Finite regions, spherical entanglement, and quantum gravity

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1 Finite regions, spherical entanglement, and quantum gravity Hal Haggard in collaboration with Eugenio Bianchi November 14th, 2013 Quantum Gravity Seminar Perimeter Institute

2 Finite region quantum gravity What is the physics of a finite region of a quantum spacetime? 1

3 Divergences... no approach to quantum gravity can claim complete success that does not explain in full and convincing detail the ultimate fate of the divergences of perturbative quantum gravity." H. Nicolai gr-qc/ The mass of the LHC scalar boson (125 GeV) narrowly avoids vacuum instability and the Landau pole of self-couplings: Currently understood physics may hold up to the Planck scale! 2

4 Polyhedral program E i = ɛ ijk e i b ek c dx b dx c = 8πl 2 Pl J i A Gauss constraint Closure relation ˆV Pol = The volume of a quantum polyhedron [Bianchi, Doná and Speziale] 3

5 Bohr-Sommerfeld Quantization of Harmonic Osc. Require: J = pdq = (n + 1 γ 2 )2π. 4

6 Tetrahedra Phase space (fixed A 1,..., A 4 ): Take volume as Hamiltonian: H = V 2 = 2 9 A 1 ( A 2 A 3 ) p = A 1 + A 2, q = ang. gen. p {q, p} = 1 Orbit areas are complete elliptic integrals, ( 4 ) 4 J(E) = a i K(m) + b i Π(αi 2, m) E i=1 i=1 5

7 V Tet = A 1 ( A 2 A 3 ) A 1 = j + 1/2 A 2 = j + 1/2 A 3 = j + 1/2 A 4 = j + 3/2 4 2 = Numerical = Bohr-Som [PRL 107, ]

8 Volume gap: pentahedra 5, 4 5, 3 5, 2 5, 1 4, 5 4, 3 4, 2 4, 1 3, 5 3, 4 3, 2 3, 1 2, 5 2, 4 2, 3 2, 1 1, 5 1, 4 1, 3 1, 2 Analytic volume: ( V Pent = αβγ ᾱ β γ ) 2 3 A 1 ( A 2 A 3 ), α A 4 ( A 3 A 2 )/ A 1 ( A 2 A 3 ) etc. Robust mechanism volume gap: chaotic dynamics & low density of states at low volume 7

9 Outstanding challenge How to identify the ground state of a general relativistic theory? Want to coordinate individual grains of space to recover Minkowski space from this quantum theory. Use entanglement 8

10 Outline 1 Entanglement and the entanglement spectrum 2 The entanglement spectrum of a sphere 3 Interest for quantum gravity 9

11 Entanglement insights Tensor networks: Condensed matter and tensor networks: Matrix Product States (MPS), Projected Entangled Pair States (PEPS), Tensor networks (TNs) Area laws. Physical corner of the Hilbert space. Spin networks are tensor networks. Casini, Huerta & Myers have extensively studied the entanglement entropy of spherical causal domains. [Casini, Huerta & Myers 11] New result the entanglement eigensystem. 10

12 Entanglement Pure state Ψ H A H B. Schmidt decomposition: λi i A i B Ψ = i with i A and i B orthonormal bases in H A and H B respectively. Leads to the reduced density matrix ρ B = Tr A Ψ Ψ = λ i i B i B, i and the entanglement entropy S E Tr ρ B log ρ B = i λ i log λ i. 11

13 Entanglement spectrum Can always write ρ B = e H E, i.e. H E log ρ B, the entanglement Hamiltonian. Already diagonalized H E : ρ B = i e ɛ i i B i B with ɛ i (i = 1, 2,... ) the entanglement spectrum. (λ i = e ɛ i ) Provides thorough understanding of entanglement. Study for spacetime fields. [Li & Haldane 08] 12

14 Boundary conditions for the remainder For simplicity I restrict to: Scalar field ϕ(x), with m = 0 on flat D = spacetime unless otherwise stated. Metric signature (, +, +, +). Results apply to any conformal field theory. 13

15 Outline 1 Entanglement and the entanglement spectrum 2 The entanglement spectrum of a sphere 3 Interest for quantum gravity 14

