Quantum footprints of symplectic rigidity
Leonid Polterovich, Tel Aviv

Luminy, October 2018
Joint work with Laurent Charles (Paris)

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

What is this lecture about?

Symplectic topology (Conley, Zehnder, Gromov, Eliashberg,
Floer, Hofer,..., 1980 – ...), discovered surprising rigidity
phenomena involving symplectic manifolds, their subsets, and theirdiffeomorphisms.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

What is this lecture about?

Symplectic topology (Conley, Zehnder, Gromov, Eliashberg,
Floer, Hofer,..., 1980 – ...), discovered surprising rigidity
phenomena involving symplectic manifolds, their subsets, and theirdiffeomorphisms.
Question: What are quantum footprints of symplectic rigidity?

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

What is this lecture about?

Symplectic topology (Conley, Zehnder, Gromov, Eliashberg,
Floer, Hofer,..., 1980 – ...), discovered surprising rigidity
phenomena involving symplectic manifolds, their subsets, and theirdiffeomorphisms.
Question: What are quantum footprints of symplectic rigidity?
Difficulty: Quantum-classical correspondence is not sharp(H.J. Groenewold, 1946)

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

What is this lecture about?

Symplectic topology (Conley, Zehnder, Gromov, Eliashberg,
Floer, Hofer,..., 1980 – ...), discovered surprising rigidity
phenomena involving symplectic manifolds, their subsets, and theirdiffeomorphisms.
Question: What are quantum footprints of symplectic rigidity?
Difficulty: Quantum-classical correspondence is not sharp(H.J. Groenewold, 1946)
Today’s story: Quantum counterpart of symplectic displacement
energy, a fundamental symplectic invariant (Hofer, 1990)

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Fidelity

H- finite dimensional complex Hilbert space

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Fidelity

H- finite dimensional complex Hilbert space
L(H)- Hermitian operators, observables

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Fidelity

H- finite dimensional complex Hilbert space
L(H)- Hermitian operators, observables
S(H) - quantum states, ρ ∈ L(H), ρ ≥ 0, trace(ρ) = 1.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Fidelity

H- finite dimensional complex Hilbert space
L(H)- Hermitian operators, observables
S(H) - quantum states, ρ ∈ L(H), ρ ≥ 0, trace(ρ) = 1.
Fidelity: θ, √σ ∈ S(H)√Φ(θ, σ) = k θ σktr .
Measures overlap between quantum states.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Fidelity

H- finite dimensional complex Hilbert space
L(H)- Hermitian operators, observables
S(H) - quantum states, ρ ∈ L(H), ρ ≥ 0, trace(ρ) = 1.
Fidelity: θ, √σ ∈ S(H)√Φ(θ, σ) = k θ σktr .
Measures overlap between quantum states.
Example: For pure states ξ, η ∈ H, |ξ| = |η| = 1,
Φ(ξ, η) = |hξ, ηi|.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Dislocation

Ft ∈ L(H) - quantum Hamiltonian.
Schrödinger equationU̇t = − ~i Ft Ut ,Ut : H → H unitary evolution, U0 = 1l, U1 = U.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Dislocation

Ft ∈ L(H) - quantum Hamiltonian.
Schrödinger equationU̇t = − ~i Ft Ut ,Ut : H → H unitary evolution, U0 = 1l, U1 = U.
Quantum Hamiltonian Ft a-dislocates a state θ ∈ S ifΦ(θ, UθU −1 ) ≤ a, a ∈ [0, 1).

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Dislocation

Ft ∈ L(H) - quantum Hamiltonian.
Schrödinger equationU̇t = − ~i Ft Ut ,Ut : H → H unitary evolution, U0 = 1l, U1 = U.
Quantum Hamiltonian Ft a-dislocates a state θ ∈ S ifΦ(θ, UθU −1 ) ≤ a, a ∈ [0, 1).
Appears e.g. in quantum computation. Margolus-Levitin (1998)
address the question about “the maximum number of distinct [i.e.,
non-overlapping] states that the system can pass through, per unitof time. For a classical computer, this would correspond to the
maximum number of operations per second.”

