Information Geometry and Quantum Fields

Home , Bures metric, Information geometry

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 √ Vacuum of Einstein-Hilbert S = 16πG d x −g(R − 2Λ) with constant negative Λ (cosmological constant).

5 32 Anti-de Sitter Spacetime

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) ∈ R × R>0

(x−µ)2 1 − p (x) = √ 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) ∈ R × R>0

(x−µ)2 1 − p (x) = √ 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) ∈ R × R>0

(x−µ)2 1 − p (x) = √ 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) ∈ R × R>0

(x−µ)2 1 − p (x) = √ 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∂jψ(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) − ψ(ξ) . R i ψ(ξ) = ln dx exp C(x) + ξ Fi(x) is a normalization factor.

Assume that {C,Fi} 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) − ψ(ξ) . R i ψ(ξ) = ln dx exp C(x) + ξ Fi(x) is a normalization factor.

Assume that {C,Fi} are lin. indep. so ξ → pξ is bijective.

gij(ξ) = ∂i∂jψ(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 |~x−~µ|2+σ2

16σ2 |~x−~µ|2−σ2 Lagrangian density: L~µ,σ(~x) = g2 (|~x−~µ|2+σ2)4

L~µ,σ(~x) < 0 for |~x − ~µ| < 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 (|~x−~µ|2+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.

16 32 16σ2 |~x−~µ|2−σ2 Lagrangian density: L~µ,σ(~x) = g2 (|~x−~µ|2+σ2)4

L~µ,σ(~x) < 0 for |~x − ~µ| < 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 (|~x−~µ|2+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. 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 |~x−~µ|2+σ2

16 32 2 2 4 16 σ4 If instead we use −φ ∂ φ − g φ , we get p~µ,σ(~x) = g2 (|~x−~µ|2+σ2)4 and the information geometry is 5d hyperbolic space.

Classical metric on States: Instantons

16σ2 |~x−~µ|2−σ2 Lagrangian density: L~µ,σ(~x) = g2 (|~x−~µ|2+σ2)4

L~µ,σ(~x) < 0 for |~x − ~µ| < σ, so not a good prob. distr.

16 32 Classical metric on States: Instantons

16σ2 |~x−~µ|2−σ2 Lagrangian density: L~µ,σ(~x) = g2 (|~x−~µ|2+σ2)4

L~µ,σ(~x) < 0 for |~x − ~µ| < 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 (|~x−~µ|2+σ2)4 and the information geometry is 5d hyperbolic space.

16 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

FM on the states of a ﬁeld theory can be sensitive to stability (scalar instanton).

17 32 Classical metric on Theories: 2d Ising Model

Geometry of space of couplings of a theory. 2d classical Ising model on a square lattice:

N N X X H = −J σi,jσi+1,j − K σi,jσi,j+1 i,j=1 i,j=1

Partition function: Z = Q P e−βH(σ) i σi=±1 1 −βH(σ) Probability distribution: pβJ,βK (σ) = Z e . This is in the exponential family!

gij = ∂i∂jf where f is the reduced free energy per site

18 32 2d Ising Model Fisher Metric Ricci Scalar

βK

1.2

1.0

0.8 ferromagnetic phase

0.6

0.4

0.2 disordered phase

0.0 βJ

0.0 0.2 0.4 0.6 0.8 1.0 1.2 (a) Ricci curvature as function of (b) 2d Ising Model phase couplings J, K (β = 1) diagram for βJ, βK ≥ 0

The scalar curvature diverges at the critical curve given by sinh(2βJ) sinh(2βK) = 1

19 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

FM on the states of a ﬁeld theory can be sensitive to stability (scalar instanton).

FM on the space of theories is sensitive to phase transitions (2d Ising model)

20 32 R d2z Well-known effective theory: S = 2π ψ∂ψ + ψ∂ψ + imψψ . 1 Can we reproduce the ln |m| divergence of the Fisher metric in the effective ﬁeld theory?

2d Isotropic Ising Model and 1d Majorana Fermions

√ 1 Set J = K = 1 and the critical temp. is βc = 2 ln 2 + 1 ≈ 0.44. gββ is the speciﬁc heat:

d2f 1 1 gββ = 2 ' ln ' ln , dβ |β − βc| |m|

tanh βc where m = 2 tanh β − 1 .

