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Information Geometry and Quantum Fields Information Geometry and Quantum Fields Kevin T. Grosvenor Julius-Maximilians Universität Würzburg, Germany SPIGL ’20, Les Houches, France July 30, 2020 Collaborators SciPost Phys. 8 (2020) 073 [arXiv:2001.02683] Johanna Erdmenger Ro Jefferson 1 32 Sources and References 2 32 Outline 1. Introduction to AdS/CFT 2. Fisher metric, Symmetries and Uniqueness 3. Classical metric on States: Instantons 4. Classical metric on Theories: Ising Model 5. Quantum metric on States: Coherent states 6. Connections and Curvature 7. Conclusions and Outlook 3 32 AdS/CFT Correspondence Holography: G. ’t Hooft (THU-93/26, ’93), L. Susskind (JMP 36, ’95) AdS/CFT: J. Maldacena (ATMP 2, ’97) Weak-strong duality. D R d E e d x φ0(x) O(x) = e−SSUGRA CFT φ(0;x)=φ0(x) 4 32 d;2 Embedding into R : −(X0)2 + (X1)2 + ··· + (Xd)2 − (Xd+1)2 = −L2 d(d−1) 1;:::;d Isometries: 2 rotations among X , one rotation between X0 and Xd+1, and 2d boosts mixing X0 and Xd+1 with X1;:::;d. Algebra of isometries is so(d; 2) . Anti-de Sitter Spacetime 2 L2 2 r2 µ ν Poincaré patch: ds = r2 dr + L2 ηµνdx dx . Constant negative curvature, maximally symmetric spacetime. 1 R d+1 p Vacuum of Einstein-Hilbert S = 16πG d x −g(R − 2Λ) with constant negative Λ (cosmological constant). 5 32 Anti-de Sitter Spacetime 2 L2 2 r2 µ ν Poincaré patch: ds = r2 dr + L2 ηµνdx dx . Constant negative curvature, maximally symmetric spacetime. 1 R d+1 p Vacuum of Einstein-Hilbert S = 16πG d x −g(R − 2Λ) with constant negative Λ (cosmological constant). d;2 Embedding into R : −(X0)2 + (X1)2 + ··· + (Xd)2 − (Xd+1)2 = −L2 d(d−1) 1;:::;d Isometries: 2 rotations among X , one rotation between X0 and Xd+1, and 2d boosts mixing X0 and Xd+1 with X1;:::;d. Algebra of isometries is so(d; 2) . 5 32 CFT fields form representations of the conformal algebra. Conformal Field Theory Translations Pµ = i@µ Rotations/boosts Jµν = 2ix[µ@ν] µ Dilatations D = ix @µ 2 ν SCT Kµ = i(x @µ − 2xµx @ν) Conformal algebra is so(d; 2) . In 1 + 1d, extended to infinite Virasoro algebra. 6 32 Conformal Field Theory Translations Pµ = i@µ Rotations/boosts Jµν = 2ix[µ@ν] µ Dilatations D = ix @µ 2 ν SCT Kµ = i(x @µ − 2xµx @ν) Conformal algebra is so(d; 2) . In 1 + 1d, extended to infinite Virasoro algebra. CFT fields form representations of the conformal algebra. 