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum speed limit

The total energy of the quantum evolution is given by `q (F ),R1`q (F ) := 0 kFt kop dt.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum speed limit

The total energy of the quantum evolution is given by `q (F ),R1`q (F ) := 0 kFt kop dt.
Quantum speed limit: universal bound on the energy required toa-dislocate a quantum state:Φ(θ, UθU −1 ) ≤ a ⇒ `q (F ) ≥ arccos(a)~Mandelstam-Tamm, 1945 “time-energy uncertainty”, Uhlmann
1992, Margolus-Levitin, 1998

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum speed limit

SPEED
LIMIT

ξ

U

π
2 h̄

Uξ
Figure: “Displacing” a pure quantum state

We explore semiclassical dislocation of semiclassical states.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Mathematical model of classical mechanics(M 2n , ω)–symplectic manifold (e.g,P2-sphere)ω– symplectic form. Locally ω = ni=1 dpi ∧ dqi .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Mathematical model of classical mechanics(M 2n , ω)–symplectic manifold (e.g,P2-sphere)ω– symplectic form. Locally ω = ni=1 dpi ∧ dqi .
M-phase space of mechanical system.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Mathematical model of classical mechanics(M 2n , ω)–symplectic manifold (e.g,P2-sphere)ω– symplectic form. Locally ω = ni=1 dpi ∧ dqi .
M-phase space of mechanical system.
Energy determines evolution: f : M × [0, 1] → R – Hamiltonian
function (energy). Hamiltonian system:(∂fq̇ = ∂p∂fṗ = − ∂qLeonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Mathematical model of classical mechanics(M 2n , ω)–symplectic manifold (e.g,P2-sphere)ω– symplectic form. Locally ω = ni=1 dpi ∧ dqi .
M-phase space of mechanical system.
Energy determines evolution: f : M × [0, 1] → R – Hamiltonian
function (energy). Hamiltonian system:(∂fq̇ = ∂p∂fṗ = − ∂qFamily of Hamiltonian diffeomorphisms
ϕt : M → M, (p(0), q(0)) 7→ (p(t), q(t))

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Mathematical model of classical mechanics(M 2n , ω)–symplectic manifold (e.g,P2-sphere)ω– symplectic form. Locally ω = ni=1 dpi ∧ dqi .
M-phase space of mechanical system.
Energy determines evolution: f : M × [0, 1] → R – Hamiltonian
function (energy). Hamiltonian system:(∂fq̇ = ∂p∂fṗ = − ∂qFamily of Hamiltonian diffeomorphisms
ϕt : M → M, (p(0), q(0)) 7→ (p(t), q(t))
Ham -group of Hamiltonian diffeomorphisms

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Mathematical model of classical mechanics(M 2n , ω)–symplectic manifold (e.g,P2-sphere)ω– symplectic form. Locally ω = ni=1 dpi ∧ dqi .
M-phase space of mechanical system.
Energy determines evolution: f : M × [0, 1] → R – Hamiltonian
function (energy). Hamiltonian system:(∂fq̇ = ∂p∂fṗ = − ∂qFamily of Hamiltonian diffeomorphisms
ϕt : M → M, (p(0), q(0)) 7→ (p(t), q(t))
Ham -group of Hamiltonian diffeomorphisms
Key feature: ϕ∗t ω = ω.
Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Small scale on symplectic manifolds

X ⊂ M displaceable if ∃ ϕ ∈ Ham : ϕX ∩ X = ∅ (Hofer, 1990)
Figure: (Non)-displaceability on the 2-sphere

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Displacement energy(M, ω) - closed symplectic manifold.
Let ft , t ∈ [0, 1] be classical Hamiltonian generating Hamiltonian
diffeomorphism ϕ ∈ Ham (M, ω). Total energy
R1`c (f ) = 0 ||ft ||dt, where ||g || := max |g |-uniform norm.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Displacement energy(M, ω) - closed symplectic manifold.
Let ft , t ∈ [0, 1] be classical Hamiltonian generating Hamiltonian
diffeomorphism ϕ ∈ Ham (M, ω). Total energy
R1`c (f ) = 0 ||ft ||dt, where ||g || := max |g |-uniform norm.
Dispacement energy of a displaceable subset X ⊂ Me(X ) := inf `c (f ) over all displacing Hamiltonians f .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Displacement energy(M, ω) - closed symplectic manifold.
Let ft , t ∈ [0, 1] be classical Hamiltonian generating Hamiltonian
diffeomorphism ϕ ∈ Ham (M, ω). Total energy
R1`c (f ) = 0 ||ft ||dt, where ||g || := max |g |-uniform norm.
Dispacement energy of a displaceable subset X ⊂ Me(X ) := inf `c (f ) over all displacing Hamiltonians f .
Fundamental invariant in modern symplectic topology (Hofer
1990,...) Yields biinvariant metric geometry of Ham.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Displacement energy(M, ω) - closed symplectic manifold.
Let ft , t ∈ [0, 1] be classical Hamiltonian generating Hamiltonian
diffeomorphism ϕ ∈ Ham (M, ω). Total energy
R1`c (f ) = 0 ||ft ||dt, where ||g || := max |g |-uniform norm.
Dispacement energy of a displaceable subset X ⊂ Me(X ) := inf `c (f ) over all displacing Hamiltonians f .
Fundamental invariant in modern symplectic topology (Hofer
1990,...) Yields biinvariant metric geometry of Ham.
Rigidity: e(X ) > 0 for all open X