21 32 2d Isotropic Ising Model and 1d Majorana Fermions

√ 1 Set J = K = 1 and the critical temp. is βc = 2 ln 2 + 1 ≈ 0.44. gββ is the speciﬁc heat:

d2f 1 1 gββ = 2 ' ln ' ln , dβ |β − βc| |m|

tanh βc where m = 2 tanh β − 1 . R d2z Well-known effective theory: S = 2π ψ∂ψ + ψ∂ψ + imψψ . 1 Can we reproduce the ln |m| divergence of the Fisher metric in the effective ﬁeld theory?

21 32 i j If ξ and ξ are masses, then h∂iS ∂jSi is 4-pt and h∂iSi is 2-pt. For a free theory, the 4-pt reduces to 2-pt fns. For the free 1d Majorana fermion theory,

Z Λ p dp Λ 1 gmm ' 2 2 ' ln ' ln 0 p + m |m| |m|

2d Isotropic Ising Model and 1d Majorana Fermions

c.f. B. P. Dolan (1998): take p to be the path integrand. In a QFT with action S, the Fisher metric on the couplings ξi is 1 g = h∂ S ∂ Si − h∂ Sih∂ Si . ij spacetime vol i j i j

22 32 2d Isotropic Ising Model and 1d Majorana Fermions

c.f. B. P. Dolan (1998): take p to be the path integrand. In a QFT with action S, the Fisher metric on the couplings ξi is 1 g = h∂ S ∂ Si − h∂ Sih∂ Si . ij spacetime vol i j i j

i j If ξ and ξ are masses, then h∂iS ∂jSi is 4-pt and h∂iSi is 2-pt. For a free theory, the 4-pt reduces to 2-pt fns. For the free 1d Majorana fermion theory,

Z Λ p dp Λ 1 gmm ' 2 2 ' ln ' ln 0 p + m |m| |m|

22 32 1+α 1−α (α) 4 R 2 2 e.g., D (p||q) = 1−α2 1 − dx p q for α ∈ R. The α = 1 limit is the Kullback-Leibler divergence, or relative (1) R p entropy D (p||q) = dx p ln q .

∂2 0 gij(ξ) = ∂ξ0i ∂ξ0j D(pξ||pξ ) ξ0=ξ

Quantum metric on States: Divergences

Divergence: bifunctional D(p||q) of two distributions p and q. D is a measure of how “different” q is from p (D = 0 iff q = p).

23 32 Quantum metric on States: Divergences

Divergence: bifunctional D(p||q) of two distributions p and q. D is a measure of how “different” q is from p (D = 0 iff q = p). 1+α 1−α (α) 4 R 2 2 e.g., D (p||q) = 1−α2 1 − dx p q for α ∈ R. The α = 1 limit is the Kullback-Leibler divergence, or relative (1) R p entropy D (p||q) = dx p ln q .

∂2 0 gij(ξ) = ∂ξ0i ∂ξ0j D(pξ||pξ ) ξ0=ξ

23 32 Quantum metric on States: Bures Metric

A quantum state is described by a density matrix ρ. p → ρ and “ R p dx = 1” → “trρ = 1”. q 1/2 1/2 Bures distance DB(ρ1, ρ2) = 2 1 − tr ρ1 ρ2ρ1 For pure states DB(|ψ1i, |ψ2i) = 2 1 − hψ1|ψ2i . The Bures metric is the leading (quadratic) term in the expansion

of DB around ρ2 ≈ ρ1.

24 32 Quantum metric on States: Coherent States

c.f. Nozaki, Ryu, Takayanagi (2012) Two free Majorana fermions {a, a†} = {b, b†} = 1.

q 1 −λa†b† Fermionic coherent state: |ψλi = 1+|λ|2 e |Ωi with λ ∈ C. One free complex scalar [a, a†] = [b, b†] = 1.

p 2 −λa†b† Bosonic coherent state: |ψλi = 1 − |λ| e |Ωi with λ ∈ C. 2 dλ dλ 2 dλ dλ Fermions: ds = (1+|λ|2)2 and Bosons: ds = (1−|λ|2)2 Fermions: 2-sphere and Bosons: 2d hyperbolic space

25 32 The Fermionic coherent state contains only |Ωi and a†b†|Ωi. 1 In this 2d space, ρλ = 2 I +n ˆλ · ~σ , where ~σ = Pauli matrices and nˆλ is a real unit 3d vector built out of λ. A general 2 × 2 special unitary matrix takes the form U = eiθnˆ·~σ for some angle θ and axis nˆ. iθnˆ·~σ Conjugation of ρλ by U = e rotates nˆλ by θ around nˆ.

