Emergence of quantum dynamics in hyperbolic dynamical systems

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Emergence of quantum dynamics in hyperbolic dynamical systems

Based onarxiv:1706.09307andarxiv:2102.11196 Slides are on myweb-page.

F. Faure(Grenoble) with M. Tsujii (Kyushu),

March 8, 2022, Roscoff

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Question

Does aquantum behaviornaturally emerges in anAnosov geodesic flowon a compact Riemannian manifold(N,g)(i.e. a model of deterministic chaos)?

What meansAnosov geodesic flow? this is for example the motion of a free (classical) particle with , constant velocity on a negatively curved manifold(N,g), (with HamiltonianH(q,p) =kpk_g onT^∗N). See movie1, movie2,movie3.

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Question

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Question

geodesic flow

compact surface N

X X

E_s E_u

q

φ^t(m)

M= (T^∗N)1=S^∗N m= (q,p)

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Question

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Does quantum behavior emerges in an Anosov geodesic flow?

What meansquantum behavior? From quantization procedureOp(.), (e.g.

Op(pj) =−i_∂q^∂

j), from the geodesic flow, we get the“wave operator”

√∆≈Op kpk_g

(with the Hodge Laplacian∆ =d^†d), that generates the wave equation, forut ∈C^∞(N),t∈R:

∂tut =i√

∆ut =⇒ ∂_t²ut =−∆ut

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Does quantum behavior emerges in an Anosov geodesic flow?

√∆≈Op kpk_g

∂tut =i√

∆ut =⇒ ∂_t²ut =−∆ut

Semi-classical analysis(WKB theory, Egorov’s Theorem etc) shows that for small wave-lengthλ1, functionu_t is approximately transported by the geodesics:

wave equation =⇒

tfixed,λ→0 geodesic flow

Ex: geometrical optics is a limit of wave optics with λ≈0.5µm. Classical Newto- nian mechanics is a limit of quantum Schrödinger mechanics. See movie

N waveut

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Does quantum behavior emerges in an Anosov geodesic flow?

√∆≈Op kpk_g

∂tut =i√

∆ut =⇒ ∂_t²ut =−∆ut

Semi-classical analysis gives:

wave equation =⇒

tfixed,λ→0 geodesic flow Curiously, our question concerns theopposite direction:

geodesic flow=⇒

??! wave equation What does it mean? are thereindicationsfor this?

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The geodesic flow in functional analysis

It will be convenient to consider thevector fieldX =P

jX_j(x)_∂x^∂

j as a derivation operator, generator of the pull back of functionsv by the flowφ^t:

e^tXu=u◦φ^t. Then itsL²-adjoint e^tX†

called “Ruelle transfer operator”, transports probabilities, e.g.

e^tX^†

δm=δ_φt(m) :particle dynamics.

Vector field stable

E0(x)

Eu(x)

unstable direction

X X

Es

Eu

Flowφ^t(m) Es(x)

M t1 u

v e^tXu

e^tXu

m

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The geodesic flow in functional analysis

It will be convenient to consider thevector fieldX =P

jX_j(x)_∂x^∂

j as a derivation operator, generator of the pull back of functionsv by the flowφ^t:

e^tXu=u◦φ^t. Then itsL²-adjoint e^tX†

called “Ruelle transfer operator”, transports probabilities, e.g.

e^tX^†

δm=δ_φt(m) :particle dynamics.

Vector field stable

E0(x)

Eu(x)

unstable direction

X X

Es

Eu

Flowφ^t(m) Es(x)

M t1 u

v e^tXu

e^tXu

m

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Ruelle resonances

A lot of contributions in mathematics: D. Ruelle, D. Bowen, M. Pollicott, H.

Rugh, Blank, Keller, V. Baladi, O. Butterley, C. Liverani, M. Tsujii, S. Gouëzel, Giullietti, F. Naud, N. Roy, F.F., J. Sjöstrand, M. Zworski, S. Dyatlov, C.