16 Spherical causal domain Cauchy development of 3-ball with boundary 2-sphere of radius R: Spatial 3-ball B Cauchy development D(B). Entangling surface the boundary 2-sphere B = S 2. Choose adapted coordinates that preserve S 2 : similar to how polar coords fix (0, 0)... 15

17 Diamond coordinates Use hyperbolas Diamond coords (λ, σ, θ, φ): sh λ t = R ch λ + ch σ sh σ r = R ch λ + ch σ with λ (, ), σ [0, ). The Minkowski metric becomes ds 2 R 2 = (ch λ + ch σ) 2 [ dλ2 + dσ 2 + sh 2 σdω 2 ], a conformal rescaling of static κ = 1 FRW. 16

18 Conformal completion Diamond coordinates can be extended to all of Minkowski space E.g. region II: sh t = R λ ch σ ch λ, sh σ r = R ch σ ch λ λ σ. All hyperbolas asymp. null: 17

19 Rindler Limit Large σ limit: t = 2Re σ sh λ = l sh λ R r = 2Re σ ch λ = l ch λ coord transformation to (left) Rindler wedge. The proper distance from right corner is l = 2Re σ. 18

20 Entanglement (or foliation) Hamiltonian µ ξ µ = λ current J µ = T µν ξν with Tµν the stress-tensor. If µ T µ = 0 the charge is conserved C = R Tµν ξ µ dσν and generates the spatial foliation discussed above: Explicitly this charge is, Cin 1 = 2R Z B r 2 dr d Ω 1 2 ~ ϕ) ~ (R2 r 2 )(ϕ 2 + ϕ + ϕ2 19

21 Density matrix from Euclidean path integral Minkowski vacuum: Spherical density matrix: Rindler density matrix: sin λe cos λe + ch σ Bipolar sh σ coords r =R cos λe + ch σ te = R Z ρb = Dϕe SE = e 2πCin 20

22 Curved background QFT Inverted strategy: study finite region of flat spacetime using QFT on a curved background L = 1 g[g µν µ ϕ ν ϕ + ( m R(x))ϕ2 ]. Action is conformally invariant under { ḡµν = Ω 2 (x)g µν ϕ = Ω(x) 1 ϕ With ḡ µν = η µν the EOM transform as ( = g µν µ ν ) ϕ = Ω 3 [ 1 6 R]ϕ. Find ϕ by finding ϕ. 21

23 Sphere modes Sphere modes: 1 u ki (x) (2k) 2 M ikλ = (ch λ + ch σ)π kj (σ)yj (θ, φ)e R 22

24 Sphere modes analytically Metric g = dλ 2 + dσ 2 + sh 2 σd Ω 2 is static κ = 1 FRW. Separate: u k (x) = χ k (λ)π kj (σ)y M J (θ, φ) with χ k (λ) = (2k) 1 2 e ikλ ( ) d 1+J Π kj (σ) = N (k, J) shj σ cos (kσ) d ch σ YJ M (θ, φ) spherical harmonics M = J, J + 1,..., J; J = 0, 1,... ; 0 < k <. Sphere modes: ϕ = Ω(x) 1 ϕ, (recall Ω = R/(ch λ + ch σ) ) ū I k(x) = (2k) 1 2 R (ch λ + ch σ)π kj (σ)y M J (θ, φ)e ikλ. 23

25 Sphere vacuum and spherons Minkowski case: ϕ(x) = d D 1 k[a k u k (x) + a k u k (x)] The vacuum satisfies a k 0 M = 0. Cut out all in-out entanglement to get sphere vacuum Sphere case: ϕ(x) = dk J,M [si k ui k + si k ui k + sii k uii The sphere vacuum satisfies sk I 0 S = sk II 0 S = 0. s I k creates spherons, excitations localized within the spherical entangling surface. k + sii k uii k ] 24