Symmetry algebra of ρλ is su(2) = so(3) .

The metric must be that of the 2-sphere.

2 1 In fact, ds = 4 dnˆλ · dnˆλ.

Symmetries of the Bures Metric

† Symmetry of a family of density matrices: ρξ → ρξ0 = UρξU for some (special) unitary matrix U.

26 32 A general 2 × 2 special unitary matrix takes the form U = eiθnˆ·~σ for some angle θ and axis nˆ. iθnˆ·~σ Conjugation of ρλ by U = e rotates nˆλ by θ around nˆ.

Symmetry algebra of ρλ is su(2) = so(3) .

The metric must be that of the 2-sphere.

2 1 In fact, ds = 4 dnˆλ · dnˆλ.

Symmetries of the Bures Metric

26 32 2 1 In fact, ds = 4 dnˆλ · dnˆλ.

Symmetries of the Bures Metric

† Symmetry of a family of density matrices: ρξ → ρξ0 = UρξU for some (special) unitary matrix U. The Fermionic coherent state contains only |Ωi and a†b†|Ωi. 1 In this 2d space, ρλ = 2 I +n ˆλ · ~σ , where ~σ = Pauli matrices and nˆλ is a real unit 3d vector built out of λ. A general 2 × 2 special unitary matrix takes the form U = eiθnˆ·~σ for some angle θ and axis nˆ. iθnˆ·~σ Conjugation of ρλ by U = e rotates nˆλ by θ around nˆ.

Symmetry algebra of ρλ is su(2) = so(3) .

The metric must be that of the 2-sphere.

26 32 Symmetries of the Bures Metric

Symmetry algebra of ρλ is su(2) = so(3) .

The metric must be that of the 2-sphere.

2 1 In fact, ds = 4 dnˆλ · dnˆλ.

26 32 The symmetry algebra of ρλ is so(2, 1) .

imθ (1, 2)-rotation by θ: Umn = e δmn. (1, 3)-boost by 1: Umn = δmn + 2 (mδm,n+1 − nδm+1,n). i (2, 3)-boost by 1: Umn = δmn + 2 (mδm,n+1 + nδm+1,n). The metric must be that of the 2d hyperbolic space.

2 1 † 1 0 0 In fact, ds = dmˆ 0 1 0 dmˆ λ for some mˆ λ. 4 λ 0 0 −1

Symmetries of the Bures Metric

1 † † n The Bosonic coherent state has n! (a b ) |Ωi for n = 0, 1,.... m+n 2 m n The (m, n)-comp. of ρλ is (ρλ)m,n = (−1) (1 − |λ| )λ λ .

27 32 2 1 † 1 0 0 In fact, ds = dmˆ 0 1 0 dmˆ λ for some mˆ λ. 4 λ 0 0 −1

Symmetries of the Bures Metric

1 † † n The Bosonic coherent state has n! (a b ) |Ωi for n = 0, 1,.... m+n 2 m n The (m, n)-comp. of ρλ is (ρλ)m,n = (−1) (1 − |λ| )λ λ .

The symmetry algebra of ρλ is so(2, 1) .

27 32 Symmetries of the Bures Metric

1 † † n The Bosonic coherent state has n! (a b ) |Ωi for n = 0, 1,.... m+n 2 m n The (m, n)-comp. of ρλ is (ρλ)m,n = (−1) (1 − |λ| )λ λ .

The symmetry algebra of ρλ is so(2, 1) .

2 1 † 1 0 0 In fact, ds = dmˆ 0 1 0 dmˆ λ for some mˆ λ. 4 λ 0 0 −1

27 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

FM on the states of a ﬁeld theory can be sensitive to stability (scalar instanton).