Guillarmou, T. Weich, Hilgert, B. Küster, J. Slipantschuk, O. Bandtlow, V. Dang, G. Rivière, T. Lefeuvre, Y. Chaubet, M. Jezequel, Y.G. Bonthonneau, M.Cekic, A.

Medanne, etc

Theorem

(Butterley-Liverani 07, Sjöstrand-F.F. 11) IfX is AnosovthenX has intrinsic discrete spectrum in some anisotropic Sobolev space, calledRuelle spectrumof X.

Theonly obvious eigenvalue isX1=0. Are there other eigenvalues? what are they?

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Ruelle resonances

Medanne, etc

Theorem

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Ruelle resonances

Medanne, etc

Theorem

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Preliminary on Ruelle spectrum: a toy model to have in mind

The simple 1 dim contracting vector field

X =−xd.

dx on R≡Es, gives∀k ∈N, Xx^k = (−k)x^k. X generates the flowφ^t(x) =e^−tx.

OnT^∗R≡T^∗E_s, the induced flow isφ˜^t(x, ξ) = (e^tx,e^−tξ).

0 e^−tx x

ξ

0 x

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Preliminary on Ruelle spectrum: a toy model to have in mind

X =−xd.

0 e^−tx x

ξ

0 x

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Preliminary on Ruelle spectrum: a toy model to have in mind

X =−xd.

0 e^−tx x

ξ

0 x

Formal spectrum:

∀k∈N, Xx^k = (−k)x^k, "Taylor" spectral projectors: T_k =x^kh1

k!δ^(k)|.i_L2. butx^k, δ^(k)∈/L²(R).

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Preliminary on Ruelle spectrum: a toy model to have in mind

X =−xd.

0 e^−tx x

ξ

0 x

InL²(R),Spec(X) =iR+¹₂,becauseX^†=−X +1⇔ X−¹₂

=− X−¹₂†

.

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Preliminary on Ruelle spectrum: a toy model to have in mind

X =−xd.

0 e^−tx x

ξ

0 x

InHW (R) :={u∈ S(R)} withkuk_H

W(R):=

D√

hξE^m FD√

hxE−m

u _L₂

, 0<h1,m≥0, the spectrumofX is:

Ess. spec. discrete Ruelle spectrum

−2 −1 0

−2m−o(1) −m−k+2+...o(1)

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Ex: geodesic flow on a surface with const. neg.

curv.

Usingsl2Ralgebra[X,U±] =±U±,[U+,U−] =2X,∆ :=−X²−X−U−U+, (ref: Selberg, M. Ratner, Flaminio-Forni 02, Dyatlov-Guillarmou-F 15), theRuelle spectrum ofX onΓ\SL₂(R)is on vertical linesz_k_,l =−¹₂−k±iq

µ_l−¹₄:

Finite rank contribution Equilibrium (mixing) z0,l

z1,l

0 Im(z)

z0,l+1

zk,l=−1/2−k±iq µl−¹₄

−5/2 γ1=−3/2 γ0=−1/2

onN= Γ\SL2(R)/SO2

µ0=1≤µ1≤µ2...

with∆ul=µlul

k,l∈N

Re(z)

¯ z0,l

Wave operator on each line, fluctuations∼e⁻¹²^te^iω^j^t

∼e⁻³²^te^iω^j^t

Ruelle eigenvalues ofX(from ppcal series) are

Rem: if X has eigenvaluea+iω∈Cthen e^tX has eigenvalue e^at

|{z}

decay

e^iωt

|{z}

oscillation

.

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Quantum behavior emerges in an Anosov geodesic flow?

For constant neg. surfaceN = Γ\SL R²

/SO R²

, we have seen the appearance of the spectrum ofq

∆−¹₄ in the Ruelle spectrum ofX, i.e some indication that

geodesic flow=⇒

??! wave equation Is it anexceptional algebraic phenomenon?

What about ageneral Anosov geodesic flowφ^t =e^tX (with non constant curvature)?

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Quantum behavior emerges in an Anosov geodesic flow?

