Kinematic Space 2: Seeing into the bulk with Integral Geometry Sam McCandlish Stanford University Work in progress with: Bartek Czech, Lampros Lamprou, Ben Mosk, James Sully 1
How do we probe the bulk? From a dual field theory localized on the boundary, we see into the bulk. Some CFT quantities involve probes extending inward from the boundary: Entanglement entropy, Wilson loops Local bulk operators however, are represented as smeared operators on the boundary How does a smeared operator see into the bulk? 2
Bulk Reconstruction Let φ be a free field in AdS = m 2 The AdS/CFT dictionary relates φ to a boundary operator O (z,x) z O(x) How do we solve this boundary value problem? A non-standard Cauchy problem! 3
Bulk Reconstruction Standard construction: bulk local operator is a smeared boundary operator [Bena 99; Hamilton, Kabat, Lifschytz, Lowe 05] (z,x) = The smearing function K is determined by brute force from the bulk mode expansion Some remaining questions: dx 0 K (x 0 z,x) O (x 0 ) Extension to other geometries? (Causal vs Entanglement wedge?) How to write as an operator at one time? Why does this see into the bulk? A non-standard Cauchy problem! = m 2 (z,x) z O(x) 4
Bulk Reconstruction A detour through kinematic space will provide insight! Real Space Kinematic Space Non-standard Standard 5 Cauchy problem Cauchy problem
Plan Bulk field reconstruction and integral geometry The X-Ray transform Structure of kinematic space Intertwining operators and kinematic fields Geodesic operators and Local bulk operators The space of CFT bilocals Generalizations and applications (work in progress): Bulk tensor operators and the modular Hamiltonian Higher dimensions MERA Beyond the vacuum Bulk interactions (1/N corrections) 6
The X-Ray Transform Kinematic space K: the space of oriented spacelike geodesics in a manifold M A point in K corresponds to a geodesic in M Real Space M Kinematic Space K γ γ 7
The X-Ray Transform The X-Ray transform: maps a function on real space to a function on kinematic space by integrating over geodesics Inversion formulas are known for some symmetric cases (hyperbolic space, flat space) M K γ γ f(x) Rf( )= dsf(x) 8
The X-Ray Transform Example: let M be H 2 Parameterize geodesics by midpoint θ and opening angle α f(x) Rf(, ) =, dsf(x) M α K θ α 9 θ
The X-Ray Transform Definition of X-Ray Transform: f(x) Rf(, ) = dsf(x) d(x, ), Inversion formula: [Helgason] f (x) = 1 1 0 d dp average Rf ( ) d(x, )=p! dp sinh p 10
Structure of Kinematic Space We want to find an equation of motion for R ( ) First, we must fix a metric on kinematic space - a distance function on the space of geodesics Kinematic Space for AdS n or H n is a highly symmetric space No distinguished geodesics: all spacelike geodesics are related by symmetry We can fix a unique metric on K using this symmetry 11
Structure of Kinematic Space Fix a unique metric on K using invariance under conformal symmetry x+dx y+dy Parameterize a geodesic by its endpoints The metric must be of the form Scaling and translation fix f µ = Inversion fixes a = 2 b ds 2 = x µ! xµ x 2 1 (x y) 2 µ 2 (x y) µ (x y) (x y) 2 ds 2 = f µ dx µ dy 1 (x y) 2 a (x y) µ (x y) (x y) 2 + b µ Note to experts: These same requirements fix the CFT two-point function of vector fields ho µ (x) O (y)i! dx µ dy x! γ y γ+dγ 12
Structure of Kinematic Space What is this metric? ds 2 = 1 (x y) 2 µ 2 (x y) µ (x y) (x y) 2! dx µ dy γ It is (2d)-dimensional - d space, d time x y For AdS3, it is ds2 x ds2: ds 2 = 1 2 du L du R u R u L 2 + 1 2 2 dv L dv R v R 2 v L 2 x 1 x 0 (u L,v L )= 2, x1 x 0 2 Left-moving ds Right-moving ds For Hyperbolic 2-space, it is the diagonal ds2 The causal structure is determined by containment of boundary causal diamonds 13 The asymptotic past of K is the boundary of AdS Boundary of AdS