FM on the space of theories is sensitive to phase transitions (2d Ising model)

The Bures metric deﬁnes a metric on quantum states (coherent states).

28 32 The exponential family is 1-ﬂat.

What physics is captured by the different curvatures? 1 R 2 2 + 1d Chern-Simons: S = 4π tr Γ ∧ dΓ + 3 Γ ∧ Γ ∧ Γ gives the e.o.m. dΓ + Γ ∧ Γ = 0. Vanishing 1-curvature could be a particular solution.

Curvature

(α) 1−α α-connection: Γij,k = h(∂i∂j`ξ + 2 ∂i`ξ ∂j`ξ)∂k`ξiξ where ` = ln p.

∂3D(α)(p ||p ) (α) ξ ξ0 Exercise: Γ = − ∂kgij ij,k ∂ξ0i ∂ξ0j ∂ξ0k ξ0=ξ α = 0 corresponds to the standard Christoffel symbols. (α) α 6= 0 is not a metric connection (ie., ∇k gij 6= 0).

29 32 1 R 2 2 + 1d Chern-Simons: S = 4π tr Γ ∧ dΓ + 3 Γ ∧ Γ ∧ Γ gives the e.o.m. dΓ + Γ ∧ Γ = 0. Vanishing 1-curvature could be a particular solution.

Curvature

(α) 1−α α-connection: Γij,k = h(∂i∂j`ξ + 2 ∂i`ξ ∂j`ξ)∂k`ξiξ where ` = ln p.

∂3D(α)(p ||p ) (α) ξ ξ0 Exercise: Γ = − ∂kgij ij,k ∂ξ0i ∂ξ0j ∂ξ0k ξ0=ξ α = 0 corresponds to the standard Christoffel symbols. (α) α 6= 0 is not a metric connection (ie., ∇k gij 6= 0). The exponential family is 1-ﬂat.

What physics is captured by the different curvatures?

29 32 Curvature

(α) 1−α α-connection: Γij,k = h(∂i∂j`ξ + 2 ∂i`ξ ∂j`ξ)∂k`ξiξ where ` = ln p.

29 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

FM on the states of a ﬁeld theory can be sensitive to stability (scalar instanton).

FM on the space of theories is sensitive to phase transitions (2d Ising model)

The Bures metric deﬁnes a metric on quantum states (coherent states).

Many concepts of curvature. What do they all mean?

30 32 Thank you very much for listening!

Outlook

What do all the curvatures mean? What physics is captured?

Can we apply these concepts to Nielsen’s ideas of geometrizing the space of states to deﬁne complexity?

Can we apply these concepts to holographic renormalization? Can we map RG ﬂow as literal “loss of information”?

How to get gravitational dynamics this way?

Recent work by Iqbal and McGreevy associating a modiﬁed 3d Ising model with a string theory. Application here? Maybe following Heckman’s info geom and string theory work?

31 32 Outlook

What do all the curvatures mean? What physics is captured?

Can we apply these concepts to Nielsen’s ideas of geometrizing the space of states to deﬁne complexity?

Can we apply these concepts to holographic renormalization? Can we map RG ﬂow as literal “loss of information”?

How to get gravitational dynamics this way?

Recent work by Iqbal and McGreevy associating a modiﬁed 3d Ising model with a string theory. Application here? Maybe following Heckman’s info geom and string theory work?

Thank you very much for listening! 31 32 Thanks for listening! References

Johanna Erdmenger, Kevin T. Grosvenor, and Ro Jefferson. “Information geometry in quantum ﬁeld theory: lessons from simple examples.” SciPost Phys. 8 (2020) 073 [arXiv:2001.02683].

Ro Jefferson. “Information Geometry (part 1/3).” https://irreverentmind.wordpress.com/2018/08/12/information-geometry-part-1-2/ (2018).

Ronald A. Fisher. “Theory of Statistical Estimation.” Mathematical Proceedings of the Cambridge Philosophical Society 22 no. 5, (1925) 700-725.

Shun-Ichi Amari, “Information Geometry of Neural Networks: An Overview.” Springer US, Boston, MA, 1997.

Matthias Blau, K. S. Narain, and George Thompson. “Instantons, the Information Metric, and the AdS/CFT Correspondence.” (2001) [arXiv:hepth-th/0108122].

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