For constant neg. surfaceN = Γ\SL R²

/SO R²

, we have seen the appearance of the spectrum ofq

∆−¹₄ in the Ruelle spectrum ofX, i.e some indication that

geodesic flow=⇒

??! wave equation Is it anexceptional algebraic phenomenon?

What about ageneral Anosov geodesic flowφ^t =e^tX (with non constant curvature)?

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“Effective classical dynamics”. Vertical bands.

Consider a generalAnosov geodesic vector fieldX onM = (T^∗N)₁. LetF →M a vector bundle andXF aderivationon C^∞(M;F)overX:

XF(fu) =X(f)u+fXF(u), ∀f ∈C^∞(M),∀u∈C^∞(M;F). Consider evolution of sections ofFk by the flow, with a generator denoted XF_k :C(M;Fk) , calledeffective classical dynamics.

Then thespectrumof the operatorXF_k in L²(M;Fk)is contained in the vertical bandBk :=

γ_k⁻, γ_k⁺

×iR,

γ2⁺ γ1⁻ γ⁺1γ0⁻ γ⁺0

B0

Re(z)

boundedresolvent boundedresolvent ω=Im(z)

γ⁻2

B2 B1

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“Effective classical dynamics”. Vertical bands.

Consider a generalAnosov geodesic vector fieldX onM = (T^∗N)₁. For everyk ∈N, we introduce thefinite rank vector bundleoverM:

Fk :=|detEs|^−1/2⊗Pol_k(E_s)⊗F →M, (1) Consider evolution of sections ofFk by the flow, with a generator denoted XF_k :C(M;Fk) , calledeffective classical dynamics.

γ_k⁻, γ_k⁺

×iR,

γ2⁺ γ1⁻ γ⁺1γ0⁻ γ⁺0

B0

Re(z)

γ⁻2

B2 B1

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“Effective classical dynamics”. Vertical bands.

γ_k⁻, γ_k⁺

×iR,

γ2⁺ γ1⁻ γ⁺1γ0⁻ γ⁺0

B0

Re(z)

γ⁻2

B2 B1

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“Effective classical dynamics”. Vertical bands.

γ_k⁻, γ_k⁺

×iR, with

γ_k^±:= lim

t→±∞log

e^tX^F^k

1/t L^∞. Ex: forX on M= Γ\SL2R, F=C, thenγ_k^±=−¹₂−k.

γ2⁺ γ1⁻ γ⁺1γ0⁻ γ⁺0

B0

Re(z)

γ⁻2

B2 B1

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“Effective classical dynamics”. Vertical bands.

γ_k⁻, γ_k⁺

×iR, We have

− d

2 +k

λmax+cF ≤γ_k⁻≤γ_k⁺≤ − d

2 +k

λmin+CF,

where 0< λ_min≤λ_max are the minimal/maximal Lyap. exponents,d=dimE_u/s.

γ2⁺ γ1⁻ γ⁺1γ0⁻ γ⁺0

B0

Re(z)

γ⁻2

B2 B1

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“Emergence of quantum dynamics”

Theorem

(M.Tsujii-F.F. 2021) For a derivationX_F on C^∞(M;F)over acontact Anosov vector fieldX onM,∀K ∈N,∀ >0,∃anisotropic Sobolev spaceH_W(M;F),

∃C >0,∀t >0, e^tX^F

|{z}

classical dynamics

=

K

M

k=0

Op_Σ e^tX^F^k

| {z }

quantum dynamics

+ Rt

|{z}

finite rank

+O_H_W

e^t(^γK+1⁺ +)

| {z }

neglictible

, (2)

Recall thatγ_K+1⁺ → −∞asK → ∞.

Op_Σ e^tX^F^k

is thegeometric quantizationof theeffective classical dynamicsX_F_k :C(Σ;F) , where Σ = (E_u⊕E_s)^⊥⊂T^∗M is the symplectic trapped set. (More details below...)