26 Spherical entanglement spectrum Curved space, traceless (improved) stress tensor T µν = 2 3 µ ϕ ν ϕ 1 6 g µν( ρ ϕ ρ ϕ) 1 3 ϕ µ ν ϕ g µν ϕ ϕ [R ab 1 2 g abr] ϕ 2. Entanglement Hamiltonian C in = B T µν ξ µ dσ ν. Entanglement spectrum C in = dk k(s I k si k 1 pt. ener. [s I zero k 2, si k]). J,M 25

27 Schmidt decomposition of Minkowski vacuum Total Hamiltonian C = C in C out C = dk k(s I J,M k si k+s II k sii k ) 0 M = dk J,M e ɛ k u I k u II k. 26

28 Outline 1 Entanglement and the entanglement spectrum 2 The entanglement spectrum of a sphere 3 Interest for quantum gravity 27

29 Spin network entanglement This suggests an intriguing perspective on spin networks. Individual nodes are like the interior of the sphere with no entanglement between a given node and its neighbors. Can we choreograph entanglement to yield the Minkowski vacuum? A wealth of condensed matter research on entanglement and tensor networks to draw from. 28

30 Enticing direction Clearly we want more nodes. The entanglement Hamiltonian for multiple diamonds in Minkowski space studied by [Casini & Huerta]. Results for fermion in D = spacetime: H = H loc + H nonloc. Two regions of radius R separated by distances (a) 1 8 R, (b) R, 1 and (c) 2000 R. What is the physical significance of the non-local mixing? 29

31 Two-point functions Leveraging conformal symmetry we can achieve a more substantial characterization of the sphere vacuum, finding its two-point functions: D + (x, x ) = Ω D 2 2 (x) D+ (x, x )Ω D 2 2 (x ) 1+1 spacetime: D + (x, x ) = 1 4π log λ2 + σ spacetime: D + (x, x ) = (ch λ + ch σ) σ(ch λ + ch σ ) 4π 2 R 2 sh ( σ) [ λ 2 + σ 2 ] 30

32 Sorkin-Johnston vacuum Causal sets provide the only fully controlled Lorentz invariant discretization of spacetime. A remarkable state, the Sorkin-Johnston vacuum: [Johnston 09, Sorkin 11] proposed vacuum for a causal set within a causal diamond. It is defined only through referencing the diamond s interior has no entanglement with outside and so does not satisfy the Reeh-Schlieder theorem. Afshordi et al have just investigated the Sorkin-Johnston vacuum in 2D analytically and numerically... [Afshordi et al 12] 31

33 Comparison... and amongst many other things they found a surprise: near the L and R diamond corners the Sorkin-Johnston two-point function is that of a static mirror in Minkowski spacetime. How does the sphere vacuum compare? In the limit λ = λ = 0, σ, σ : D + 2D 1 4π log (log l l ) D + 4D 1 2π(l 2 l 2 ) log l l The Rindler two-point functions. [Troost & Van Dam 79] (Recall l = 2Re σ is the proper distance from the right corner.) = the sphere and Sorkin-Johnston vacuums differ in 2D. Is this still true in 4D? Have the tools to completely characterize these differences. 32

34 Reeh-Schlieder theorem If quantum gravity cuts off the continuum what becomes of the Reeh-Schlieder theorem? A special, initial pure state of two q-bits, Ψ = 1 2 ( ) can be used to steer one of the q-bits onto whole bloch sphere. More generally: Schmidt rank of the decomp. Ψ = i λi i A i B is at most dimh A. There is no way to steer onto all of H B. In what sense, if any, does the Reeh-Schlieder theorem hold for a large but finite #d.o.f.? 33

35 Conclusions Looking to engineer the Minkowski vacuum and its entanglement from spin network superposition. Interesting connections to causal sets. What is the nature of the Sorkin-Johnston vacuum in 4D? The fate of the Reeh-Schlieder theorem for a large but finite number of degrees of freedom needs to be understood. Numerous possibilities More general entangling surfaces Continue to draw from condensed matter Anomalies 34

36 Credits Spherical spin network: Z. Merali, The origins of space and time, Nature News, Aug. 28, 2013 Special thanks to the Cracow School of Theoretical Physics, LIII Course, 2013 for a lovely school and setting while this work was begun. And to the Perimeter Institute for their gracious hosting of visitors and support during the continuation of this work. 35