General Relativity is Not a Field Theory Cauchy-Kowalevskaya – Field Theory Has Finite # DOF per Space Point Cauchy-Kowalevskaya – Field Theory Has Finite # DOF per Space Point Arnowitt Deser Misner: Space is Dynamical in GR Arnowitt Deser Misner: Space is Dynamical in GR Hawking-Penrose: Generic Initial Conditions Lead to Singularities Hawking-Penrose: Generic Initial Conditions Lead to Singularities No Global Existence Theorems No Global Existence Theorems Cosmic Censorship Conjecture in Various Space-times With Fixed Asymptotic Structure (e.g. Asymptotically Flat): Cosmic Censorship Conjecture in Various Space-times With Fixed Asymptotic Structure (e.g. Asymptotically Flat): All Singularities are Black Holes. All Singularities are Black Holes.

Geometry of a Black Hole Asymptotically Stationary Outside R S Asymptotically Stationary Outside R S Shrinking d-2 Area and Infinite Length in time ~ R S In Region Inside R S Shrinking d-2 Area and Infinite Length in time ~ R S In Region Inside R S Interior and Exterior Become Causally Disconnected Interior and Exterior Become Causally Disconnected For Large Black Holes, Relation Between R S and M Depends on R : For Large Black Holes, Relation Between R S and M Depends on R : ds 2 = - f(r) dt 2 + dr 2 / f(r) + r 2 d W 2 ds 2 = - f(r) dt 2 + dr 2 / f(r) + r 2 d W 2 (1 – 2c d M/r d-3 +/- (r/R) 2 ) = f(r) (1 – 2c d M/r d-3 +/- (r/R) 2 ) = f(r) 3R -2 = -/+ L L P 2 L P = cm h/2 p = c = 1 3R -2 = -/+ L L P 2 L P = cm h/2 p = c = 1

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

Particle Scattering at b < 2 E cm G = R S (E) Leads to Black Hole Formation (Penrose, Amati-Veneziano, Matschull, TB Fischler, Giddings-Eardley etc.) Particle Scattering at b < 2 E cm G = R S (E) Leads to Black Hole Formation (Penrose, Amati-Veneziano, Matschull, TB Fischler, Giddings-Eardley etc.) Suggests that probes of distances smaller than L P ~ cm. fail, and instead create larger and larger black holes – the UV/IR correspondence Suggests that probes of distances smaller than L P ~ cm. fail, and instead create larger and larger black holes – the UV/IR correspondence

Black Hole Entropy Formula 4S L P d-2 = A of Bekenstein and Hawking Leads to Association of DOF With Boundary d – 2 Surfaces: Thorn, ‘t Hooft, Susskind, Fischler, Bousso Black Hole Entropy Formula 4S L P d-2 = A of Bekenstein and Hawking Leads to Association of DOF With Boundary d – 2 Surfaces: Thorn, ‘t Hooft, Susskind, Fischler, Bousso Feynman & Wilson: Definition of a Quantum Theory Comes From High Energy Feynman & Wilson: Definition of a Quantum Theory Comes From High Energy Asymptotic Darkness: Black Holes Dominate High Energy Spectrum and Definition of Hamiltonian Asymptotic Darkness: Black Holes Dominate High Energy Spectrum and Definition of Hamiltonian Vacua: Different IR Superselection Sectors of Single Local Field Theory Vacua: Different IR Superselection Sectors of Single Local Field Theory This Concept is Not Applicable in Quantum Theories of Gravity This Concept is Not Applicable in Quantum Theories of Gravity

Trying to Create Other Vacua Guth Farhi: Create Local Region of Meta-stable dS space, which can Inflate, in Asymptotically flat background. Instead find Black Hole: inflating region separated from exterior by BH singularity. Guth Farhi: Create Local Region of Meta-stable dS space, which can Inflate, in Asymptotically flat background. Instead find Black Hole: inflating region separated from exterior by BH singularity. TB: Same true for large regions of zero c.c. vacuum separated from another by potential barrier. R S = M = T R 2 >> R Also true for regions on moduli space separated by f > m P TB: Same true for large regions of zero c.c. vacuum separated from another by potential barrier. R S = M = T R 2 >> R Also true for regions on moduli space separated by f > m P Matrix Theory and AdS/CFT Confirm Idea that Change of Vacuum Corresponds to Change of Parameters in the Hamiltonian Matrix Theory and AdS/CFT Confirm Idea that Change of Vacuum Corresponds to Change of Parameters in the Hamiltonian