6 32 AdS/CFT Textbook 7 32 Quantum Information Theory in Holography Entanglement probes of the bulk. [0603011 Ryu, Takayanagi; 0705.0016 Hubeny, Rangamani, Takayanagi; 1408.3203 Engelhardt, Wall] Quantum error correction =) bulk reconstruction. [1411.7041 Almheiri, Dong, Harlow] Tensor networks =) bulk-boundary maps. [1601.01694 Hayden et. al.] Holographic distance measures. [pure: 1507.0755 Lashkari, van Raamsdonk; mixed: 1701.02319 Banerjee, Erdmenger, Sarkar] 8 32 Previous Info Geom + Physics work Ruppeiner (’95,’96) - stat phys and thermo (e.g., ideal gas) Brody & Hook (2008) - stat phys and thermo (e.g., vdW gas) Janke et. al. (2002, 2003) - 1d Ising & Spherical models Dolan, Johnston, Kenna (2002) - Potts model Amari (’97) - Neural networks Ke & Nielsen (2016) - machine learning Heckman (2013) - string theory Blau, Narain, Thompson (2001) - YM instantons in 3+1-d 9 32 R Fisher metric: gij(ξ) ≡ dx pξ(x) @i ln pξ(x) @j ln pξ(x) R R q q Fact 1: gij (ξ) = − dx pξ(x) @i@j ln pξ(x) and gij (ξ) = 4 dx @i pξ(x) @j pξ(x) Fact 2: gij is positive semi-definite. Fisher Metric x x x pξ(x) ≥ 0 p R R dx pξ(x) = 1 ξ injective @ ξ −−−−! pξ smooth, @i ≡ @ξi 10 32 R R q q Fact 1: gij (ξ) = − dx pξ(x) @i@j ln pξ(x) and gij (ξ) = 4 dx @i pξ(x) @j pξ(x) Fact 2: gij is positive semi-definite. Fisher Metric x x x pξ(x) ≥ 0 p R R dx pξ(x) = 1 ξ injective @ ξ −−−−! pξ smooth, @i ≡ @ξi R Fisher metric: gij(ξ) ≡ dx pξ(x) @i ln pξ(x) @j ln pξ(x) 10 32 Fisher Metric x x x pξ(x) ≥ 0 p R R dx pξ(x) = 1 ξ injective @ ξ −−−−! pξ smooth, @i ≡ @ξi R Fisher metric: gij(ξ) ≡ dx pξ(x) @i ln pξ(x) @j ln pξ(x) R R q q Fact 1: gij (ξ) = − dx pξ(x) @i@j ln pξ(x) and gij (ξ) = 4 dx @i pξ(x) @j pξ(x) Fact 2: gij is positive semi-definite. 10 32 dµ2 + 2dσ2 ds2 = σ2 2 2 1 σ2 2 dµ +dσ Alternative example: pµ,σ (x) = gives ds = . πσ (x−µ)2+σ2 2σ2 Comments: 1. It is not true that all free theories have flat Fisher metrics. 2. This hyperbolic space is a priori unrelated to AdS/CFT. Gaussian Distribution A simple example (1d Gaussians): Random variable space is X = R. Parameter space is Ξ = upper half-plane = (µ, σ) 2 R × R>0 (x−µ)2 1 − p (x) = p e 2σ2 µ,σ 2π σ 11 32 2 2 1 σ2 2 dµ +dσ Alternative example: pµ,σ (x) = gives ds = . πσ (x−µ)2+σ2 2σ2 Comments: 1. It is not true that all free theories have flat Fisher metrics. 2. This hyperbolic space is a priori unrelated to AdS/CFT. Gaussian Distribution A simple example (1d Gaussians): Random variable space is X = R. Parameter space is Ξ = upper half-plane = (µ, σ) 2 R × R>0 (x−µ)2 1 − p (x) = p e 2σ2 µ,σ 2π σ dµ2 + 2dσ2 ds2 = σ2 11 32 Comments: 1. It is not true that all free theories have flat Fisher metrics. 2. This hyperbolic space is a priori unrelated to AdS/CFT. Gaussian Distribution A simple example (1d Gaussians): Random variable space is X = R. Parameter space is Ξ = upper half-plane = (µ, σ) 2 R × R>0 (x−µ)2 1 − p (x) = p e 2σ2 µ,σ 2π σ dµ2 + 2dσ2 ds2 = σ2 2 2 1 σ2 2 dµ +dσ Alternative example: pµ,σ (x) = gives ds = . πσ (x−µ)2+σ2 2σ2 11 32 Gaussian Distribution A simple example (1d Gaussians): Random variable space is X = R. Parameter space is Ξ = upper half-plane = (µ, σ) 2 R × R>0 (x−µ)2 1 − p (x) = p e 2σ2 µ,σ 2π σ dµ2 + 2dσ2 ds2 = σ2 2 2 1 σ2 2 dµ +dσ Alternative example: pµ,σ (x) = gives ds = . πσ (x−µ)2+σ2 2σ2 Comments: 1. It is not true that all free theories have flat Fisher metrics. 