Intertwining Operators We want to find two equations of motion for R ( ) We will find: R AdS3 = ds ds R Also, ( dsl dsr ) Rf =0 AdS3 = m 2 AdS3 ds ds R = m 2 R ( dsl dsr ) R =0 ds2 x ds2 Non-standard Cauchy problem Standard Cauchy problem 14
Intertwining Operators Rf( ) First step: write as an integral over all of space Rf ( )= ds f (x) = d d xf(x) F (d (x, )) For Euclidean space: For Lorentzian space: F (x) = F (x) = (x) x d 2 S d 2 (x) x d 2 log xs d 3 d(x, ) Can show: ( AdS3 + ds ds ) F (d (x, )) = 0 Comes from relationship to conformal Casimir operator Integrating by parts proves the intertwinement relation: R AdS3 = ds ds R 15
Intertwining Operators Rf ( )= ds f (x) = d d xf(x) F (d (x, )) Can also show: ( dsl dsr ) F (d (x, )) = 0 Directly implies ( dsl dsr ) Rf =0 Comes from a redundancy in the Radon transform f is a function of 3 variables, but Rf is a function of 4 variables The constraint reduces this dimensionality by 1 E.g., can determine boosted geodesics in terms of unboosted geodesics Similar result in flat space: John s equation 16
Geodesic Operators Now, solve the Cauchy problem to determine the geodesic operators Equations of Motion: ( dsl + dsr ) R = m 2 R Boundary Conditions: (z,x) z O(x) AdS3 = m 2 ( dsl dsr ) R =0 R (x, y) c (y x) O x + y 2 x y The geodesic operator depends only on the boundary values in the causal diamond it subtends! 17
Geodesic Operators Now, solve the Cauchy problem to determine the geodesic operators γ x y The result: a smeared operator on the causal diamond (see previous talk for details) R ( )= ds (x) = d 2 x 0 y x 0 x 0 x y x 2 O (x 0 ) 18
Geodesic Operators The result: a smeared operator on the causal diamond (see previous talk for details) R ( )= ds (x) = d 2 x 0 y x 0 x 0 x y x 2 O (x 0 ) Note: We have two choices for the causal diamond two representations of the geodesic operator y x 19
How can we obtain local bulk operators? Invert the X-ray transform on a time-slice. f (x) = Local Bulk Operators 1 1 0 d dp We integrate over all geodesics on a slice (center) = 1 2 average Rf ( ) d(x, )=p For each geodesic, choose a diamond! dp sinh p d d tan d R (, ) d α θ We only need to integrate over half of kinematic space for the time-slice α θ 20
Local Bulk Operators Choosing which half of kinematic space determines which smearing representation of the bulk operator we obtain Global smearing Poincaré smearing 21
Local Bulk Operators Let s obtain the global smearing function (center) = 1 half K d d tan d R (, ) d In global coordinates, becomes R (, ) = d d R apple (cos cos ( + )) (cos cos ( )) 2 1 cos (2 ) /2 1 O (, + ) The result matches the HKLL result: (center) = strip d 2 x h (cos ) HKLL Result 2 log cos + (cos ) Divergent piece i 2 log O (x) The divergent piece vanishes its Fourier modes have no overlap with the bulk mode expansion (see HKLL 2006) 22
The Operator Product Expansion Consider a CFT in d dimensions. A product of local primary operators can be written as a sum over the primary operators in the theory: O 1 (x) O 2 (y) = X k C 12k 1+#@ +#@ 2 +... O k (x) Fixed by conformal invariance X k [O 1 (x) O 2 (y)] k The coefficients look a lot like a Taylor expansion. Can we undo it? D E [O 1 (x) O 2 (y)] k = N 12k d d z O 1 (x) O 2 (y) Õµ... k (z) O kµ... (z) A shadow operator with k gives the correct conformal transformation properties, but also includes unwanted pieces [Simmons-Duffin 2014] k = d What integration region? How to normalize? Is this a useful OPE? 23