AdS/CFT Quantum Theory of AdS d X Y is CFT on Conformal Boundary R X S d-2 : Quantum Theory of AdS d X Y is CFT on Conformal Boundary R X S d-2 : ds 2 = - (1 + r 2 / R 2 ) dt 2 + dr 2 /(1 + r 2 / R 2 ) + r 2 d W 2 ds 2 = - (1 + r 2 / R 2 ) dt 2 + dr 2 /(1 + r 2 / R 2 ) + r 2 d W 2 BH Entropy Formula fits CFT Entropy S = c (TR) d-2 BH Entropy Formula fits CFT Entropy S = c (TR) d-2 If c = (RM P ) k k = (d 2 - 3d – 6)/(d – 1) If c = (RM P ) k k = (d 2 - 3d – 6)/(d – 1) C.C. counts # of Degrees of Freedom in CFT C.C. counts # of Degrees of Freedom in CFT

Large RM P Insufficient for Low Curvature Space-time Scalar field in Euclidean AdS d : Scalar field in Euclidean AdS d : f ~ r - D f ~ r - D D = (1/2) [(d – 1) +/- ((d – 1) 2 + (m R) 2 ) 1/2 ] D = (1/2) [(d – 1) +/- ((d – 1) 2 + (m R) 2 ) 1/2 ] Zero mass, marginal op. ; negative m 2, real D, B reitenlohner Freedman allowed tachyon. Zero mass, marginal op. ; negative m 2, real D, B reitenlohner Freedman allowed tachyon. Normal CFT spectrum of primaries: exponentially growing number of bulk fields. “R is string scale” (generally no weakly coupled string interpretation). Normal CFT spectrum of primaries: exponentially growing number of bulk fields. “R is string scale” (generally no weakly coupled string interpretation). Only known examples with gap in dimension spectrum are exactly supersymmetric CFTs Only known examples with gap in dimension spectrum are exactly supersymmetric CFTs

Holographic Renormalization Group Asymptotic values of moduli are lines of fixed points Asymptotic values of moduli are lines of fixed points d dimensional SUGRA (and gauged SUGRA) with scalars has potential with multiple AdS minima. Static domain wall solutions interpolating between minima (Poincare patch) are mapped by AdS/CFT to RG flows between Super-conformal field theories. Two minima have different “c” (c thm becomes area thm). Domain walls interpolate between theories, not vacua. d dimensional SUGRA (and gauged SUGRA) with scalars has potential with multiple AdS minima. Static domain wall solutions interpolating between minima (Poincare patch) are mapped by AdS/CFT to RG flows between Super-conformal field theories. Two minima have different “c” (c thm becomes area thm). Domain walls 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 planar order (cf. Scherk-Schwarz in “flat space”) Non Susic Orbifolds of e.g. SU(N) MSYM are not conformal beyond planar order (cf. Scherk-Schwarz in “flat space”) Horowitz-Hertog: Coleman de Lucia instanton for “decay of SUSic AdS” corresponds to perturbation of SCFT by SUSY violating marginal operator which is unbounded from below (subtle details of IR b.c.) Horowitz-Hertog: Coleman de Lucia instanton for “decay of SUSic AdS” corresponds to perturbation of SCFT by SUSY violating marginal operator which is unbounded from below (subtle details of IR b.c.)