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Flexibility
Counterpoint: If Vol(X ) < 21 · Vol(M), for all  > 0, δ > 0 there
exists ft such that
Vol(ϕX ∩ X ) < , `c (f ) < δ.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Flexibility
Counterpoint: If Vol(X ) < 21 · Vol(M), for all  > 0, δ > 0 there
exists ft such that
Vol(ϕX ∩ X ) < , `c (f ) < δ.

No measure-theoretic symplectic rigidity

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Flexibility
Counterpoint: If Vol(X ) < 21 · Vol(M), for all  > 0, δ > 0 there
exists ft such that
Vol(ϕX ∩ X ) < , `c (f ) < δ.

No measure-theoretic symplectic rigidity
Based on Katok’s lemma, 1970, Ostrover-Wagner, 2005.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-1(M, ω)- closed Kähler manifold, quantizable: [ω]/(2π) ∈ H 2 (M, Z)

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-1(M, ω)- closed Kähler manifold, quantizable: [ω]/(2π) ∈ H 2 (M, Z)
L- a holomorphic Hermitian line bundle over M
Curvature of Chern connection = iω.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-1(M, ω)- closed Kähler manifold, quantizable: [ω]/(2π) ∈ H 2 (M, Z)
L- a holomorphic Hermitian line bundle over M
Curvature of Chern connection = iω.
H~ := H 0 (M, L⊗k ) ⊂ V~ := L2 (M, L⊗k ).

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-1(M, ω)- closed Kähler manifold, quantizable: [ω]/(2π) ∈ H 2 (M, Z)
L- a holomorphic Hermitian line bundle over M
Curvature of Chern connection = iω.
H~ := H 0 (M, L⊗k ) ⊂ V~ := L2 (M, L⊗k ).
Π~ : V~ → H~ – the orthogonal projection.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-1(M, ω)- closed Kähler manifold, quantizable: [ω]/(2π) ∈ H 2 (M, Z)
L- a holomorphic Hermitian line bundle over M
Curvature of Chern connection = iω.
H~ := H 0 (M, L⊗k ) ⊂ V~ := L2 (M, L⊗k ).
Π~ : V~ → H~ – the orthogonal projection.
The Toeplitz operator: T~ (f )(s) := Π~ (fs), f ∈ C ∞ (M), s ∈ H~ .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-2
Hyperplane Ez ⊂ H, Ez := {s ∈ H~ : s(z) = 0}.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-2
Hyperplane Ez ⊂ H, Ez := {s ∈ H~ : s(z) = 0}.
Kodaira embedding M → P(H~∗ ), z 7→ Ez

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-2
Hyperplane Ez ⊂ H, Ez := {s ∈ H~ : s(z) = 0}.
Kodaira embedding M → P(H~∗ ), z 7→ Ez
Pz,~ – orthogonal projector of H~ to Ez⊥ coherent state projector

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-2
Hyperplane Ez ⊂ H, Ez := {s ∈ H~ : s(z) = 0}.
Kodaira embedding M → P(H~∗ ), z 7→ Ez
Pz,~ – orthogonal projector of H~ to Ez⊥ coherent state projector
There exists Rawnsley function R~ ∈ C ∞ (M):Z
T~ (f ) =f (x)R~ (x)Px,~ dVol(x) ,M