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Consequences

∀C >0,∀ >0,∃ω>0, s.t. the Ruelle discrete spectrum ofX for Re(z)>−C,|Im(z)|> ω,iscontained in vertical bands

Re(z)∈ [

k≥0

γ_k⁻−, γ_k⁺+

with a precisedensity ^rank(F_(2π)^k^)Vol(M)d+1 ω^d (calledWeyl law).

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γ⁺₂ γ⁻₁ γ₁⁺γ⁻₀ γ₀⁺ B0

Re(z) H

ˇ γ0

−2λmaxr+C⁰

−λminr+C

B2 B1

boundedresolvent

ωε

ω=Im(z)

Essential spectrum

r1

Intrinsic discrete spectrum

Spectrum that controls emerging behavior

boundedresolvent boundedresolvent

Related papers: Cekic-Guillarmou 20: 1st band for contact Anosov flow in dim. 3 using horocycle operators.

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A special and interesting bundle F = |detE

_s

|

^1/2

→ M

For the first bandk =0 we get thetrivial bundle:

F0=|detE_s|^−1/2⊗Pol_k=0(E_s)⊗F =C

From (2), fort 1, the operator e^tX^F is well described by the dominant term Op_Σ e^tX^F⁰

that is the “quantization” of the dynamicsX itself.

The eigenvalues ofXF accumulate on the imaginary axisRe(z) =γ₀^±=0 with density given by the Weyl law, separated by a uniform spectral gapγ₁⁺<0.

EquilibriumGibbs Re(z) ω=Im(z)

spectral gap

B0

asymptotic

γ1⁺

−C

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A special and interesting bundle F = |detE

_s

|

^1/2

→ M

spectral gap

B0

asymptotic

γ1⁺

−C

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A special and interesting bundle F = |detE

_s

|

^1/2

→ M

spectral gap

B0

asymptotic

γ1⁺

−C

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

Σ = (Eu⊕Es)^⊥ symplectic Trapped set

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

e^tX^F is aFourier integral operator: in the limit of high frequencies, evolution of functions by the pull-back operatore^tX^F is well described on the cotangent bundle T^∗M with the induced flowφ˜^t:= (dφ^t)^∗,t ∈R.

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

We introduce a specific metric g on T^∗M, compatible with the symplectic form and then we define an L²-isometric “wave-packet transform” T : C^∞(M;F) → S(T^∗M;F), that allows to use micro-local analysis on T^∗M for the pull back operatore^tX^F. The unit boxes for the metricg correspond to the effective size of wave-packets and reflect theuncertainty principle.

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

The dynamics φ˜^t is a “scattering dynamics” on the trapped set Σ = R^∗A ⊂ T^∗M (Liouville 1-form) and Σ is symplectic and normally hyperbolic. As a consequence, for large time |t| 1, the outer part of the trapped set Σ has a negligible contribution, because information escapes away (i.e. the Sobolev norm measured by the specific weight W decays). So only the dynamics in a vicinity of Σplays a role for our purpose.

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

We will hence consider a vicinity ofΣof a giveng−sizeω^µ/2, with some 0< µ <1.

Remark that the projection on M has sizeω^{−(1−µ)/2} 1 if ω 1. This will allow us to use thelinearization of the dynamicsφ˜^t as a local approximation.

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

In this vicinity of Σ there is a micro-local decoupling between the directions tangent to Σ and those (symplectically)normal to Σ, represented by a normal vector bundle denotedN→Σ. Forρ∈Σ,

T_ρT^∗M= T_ρΣ

|{z}

Tangent

⊕⊥(N_u(ρ)⊕N_s(ρ))

| {z }

normal

:invariant decomp.

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

The dynamics on the normal direction N is hyperbolic and responsible for the emergence of polynomial functions along the stable directionN_s ≡E_s idemX =

−x^d._dx, Xx^k = (−k)x^k on R. We introduce approximate projectors Op_Σ(T_k) that restricts functions to the symplectic trapped set Σ and that are polynomial valued alongN_s with degreek. This projector plays a similar role as the Bergman projector (or Szegö projector) in geometric quantization. What remains for large time, is aneffective Hilbert spaceof quantum waves that live on the trapped set Σ, valued in the vector bundleFk.