The Holographic Principle In asymptotically flat and AdS (including approximately AdS) space-times, theory is defined on conformal boundary. In asymptotically flat and AdS (including approximately AdS) space-times, theory is defined on conformal boundary. Boundary correlators in AdS, S-matrix in AF only gauge invariant objects. Boundary correlators in AdS, S-matrix in AF only gauge invariant objects. In rigorously established examples, small c.c. realized only with exact (AF) or asymptotically exact (AAdS) SUSY In rigorously established examples, small c.c. realized only with exact (AF) or asymptotically exact (AAdS) SUSY

An Approach to Local Description of Quantum Gravity & Cosmology Causal Diamond: Intersection of Interior of Forward Light Cone of P and Backward Light- cone of Q in Future of P. Region under experimental control of time-like observer travelling between P and Q. Causal Diamond: Intersection of Interior of Forward Light Cone of P and Backward Light- cone of Q in Future of P. Region under experimental control of time-like observer travelling between P and Q. The Holographic Screen of the Causal Diamond of a Local Observer Has Finite Area: Local Physics Has Inherent Quantum Ambiguity – Quantum Origin of General Covariance The Holographic Screen of the Causal Diamond of a Local Observer Has Finite Area: Local Physics Has Inherent Quantum Ambiguity – Quantum Origin of General Covariance

Holographic Cosmology H n (x) Hilbert Space of Observer n Time Steps From Big Bang H n (x) Hilbert Space of Observer n Time Steps From Big Bang Dim [ H n (x)] = Dim [ K ] n K Irrep. Of Pixel Algebra Defined Below Dim [ H n (x)] = Dim [ K ] n K Irrep. Of Pixel Algebra Defined Below Equal Area Step Time Slicing. Equal Area Step Time Slicing. Dynamics Takes Place in H nmax (x) for Maximal Area Slice, But Dynamics Takes Place in H nmax (x) for Maximal Area Slice, But H(n,k,x) = H in (k,k,x) + H out (n,k,x) H(n,k,x) = H in (k,k,x) + H out (n,k,x) Enforces Concept of Particle Horizon: D.O.F. Inside Horizon Do Not Interact With Those Outside Until Horizon Expands Enforces Concept of Particle Horizon: D.O.F. Inside Horizon Do Not Interact With Those Outside Until Horizon Expands

Degrees of Freedom of Quantum Gravity S a (y) Real Components of d – 2 Spinor Determines Orientation of Holoscreen at y via S T g m1 … mk S 1<k<d -2 (Cartan – Penrose) S a (y) Real Components of d – 2 Spinor Determines Orientation of Holoscreen at y via S T g m1 … mk S 1<k<d -2 (Cartan – Penrose) S a I (m) S b J (n ) + S b J (n) S a I (m) = d ab d mn M IJ S a I (m) S b J (n ) + S b J (n) S a I (m) = d ab d mn M IJ m,n pixelation of holoscreen. I,J refer to compact dimensions m,n pixelation of holoscreen. I,J refer to compact dimensions DOF of Supersymmetric Massless Particles Penetrating Pixels of Holoscreen DOF of Supersymmetric Massless Particles Penetrating Pixels of Holoscreen 16 Real Components per pixel implies graviton in spectrum 16 Real Components per pixel implies graviton in spectrum

The Dense Black Hole Fluid A full holographic cosmology introduces a spatial lattice of observers with the topology of d-1 Euclidean space A full holographic cosmology introduces a spatial lattice of observers with the topology of d-1 Euclidean space Nearest neighbor observers have overlap Hilbert space of dimension (dim K) n-1 at Time Step n. Dynamics must agree on overlap. Only known solution of these difficult conditions. H n (x) is the same random Hamiltonian for each x. Chosen from a distribution with free fermion spectrum for large n. Nearest neighbor observers have overlap Hilbert space of dimension (dim K) n-1 at Time Step n. Dynamics must agree on overlap. Only known solution of these difficult conditions. H n (x) is the same random Hamiltonian for each x. Chosen from a distribution with free fermion spectrum for large n. Gives rise to Emergent Space-Time Geometry : Flat FRW with p = r. Gives rise to Emergent Space-Time Geometry : Flat FRW with p = r. Horizon Filling Black Hole for Every Observer at Every Time Horizon Filling Black Hole for Every Observer at Every Time

