choose c Appropriately) Density of States ~ e – p R2 T, R In Planck Units r = e- 2pR H Is Vacuum Density Matrix.

Black Holes as Excitations of Black Holes as Excitations of Fermionic PixelsFermionic Pixels

Factor space of states with Factor space of states with cciiA A | | BH > = 0 with BH > = 0 with ii

Bounded by NBounded by N-- < N/3 < N/31/21/2 , A by N , A by N+ + ([N([N-- + N + N++]]2 2 - N - N+ +

NN- - = N= N22): right entropy for black hole of ): right entropy for black hole of

Schwarzschild radius ~ NSchwarzschild radius ~ N-- in Planck units. in Planck units.

PP0 0 = (ln 2 /2 = (ln 2 /2 pp))1/2 1/2 MMP P (N(N22 - 2 - 2 NN )(N )(N22 - - NN))1/21/2

N N Fermion Number OperatorFermion Number Operator <<P<<P00>> = M>> = MBHBH in BH Ensemble, With Small in BH Ensemble, With Small

Fluctuations (BH States Not All Eigenstates w/ Fluctuations (BH States Not All Eigenstates w/ Same Eigenvale)Same Eigenvale)

Supersymmetric Particles as Supersymmetric Particles as Excitations of Fermionic PixelsExcitations of Fermionic Pixels

Heuristic Argument: Maximal Field Theoretic Heuristic Argument: Maximal Field Theoretic Entropy in Single Horizon Comes from ~ NEntropy in Single Horizon Comes from ~ N3/2 3/2

Particles With Momenta < NParticles With Momenta < N-1/2 -1/2 and is o(Nand is o(N3/2 3/2 ): ): NN1/2 1/2 Independent Horizon VolumesIndependent Horizon Volumes

Same Counting Comes From Using Pixel Ops. Same Counting Comes From Using Pixel Ops. In Band of Matrix Made of Blocks of Size NIn Band of Matrix Made of Blocks of Size N1/21/2 : : Matrix Theory Like Counting of Particle Matrix Theory Like Counting of Particle Momenta and Explanation of Particle Statistics.Momenta and Explanation of Particle Statistics.

Suggests Corrections [P , Q ] ~ N Suggests Corrections [P , Q ] ~ N -1/2-1/2 m mPP ~ ~ LL1/41/4 : : mm3/2 3/2 ~ ~ LL1/41/4

Matrix Theory Decomposition of dS Matrix Theory Decomposition of dS Pixel OperatorsPixel Operators

Each Band: Field Theory D.O.F. of Single Each Band: Field Theory D.O.F. of Single HorizonHorizon

Maximal FT Entropy K ~ NMaximal FT Entropy K ~ N1/2 1/2 ~ ~ LL-1/4-1/4

Permutation Symmetry of Blocks in a Band: Permutation Symmetry of Blocks in a Band: Particle StatisticsParticle Statistics

Exchanges of Bands – Discrete Analog of dS Exchanges of Bands – Discrete Analog of dS Transformations Not in R X SO(3)??Transformations Not in R X SO(3)??

TB, Fiol, Morisse: SUSY particles in limit. Chiral TB, Fiol, Morisse: SUSY particles in limit. Chiral mults. Not Graviton. Need More Pixel Ops. mults. Not Graviton. Need More Pixel Ops. Compact DimensionsCompact Dimensions

ConclusionsConclusions Supersymmetric Quantum Theories of Gravity Abound and make Supersymmetric Quantum Theories of Gravity Abound and make

Beautiful Quantitative Predictions About Imaginary Worlds, Some of Beautiful Quantitative Predictions About Imaginary Worlds, Some of Which Have Properties Tantalizingly Close to Our Own (Heterotic Which Have Properties Tantalizingly Close to Our Own (Heterotic Strings on CY3, 11D SUGRA on G2)Strings on CY3, 11D SUGRA on G2)

Observables are defined as generalized scattering amplitudes on Observables are defined as generalized scattering amplitudes on infinite asymptotic boundariesinfinite asymptotic boundaries

The real world is not supersymmetric and may not have such infinite The real world is not supersymmetric and may not have such infinite boundaries (acceleration of the universe). boundaries (acceleration of the universe).

The challenge is to find a consistent quantum gravitational system The challenge is to find a consistent quantum gravitational system which violates SUSY in a space-time of low curvature, and to which violates SUSY in a space-time of low curvature, and to understand the relation between the splitting in supermultiplets and understand the relation between the splitting in supermultiplets and the c.c.the c.c.

Beginnings of a Quantum Theory of dS Space Which May Solve Beginnings of a Quantum Theory of dS Space Which May Solve This Problem Exist. Route to Derivation of mThis Problem Exist. Route to Derivation of m3/23/2 ~ ~ LL1/4 1/4 Clear. Need Clear. Need to Understand Compactificationto Understand Compactification