2. This hyperbolic space is a priori unrelated to AdS/CFT. 11 32 gij(ξ) = @i@j (ξ) R Fact: gij (ξ) = h(Fi − hFiiξ)(Fj − hFj iξiξ where hfiξ ≡ dx f(x) pξ(x). Exponential Family Generalization of the Gaussian (Exponential Family): i pξ(x) = exp C(x) + ξ Fi(x) − (ξ) . R i (ξ) = ln dx exp C(x) + ξ Fi(x) is a normalization factor. Assume that fC; Fig are lin. indep. so ξ ! pξ is bijective. 12 32 Exponential Family Generalization of the Gaussian (Exponential Family): i pξ(x) = exp C(x) + ξ Fi(x) − (ξ) . R i (ξ) = ln dx exp C(x) + ξ Fi(x) is a normalization factor. Assume that fC; Fig are lin. indep. so ξ ! pξ is bijective. gij(ξ) = @i@j (ξ) R Fact: gij (ξ) = h(Fi − hFiiξ)(Fj − hFj iξiξ where hfiξ ≡ dx f(x) pξ(x). 12 32 Gaussian example: µ!µ+c −−−−−−−−! −−−−!x!x+c (µ,σ)!λ(µ,σ) −−−−−−−−! −−−−−!x!λx Symmetries ξ ! ξ~(ξ) is a symmetry of p if it can be undone by x ! x~(x) pξ~(x) dx = pξ(~x) dx~ Fact: a symmetry of p is also a symmetry of gij . 13 32 Symmetries ξ ! ξ~(ξ) is a symmetry of p if it can be undone by x ! x~(x) pξ~(x) dx = pξ(~x) dx~ Fact: a symmetry of p is also a symmetry of gij . Gaussian example: µ!µ+c −−−−−−−−! −−−−!x!x+c (µ,σ)!λ(µ,σ) −−−−−−−−! −−−−−!x!λx 13 32 Non-Uniqueness Infinitely many distributions give the same information geometry [Clingman, Murugan and Shock 2015] E.g., 1d Gaussian and Cauchy-Lorentz give 2d hyperbolic space Cauchy-Lorentz is not in the exponential family Weirder example in Clingman, Murugan, Shock (2015) of a 3d distribution that gives 2d hyperbolic space, but has none of its symmetries. 14 32 Lessons Many statistical models S can lead to the same Fisher metric (Gaussian and Cauchy-Lorentz). FM inherits the symmetries of S but can have more 15 32 R 4 µ 2 4 Simpler model: d x @µφ @ φ − g φ (Euclidean signature) 2 σ2 Exact solution to e.o.m.: φ~µ,σ(~x) = gσ j~x−~µj2+σ2 16σ2 j~x−~µj2−σ2 Lagrangian density: L~µ,σ(~x) = g2 (j~x−~µj2+σ2)4 L~µ,σ(~x) < 0 for j~x − ~µj < σ, so not a good prob. distr. 2 2 4 16 σ4 If instead we use −φ @ φ − g φ , we get p~µ,σ(~x) = g2 (j~x−~µj2+σ2)4 and the information geometry is 5d hyperbolic space. Classical metric on States: Instantons Blau, Narain, Thompson (2001) - YM instantons in 3+1-d Uses prescription of Hitchin (1990): probability distribution is taken to be the Lagrangian density evaluated on the state.
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