Asymptotically Flat Space Super- Poincare Invariant? Superstring/M-theory Provides Ample Evidence This is True Superstring/M-theory Provides Ample Evidence This is True Multi-parameter Web of Supersymmetric Theories in SpaceTime d = 4 … 11 Multi-parameter Web of Supersymmetric Theories in SpaceTime d = 4 … 11 Strange Dualities and Connections (11D Theory Compactified on K3 4-folds = 10D Heterotic String Compactified on 3-torus etc. ) Explanation of Origin of Gauge Theory and Chirality Strange Dualities and Connections (11D Theory Compactified on K3 4-folds = 10D Heterotic String Compactified on 3-torus etc. ) Explanation of Origin of Gauge Theory and Chirality No Consistent AF Space-time w/o SUSY No Consistent AF Space-time w/o SUSY

These are mathematical theories of quantum gravity, but don’t describe the real world These are mathematical theories of quantum gravity, but don’t describe the real world Exact SUSY, Poincare Invariance, Massless Spin Zero Particles Exact SUSY, Poincare Invariance, Massless Spin Zero Particles No Cosmology No Cosmology Asymptotically Anti-deSitter (negative c.c.) String Theories Lead to Similar Conclusions. Asymptotically Anti-deSitter (negative c.c.) String Theories Lead to Similar Conclusions. AdS/CFT Gives Rigorous Evidence for UV/IR Connection Between Black Hole Spectrum and c.c.. AdS/CFT Gives Rigorous Evidence for UV/IR Connection Between Black Hole Spectrum and c.c..

The Real World Has(?) Positive L Evidence From Distant Supernovae, Ages of Globular Clusters/Universe, Large Scale Structure, Cosmic Microwave Background Evidence From Distant Supernovae, Ages of Globular Clusters/Universe, Large Scale Structure, Cosmic Microwave Background If True: Holographic Principle Implies Finite Number (ln N = ) of Quantum States (TB – Fischler) If True: Holographic Principle Implies Finite Number (ln N = ) of Quantum States (TB – Fischler) (1 – 2c d M/r d-3 -(r/R) 2 ) = 0 (1 – 2c d M/r d-3 -(r/R) 2 ) = 0 No Exact Scattering Theory as in Conventional String Theory No Exact Scattering Theory as in Conventional String Theory

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

The Quantum Theory of de Sitter (dS) Space Holographic Principle Implies Finite Number of States (TB Fischler) Holographic Principle Implies Finite Number of States (TB Fischler) Holoscreen Variables c i A : Holoscreen Variables c i A : [c i A, c* B k ] + = d i k d A B 2 N(N + 1) States [c i A, c* B k ] + = d i k d A B 2 N(N + 1) States Spinor Bundle Over Fuzzy 2 Sphere Spinor Bundle Over Fuzzy 2 Sphere N ~ R dS N ~ R dS Static Hamiltonian H : Everything dS d(cays): Spectrum e [0, c T ] T = 1/2 p R Static Hamiltonian H : Everything dS d(cays): Spectrum e [0, c T ] T = 1/2 p R

So What Are Ordinary Energies? : P 0 So What Are Ordinary Energies? : P 0 [P 0, H] ~ f(P 0 /R) (bounded) : P 0 Resolves Degeneracy of H. Low P 0 Eigenstates Approximately Stable. Particle and Black Hole Masses, etc. [P 0, H] ~ f(P 0 /R) (bounded) : P 0 Resolves Degeneracy of H. Low P 0 Eigenstates Approximately Stable. Particle and Black Hole Masses, etc. Semiclassical Result That r Is Thermal In P 0 Is Reproduced If P 0 Eigenvalue is Related To Entropy Deficit of Eigenspace Semiclassical Result That r Is Thermal In P 0 Is Reproduced If P 0 Eigenvalue is Related To Entropy Deficit of Eigenspace This Relation Valid For Small Black Holes S = S dS - 2 p RP 0 This Relation Valid For Small Black Holes S = S dS - 2 p RP 0 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 R 2 T, R In Planck Units r = e - 2 p R H Is Vacuum Density Matrix.