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Ideas of the proof

flow

Lifted flow X

Ns(ρ)

m Es

T^∗M M

Eu

Sizeω^{−(1−µ)/2}1

Nu(ρ)

0

ρ=ωA(m)

< ω >^µ/2 ω

φ^t

φ˜^t N(ρ)

Rem: the main geometrical object considered in this paper is this fibration N_s →Σ→M. Beware that for a geodesic flow on(N,g), this fibration sequence continues withM =T₁^∗N → N.

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What is the meaning of going beyond the equilibrium description for the Ruelle spectrum?

Illustration: the ocean is quite, deep, flat, gentle ≡Equilibrium state, but with a better look, the behavior at the surface may be furious, wavy, and never stop to move. Are they neglictible?

Some very speculative question: can quantum dynamics in the physics world emerges from an underlying chaotic deterministic yet unknown system?

Thank you for your attention!

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(*) Wave packets

Local flow box coordinates onM: y = (x,z)∈Rⁿ×Rs.t. X = _∂z^∂ and dual coordinatesη = (ξ, ω)∈Rⁿ×Ron T_y^∗M.

η^−α

δ x z

X

flow direction

M m

ξ

ω δ⁻¹

0 ω=0

η η^α

T^∗mM

Let ¹₂≤α <1 and 0< δ 1. Wave packet functionis:

ϕ_(y_,η) y⁰

≈

|η|1aexp iη.y⁰−

x⁰−x hηi^−α

2

−

z⁰−z δ

2!

,

ϕ_(y,η)

L²(M) ≈

|η|11

Metricg onT^∗M, compatible withΩ =dy∧dη:

gy,η= dx

hηi^−α 2

+ dξ

hηi^α 2

+ dz

δ 2

+ dω

δ⁻¹ 2

Rem: α≥¹₂ ⇔g remains equivalent uniformly/η after change of flow box coordinates.

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(*) Wave packets

η^−α

δ x z

X

flow direction

M m

ξ

ω δ⁻¹

0 ω=0

η η^α

T^∗mM

ϕ_(y_,η) y⁰

≈

x⁰−x hηi^−α

2

−

z⁰−z δ

2!

,

ϕ_(y,η)

L²(M) ≈

|η|11

gy,η= dx

hηi^−α 2

+ dξ

hηi^α 2

+ dz

δ 2

+ dω

δ⁻¹ 2

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(*) Wave packets

η^−α

δ x z

X

flow direction

M m

ξ

ω δ⁻¹

0 ω=0

η η^α

T^∗mM

ϕ_(y_,η) y⁰

≈

x⁰−x hηi^−α

2

−

z⁰−z δ

2!

,

ϕ_(y,η)

L²(M) ≈

|η|11

gy,η= dx

hηi^−α 2

+ dξ

hηi^α 2

+ dz

δ 2

+ dω

δ⁻¹ 2

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(*) Wave packet transform

(or FBI, wavelet, Bargmann, Anti-Wick... transform)

(Abuse of notations that forget charts and partitions of unity.) T :

(C^∞(M) → S(T^∗M)

u(y⁰) →(Tu) (y, η) :=hϕy,η,ui_L2(M)

Lemma (fundamental 1.

”Resolution of identity”

)

T^∗◦ T =Id

Bergmann projector

L²(M) T

P=T T^†

L²(T^∗M) u

v

Im(T) T^†

Remarks: ∀u∈C^∞(M), u(y⁰) =R

T^∗Mϕ_y,η(y⁰)hϕ_y,η,ui_(2π)^dydηn+1. T :L²(M)→Im(T)⊂L²(T^∗M)is an isomorphism. Hencewe “lift the analysis toT^∗M”.

Π =T ◦ T^∗:L²(T^∗M)→Im(T)is an orthogonal projector.