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Berezin-Toeplitz quantization-2
Hyperplane Ez ⊂ H, Ez := {s ∈ H~ : s(z) = 0}.
Kodaira embedding M → P(H~∗ ), z 7→ Ez
Pz,~ – orthogonal projector of H~ to Ez⊥ coherent state projector
There exists Rawnsley function R~ ∈ C ∞ (M):Z
T~ (f ) =f (x)R~ (x)Px,~ dVol(x) ,MDefinition: For classical state τ (probability measure on M)
Z
Q~ (τ ) =Px,~ dτ (x) ∈ S(H~ )M“classical” quantum state , Giraud-Braun-Braun 2008

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Displacement yields dislocation
ft -classical Hamiltonian, t ∈ [0, 1], τ -classical state.
Ft = T~ (ft )- quantum Hamiltonian, θ = Q~ (τ ) - quantum state.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Displacement yields dislocation
ft -classical Hamiltonian, t ∈ [0, 1], τ -classical state.
Ft = T~ (ft )- quantum Hamiltonian, θ = Q~ (τ ) - quantum state.
Theorem (Charles-P., 2016)
If ft displaces supp(τ ) ⇒ Ft O(~∞ )-dislocates θ.

Figure: ϕ-time-one-map of the flow of ft

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Dislocation yields displacement

Assume τ has smooth density, ft,~ depends on ~and bounded with 4 derivatives, dim M = 2n.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Dislocation yields displacement

Assume τ has smooth density, ft,~ depends on ~and bounded with 4 derivatives, dim M = 2n.
Theorem (Charles-P., 2016, sketch)
If Ft,~ o(~n )-dislocates θ ⇒ ft,~ displaces supp(τ ) and`q (Ft,~ ) ≥ e(supp(τ )) .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Dislocation yields displacement

Assume τ has smooth density, ft,~ depends on ~and bounded with 4 derivatives, dim M = 2n.
Theorem (Charles-P., 2016, sketch)
If Ft,~ o(~n )-dislocates θ ⇒ ft,~ displaces supp(τ ) and`q (Ft,~ ) ≥ e(supp(τ )) .
Conclusion: Speed limit becomes more restrictive ∼ 1 than theuniversal bound ∼ ~.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Rigidity vs. flexibility on quantum side

Theorem (Charles-P., 2016)
Assume Vol(supp(τ )) < 21 · Vol(M). Then ∀, δ > 0 there exists ft
such that Ft -dislocates θ and `q (Ft ) < δ.
Conclusion: Competition between rigidity (`q > e) vs. flexibility(`q < δ) is governed by the rate of dislocation .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Rigidity vs. flexibility on quantum side

Theorem (Charles-P., 2016)
Assume Vol(supp(τ )) < 21 · Vol(M). Then ∀, δ > 0 there exists ft
such that Ft -dislocates θ and `q (Ft ) < δ.
Conclusion: Competition between rigidity (`q > e) vs. flexibility(`q < δ) is governed by the rate of dislocation .

RATE OF DISLOCATION

Leonid Polterovich, Tel Aviv University

RIGIDITYFLEXIBILITYo(~n )



Quantum footprints of symplectic rigidity

Remainders of BT quantization
M. Bordemann-E. Meinrenken-M. Schlichenmaier, 1994, based ontechnology of V. Guillemin-L. Boutet de Monvel.
Important tool in our work.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Remainders of BT quantization
M. Bordemann-E. Meinrenken-M. Schlichenmaier, 1994, based ontechnology of V. Guillemin-L. Boutet de Monvel.
Important tool in our work.
(P1) (norm correspondence) kf k − O(~) ≤ kT~ (f )kOp ≤ kf k;Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Remainders of BT quantization
M. Bordemann-E. Meinrenken-M. Schlichenmaier, 1994, based ontechnology of V. Guillemin-L. Boutet de Monvel.
Important tool in our work.
(P1) (norm correspondence) kf k − O(~) ≤ kT~ (f )kOp ≤ kf k;(P2) (the correspondence principle)k − (i/~)[T~ (f ), T~ (g )] − T~ ({f , g })kOp = O(~) ;Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Remainders of BT quantization
M. Bordemann-E. Meinrenken-M. Schlichenmaier, 1994, based ontechnology of V. Guillemin-L. Boutet de Monvel.
Important tool in our work.
(P1) (norm correspondence) kf k − O(~) ≤ kT~ (f )kOp ≤ kf k;(P2) (the correspondence principle)k − (i/~)[T~ (f ), T~ (g )] − T~ ({f , g })kOp = O(~) ;(P3) (quasi-multiplicativity) kT~ (fg ) − T~ (f )T~ (g )kOp = O(~);