Black Holes as Excitations of Fermionic Pixels Factor space of states with c i A | BH > = 0 with i Bounded by N - = 0 with i Bounded by N - < N/3 1/2, A by N + ([N - + N + ] 2 - N + N - = N 2 ): right entropy for black hole of Schwarzschild radius ~ N - in Planck units. P 0 = (ln 2 /2 p ) 1/2 M P (N N )(N 2 - N ) 1/2 P 0 = (ln 2 /2 p ) 1/2 M P (N N )(N 2 - N ) 1/2 N Fermion Number Operator N Fermion Number Operator > = M BH in BH Ensemble, With Small Fluctuations (BH States Not All Eigenstates w/ Same Eigenvale) > = M BH in BH Ensemble, With Small Fluctuations (BH States Not All Eigenstates w/ Same Eigenvale)

Supersymmetric Particles as Excitations of Fermionic Pixels Heuristic Argument: Maximal Field Theoretic Entropy in Single Horizon Comes from ~ N 3/2 Particles With Momenta < N -1/2 and is o(N 3/2 ): N 1/2 Independent Horizon Volumes Heuristic Argument: Maximal Field Theoretic Entropy in Single Horizon Comes from ~ N 3/2 Particles With Momenta < N -1/2 and is o(N 3/2 ): N 1/2 Independent Horizon Volumes Same Counting Comes From Using Pixel Ops. In Band of Matrix Made of Blocks of Size N 1/2 : Matrix Theory Like Counting of Particle Momenta and Explanation of Particle Statistics. Same Counting Comes From Using Pixel Ops. In Band of Matrix Made of Blocks of Size N 1/2 : Matrix Theory Like Counting of Particle Momenta and Explanation of Particle Statistics. Suggests Corrections [P, Q ] ~ N -1/2 m P ~ L 1/4 : m 3/2 ~ L 1/4 Suggests Corrections [P, Q ] ~ N -1/2 m P ~ L 1/4 : m 3/2 ~ L 1/4

Matrix Theory Decomposition of dS Pixel Operators

Each Band: Field Theory D.O.F. of Single Horizon Each Band: Field Theory D.O.F. of Single Horizon Maximal FT Entropy K ~ N 1/2 ~ L -1/4 Maximal FT Entropy K ~ N 1/2 ~ L -1/4 Permutation Symmetry of Blocks in a Band: Particle Statistics Permutation Symmetry of Blocks in a Band: Particle Statistics Exchanges of Bands – Discrete Analog of dS Transformations Not in R X SO(3)?? Exchanges of Bands – Discrete Analog of dS Transformations Not in R X SO(3)?? TB, Fiol, Morisse: SUSY particles in limit. Chiral mults. Not Graviton. Need More Pixel Ops. Compact Dimensions TB, Fiol, Morisse: SUSY particles in limit. Chiral mults. Not Graviton. Need More Pixel Ops. Compact Dimensions

Conclusions Supersymmetric Quantum Theories of Gravity Abound and make Beautiful Quantitative Predictions About Imaginary Worlds, Some of Which Have Properties Tantalizingly Close to Our Own (Heterotic Strings on CY3, 11D SUGRA on G2) Supersymmetric Quantum Theories of Gravity Abound and make Beautiful Quantitative Predictions About Imaginary Worlds, Some of Which Have Properties Tantalizingly Close to Our Own (Heterotic Strings on CY3, 11D SUGRA on G2) Observables are defined as generalized scattering amplitudes on infinite asymptotic boundaries Observables are defined as generalized scattering amplitudes on infinite asymptotic boundaries The real world is not supersymmetric and may not have such infinite boundaries (acceleration of the universe). The real world is not supersymmetric and may not have such infinite boundaries (acceleration of the universe). The challenge is to find a consistent quantum gravitational system which violates SUSY in a space-time of low curvature, and to understand the relation between the splitting in supermultiplets and the c.c. The challenge is to find a consistent quantum gravitational system which violates SUSY in a space-time of low curvature, and to understand the relation between the splitting in supermultiplets and the c.c. Beginnings of a Quantum Theory of dS Space Which May Solve This Problem Exist. Route to Derivation of m 3/2 ~ L 1/4 Clear. Need to Understand Compactification Beginnings of a Quantum Theory of dS Space Which May Solve This Problem Exist. Route to Derivation of m 3/2 ~ L 1/4 Clear. Need to Understand Compactification