The Operator Product Expansion The fix: integrate over a causal diamond D [O 1 (x) O 2 (y)] k = N 12k d 2 z O 1 (x) O 2 (y) Õµ... k Is this projected OPE related to the AdS3 geodesic operator? This object obeys a conformal Casimir equation E (z) O kµ... (z) (L 1 + L 2 ) 2 [O 1 (x) O 2 (y)] k = C 2 k [O 1 (x) O 2 (y)] k This Casimir equation is just a kinematic space wave equation if It also obeys the constraint equation: ds ds m 2 [O 1 (x) O 1 (y)] k =0 O 1 = O 2 m 2 = C 2 k ( dsl dsr )[O 1 (x) O 1 (y)] k = h h [O1 (x) O 2 (y)] k 24 Is has boundary conditions for small separation of the operators: x + y [O 1 (x) O 2 (y)] k (y x) k 1 2 O k 2
The Operator Product Expansion The projected OPE obeys the same equations of motion as the geodesic operator for AdS3, with the same boundary conditions They must be equal! A product of boundary operators is localized on a geodesic O 1 (x)o 1 (y) = [O 1 (x) O 1 (y)] k / A bulk local operator can be written as a projected smeared bilocal k (center) = 1 half K 1 (x y) 2 1 X k d d tan d d 1 (x y) 2 1 C 11k x k x!y y ds k + tensors, local descendants (1 cos ( 2 1 )) 2 1 [O 1 ( ) O 1 ( + )] k 25
Work in Progress 26
Bulk Tensor Fields What do we do with tensor operators in the bulk? [O 1 (x) O 2 (y)] k = N 12k A hint from the modular Hamiltonian: d 2 x D O 1 (x 0 d 2 z D O 1 (x) O 2 (y) Õµ... k R) O 1 (x 0 + R) T E µ (x) T µ (x) = 2 E (z) O kµ... (z) dx (x x 0) 2 R 2 2R T 00 (x) The modular Hamiltonian is a kinematic operator! H mod = A = ds g µ ˆv µˆv [Nozaki, Numasawa, Prudenziati, Takayanagi] [de Boer, Myers, Heller, Neiman] [Lashkari, McDermott, van Raamsdonk] [Swingle, van Raamsdonk] Entanglement equation of motion Einstein s equations 27 Take the OPE of twist operators to derive the modular Hamiltonian n (x) n (y) = 1 (y x) c 6(n 1 n) (1 + (1 n) H mod + other operators) Suppressed by (1 n) or supported on multiple orbifold sheets
Higher Dimensions What do we do in higher dimensions? Spacelike separated points correspond to geodesics Timelike separated points correspond to minimal surfaces Define the Radon transform of a function - its integral over the minimal surface The previous discussion holds - a diamond-smeared boundary operator is a bulk surface operator The (d-1) boosts provide (d-1) constraints The modular Hamiltonian is again a kinematic operator 28 Timelike kinematic space for is H d ds d [deboer, Heller, Myers, Neiman 2015]
Tensor Networks The MERA tensor network for a 2D CFT ground state naturally lives on ds 2 more generally, on 2D kinematic space [Beny 2011; Czech, Lamprou, McCandlish, Sully 2015] Each tensor is associated with a boundary ball in its asymptotic past Does kinematic space tell us how to generalize MERA to higher dimensions, including time? 29
Beyond the Vacuum The previous discussion holds if we consider a quotient of AdS Kinematic space is also a quotient of the AdS kinematic space Non-minimal entwinement geodesics come from winding diamond operators 30
Interactions How do 1/N corrections appear in this description? The X-Ray (or Radon) transform transforms the free equation of motion: X m 2 =0 =) K m 2 =0 ( = R ) If we add local bulk interactions, we get a nonlocal interaction in kinematic space (similar to momentum space) X m 2 = 3 =) K m 2 = R R 1 3 Can we include Virasoro descendants in 2D? 31
Entanglement Wedge Geodesic operators are Rindler-wedge operators How can we get local Rindler-wedge operators? How can we invert the X-ray transform with limited data? Is the Ryu-Takayanagi transition a phase transition in the limited data inversion formula for the X-ray transform? vs 32