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Remainders of BT quantization
M. Bordemann-E. Meinrenken-M. Schlichenmaier, 1994, based ontechnology of V. Guillemin-L. Boutet de Monvel.
Important tool in our work.
(P1) (norm correspondence) kf k − O(~) ≤ kT~ (f )kOp ≤ kf k;(P2) (the correspondence principle)k − (i/~)[T~ (f ), T~ (g )] − T~ ({f , g })kOp = O(~) ;(P3) (quasi-multiplicativity) kT~ (fg ) − T~ (f )T~ (g )kOp = O(~);(P4) (trace correspondence)Rntrace(T~ (f )) − (2π~)−n M f ωn! = O(~−(n−1) ),for all f , g ∈ C ∞ (M).

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.
Involves sharp remainder bounds for BT quantization (Charles-P.,
2016; Charles, 2017)
Earlier work: T. Barron-X. Ma-G. Marinescu-M. Pinsonnault, 2014

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.
Involves sharp remainder bounds for BT quantization (Charles-P.,
2016; Charles, 2017)
Earlier work: T. Barron-X. Ma-G. Marinescu-M. Pinsonnault, 2014
Notation: |f |i - C i -norm.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.
Involves sharp remainder bounds for BT quantization (Charles-P.,
2016; Charles, 2017)
Earlier work: T. Barron-X. Ma-G. Marinescu-M. Pinsonnault, 2014
Notation: |f |i - C i -norm.
(P1) ≤ α|f |2 ~;Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.
Involves sharp remainder bounds for BT quantization (Charles-P.,
2016; Charles, 2017)
Earlier work: T. Barron-X. Ma-G. Marinescu-M. Pinsonnault, 2014
Notation: |f |i - C i -norm.
(P1) ≤ α|f |2 ~;(P2) ≤ β(|f |1 · |g |3 + |f |2 · |g |2 + |f |3 · |g |1 )~ .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.
Involves sharp remainder bounds for BT quantization (Charles-P.,
2016; Charles, 2017)
Earlier work: T. Barron-X. Ma-G. Marinescu-M. Pinsonnault, 2014
Notation: |f |i - C i -norm.
(P1) ≤ α|f |2 ~;(P2) ≤ β(|f |1 · |g |3 + |f |2 · |g |2 + |f |3 · |g |1 )~ .
(P3) ≤ γ(|f |0 · |g |2 + |f |1 · |g |1 + |f |2 · |g |0 )~ .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.
Involves sharp remainder bounds for BT quantization (Charles-P.,
2016; Charles, 2017)
Earlier work: T. Barron-X. Ma-G. Marinescu-M. Pinsonnault, 2014
Notation: |f |i - C i -norm.
(P1) ≤ α|f |2 ~;(P2) ≤ β(|f |1 · |g |3 + |f |2 · |g |2 + |f |3 · |g |1 )~ .
(P3) ≤ γ(|f |0 · |g |2 + |f |1 · |g |1 + |f |2 · |g |0 )~ .

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Zooming into small scales
Theorems 1,2 extend to dislocation of semiclassical states which“occupy” a ball of radius ~ε , ε ∈ [0, 1/2) in the phase space. The
speed limit on such a scale is ∼ ~2ε which, again, is more
restrictive than the universal quantum speed limit ∼ ~.
Involves sharp remainder bounds for BT quantization (Charles-P.,
2016; Charles, 2017)
Earlier work: T. Barron-X. Ma-G. Marinescu-M. Pinsonnault, 2014
Notation: |f |i - C i -norm.
(P1) ≤ α|f |2 ~;(P2) ≤ β(|f |1 · |g |3 + |f |2 · |g |2 + |f |3 · |g |1 )~ .
(P3) ≤ γ(|f |0 · |g |2 + |f |1 · |g |1 + |f |2 · |g |0 )~ .
Rigidity of remainders: (Charles-P., 2016) α, β, γ cannot be
small simultaneously
Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum “proof” of e(B) > 0
B(r ) ⊂ R2n – ball of radius r . To prove: e(B(r )) & r 2 .
√WLOG choose r = ~ - quantum scale.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum “proof” of e(B) > 0
B(r ) ⊂ R2n – ball of radius r . To prove: e(B(r )) & r 2 .
√WLOG choose r = ~ - quantum scale.
Let τ be the “squeezed” coherent state occupying B.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum “proof” of e(B) > 0
B(r ) ⊂ R2n – ball of radius r . To prove: e(B(r )) & r 2 .
√WLOG choose r = ~ - quantum scale.
Let τ be the “squeezed” coherent state occupying B.

If time 1 map φ of classical Hamiltonian f displaces B, thequantum Hamiltonian F dislocates τ , so by universal speed limit√`c (f ) ≈ `q (F ) & ~ ⇒ e(B( ~)) & ~. QED

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum “proof” of e(B) > 0
B(r ) ⊂ R2n – ball of radius r . To prove: e(B(r )) & r 2 .
√WLOG choose r = ~ - quantum scale.
Let τ be the “squeezed” coherent state occupying B.

If time 1 map φ of classical Hamiltonian f displaces B, thequantum Hamiltonian F dislocates τ , so by universal speed limit√`c (f ) ≈ `q (F ) & ~ ⇒ e(B( ~)) & ~. QEDHOW COME?? Such a proof cannot exist - not hard enough.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum “proof” of e(B) > 0
B(r ) ⊂ R2n – ball of radius r . To prove: e(B(r )) & r 2 .
√WLOG choose r = ~ - quantum scale.
Let τ be the “squeezed” coherent state occupying B.

If time 1 map φ of classical Hamiltonian f displaces B, thequantum Hamiltonian F dislocates τ , so by universal speed limit√`c (f ) ≈ `q (F ) & ~ ⇒ e(B( ~)) & ~. QEDHOW COME?? Such a proof cannot exist - not hard enough.
√Resolution: Remainders of quantization are large on scale ∼ ~Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum footprints of symplectic rigidity: outlook
Quantum indeterminism ↔ symplectic quasi-states(Entov-P.,2006), positive functionals linear on (Poisson)commutative subalgebras of C (M) but not on the whole space.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum footprints of symplectic rigidity: outlook
Quantum indeterminism ↔ symplectic quasi-states(Entov-P.,2006), positive functionals linear on (Poisson)commutative subalgebras of C (M) but not on the whole space.
Gleason’s theorem (1957): dim H ≥ 3 ⇒ every quantum
quasi-state linear, cf. discussion on hidden variables in quantum
mechanics (also,Groenewold, 1946).

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum footprints of symplectic rigidity: outlook
Quantum indeterminism ↔ symplectic quasi-states(Entov-P.,2006), positive functionals linear on (Poisson)commutative subalgebras of C (M) but not on the whole space.
Gleason’s theorem (1957): dim H ≥ 3 ⇒ every quantum
quasi-state linear, cf. discussion on hidden variables in quantum
mechanics (also,Groenewold, 1946).
Anti-Gleason phenomenon in classical mechanics: quasi-statescoming from Floer theory.

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum footprints of symplectic rigidity: outlook
Quantum indeterminism ↔ symplectic quasi-states(Entov-P.,2006), positive functionals linear on (Poisson)commutative subalgebras of C (M) but not on the whole space.
Gleason’s theorem (1957): dim H ≥ 3 ⇒ every quantum
quasi-state linear, cf. discussion on hidden variables in quantum
mechanics (also,Groenewold, 1946).
Anti-Gleason phenomenon in classical mechanics: quasi-statescoming from Floer theory.
Rigidity of Poisson brackets ↔ noise production in phase-spaceregistration of a quantum particle (P., 2014, Charles-P., 2016)

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

Quantum footprints of symplectic rigidity: outlook
Quantum indeterminism ↔ symplectic quasi-states(Entov-P.,2006), positive functionals linear on (Poisson)commutative subalgebras of C (M) but not on the whole space.
Gleason’s theorem (1957): dim H ≥ 3 ⇒ every quantum
quasi-state linear, cf. discussion on hidden variables in quantum
mechanics (also,Groenewold, 1946).
Anti-Gleason phenomenon in classical mechanics: quasi-statescoming from Floer theory.
Rigidity of Poisson brackets ↔ noise production in phase-spaceregistration of a quantum particle (P., 2014, Charles-P., 2016)
Symplectic capacities, monotone invariants based on periodic
orbits ↔ Gutzwiller type trace formula (A. Uribe, 2016); in
progress Charles, Le Floch, P., Uribe

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity

The End

Leonid Polterovich, Tel Aviv University

Quantum footprints of symplectic rigidity