Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering - - PowerPoint PPT Presentation

Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems

Wojciech Dybalski1 joint work with Sven Bachmann2 and Pieter Naaijkens3

1Technical University of Munich

2University of British Columbia 3RWTH Aachen University

ICMP, AHP Journal Prize session, 27/07/2018

Wojciech Dybalski Scattering theory for spin systems

Introduction

Our observation: For a class of gapped quantum spin systems satisfying Lieb-Robinson bounds, admitting single-particle states Haag-Ruelle scattering theory can be developed in a natural, model independent manner.

Wojciech Dybalski Scattering theory for spin systems

Introduction

Our observation: For a class of gapped quantum spin systems satisfying Lieb-Robinson bounds, admitting single-particle states Haag-Ruelle scattering theory can be developed in a natural, model independent manner. Comparison with the literature

1 Haag-Ruelle scattering theory for Euclidean lattice quantum

field theories. [Barata-Fredenhagen 91, Barata 91, 92, Auil-Barata 01,05]

2 Scattering theory for quantum spin systems relying on

properties of concrete Hamiltonians. [Hepp 65, Graf-Schenker 97, Malyshev 78, Yarotsky 04... ]

Wojciech Dybalski Scattering theory for spin systems

Introduction

Our observation: For a class of gapped quantum spin systems satisfying Lieb-Robinson bounds, admitting single-particle states Haag-Ruelle scattering theory can be developed in a natural, model independent manner. Comparison with the literature

1 Haag-Ruelle scattering theory for Euclidean lattice quantum

field theories. [Barata-Fredenhagen 91, Barata 91, 92, Auil-Barata 01,05]

2 Scattering theory for quantum spin systems relying on

properties of concrete Hamiltonians. [Hepp 65, Graf-Schenker 97, Malyshev 78, Yarotsky 04... ]

Wojciech Dybalski Scattering theory for spin systems

Outline

1

Scattering in Quantum Mechanics

2

Scattering in QFT and spin systems

3

The problem of asymptotic completeness

4

Conclusions and outlook

Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 Hilbert space: H := L2(R3, dx) 2 Hamiltonian: H = − 1

2∆ + V (x)

3 Schrödinger equation: i∂tΨt = HΨt 4 Time evolution: Ψt := e−itHΨt=0 Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 Hilbert space: H := L2(R3, dx) 2 Hamiltonian: H = − 1

2∆ + V (x)

3 Schrödinger equation: i∂tΨt = HΨt 4 Time evolution: Ψt := e−itHΨt=0 V x y v v Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 Hilbert space: H := L2(R3, dx) 2 Hamiltonian: H = − 1

2∆ + V (x)

3 Schrödinger equation: i∂tΨt = HΨt 4 Time evolution: Ψt := e−itHΨt=0 V x y v v Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 There are states Ψout ∈ H of the particle in potential V which

for large times evolve like states of the free particle.

2 For any such Ψout there exists Ψ ∈ H s.t. 3 Def: Ψout := limt→∞ eitHe−itH0Ψ is the scattering state. 4 Def: W out := limt→∞ eitHe−itH0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 There are states Ψout ∈ H of the particle in potential V which

for large times evolve like states of the free particle.

2 For any such Ψout there exists Ψ ∈ H s.t.

lim

t→∞ e−itHΨout − e−itH0Ψ = 0

3 Def: Ψout := limt→∞ eitHe−itH0Ψ is the scattering state. 4 Def: W out := limt→∞ eitHe−itH0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 There are states Ψout ∈ H of the particle in potential V which

for large times evolve like states of the free particle.

2 For any such Ψout there exists Ψ ∈ H s.t.

lim

t→∞ Ψout − eitHe−itH0Ψ = 0

3 Def: Ψout := limt→∞ eitHe−itH0Ψ is the scattering state. 4 Def: W out := limt→∞ eitHe−itH0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 There are states Ψout ∈ H of the particle in potential V which

for large times evolve like states of the free particle.

2 For any such Ψout there exists Ψ ∈ H s.t.

lim

t→∞ Ψout − eitHe−itH0Ψ = 0

3 Def: Ψout := limt→∞ eitHe−itH0Ψ is the scattering state. 4 Def: W out := limt→∞ eitHe−itH0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics

1 There are states Ψout ∈ H of the particle in potential V which

for large times evolve like states of the free particle.

2 For any such Ψout there exists Ψ ∈ H s.t.

lim

t→∞ Ψout − eitHe−itH0Ψ = 0

3 Def: Ψout := limt→∞ eitHe−itH0Ψ is the scattering state. 4 Def: W out := limt→∞ eitHe−itH0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Cook’s method

1 Let Ψt := eitHe−itH0Ψ. 2 Suppose we can show

∂tΨt = eitHV e−itH0Ψ ∈ L1(R, dt).

3 Then limt→∞ Ψt =

∞

t0 (∂τΨτ)dτ + Ψt0 exists.

Wojciech Dybalski Scattering theory for spin systems

Cook’s method

1 Let Ψt := eitHe−itH0Ψ. 2 Suppose we can show

∂tΨt = eitHV e−itH0Ψ ∈ L1(R, dt).

3 Then limt→∞ Ψt =

∞

t0 (∂τΨτ)dτ + Ψt0 exists.

Wojciech Dybalski Scattering theory for spin systems

Cook’s method

1 Let Ψt := eitHe−itH0Ψ. 2 Suppose we can show

∂tΨt = eitHV e−itH0Ψ ∈ L1(R, dt).

3 Then limt→∞ Ψt =

∞

t0 (∂τΨτ)dτ + Ψt0 exists.

Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞).

Lieb-Robinson bounds: [τt(A), B] ≤ CA,Beλ(vLRt−d(A,B)), A, B ∈ A local.

5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A.

p Sp U Ω H P Σ(p)

Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A.

p Sp U Ω H P Σ(p) p Sp U Ω H −π π Σ(p)

Wojciech Dybalski Scattering theory for spin systems

Examples

1 QFT: λφ4 theory for 1 and 2 space dimensions:

L = 1 2∂µφ∂µφ − 1 2m2φ2 − λ 4!φ4

2 Spin systems: Ising model in transverse magnetic field for any

space dimension: H = −1 2

• i

(σ(z)

i

− 1) − ε

• i,j

σ(x)

i

σ(x)

j

p Sp U Ω H P Σ(p) p Sp U Ω H −π π Σ(p)

Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A.

p Sp U Ω H P Σ(p) p Sp U Ω H −π π Σ(p)

Let B∗

1, . . . , B∗ n ∈ B be s.t. B∗ i Ω ∈ 1

lU(∆i)H are single-particle states.

Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A.

i

Sp U Ω H P Σ(p) p

∆ Β∗ i i

Sp U Ω H −π π Σ(p) p

∆ * Bi

Let B∗

1, . . . , B∗ n ∈ B be s.t. B∗ i Ω ∈ 1

lU(∆i)H are single-particle states.

Wojciech Dybalski Scattering theory for spin systems

Arveson spectrum

Let (A, τ) be a C ∗-dynamical system. Definition The Arveson spectrum of A ∈ A is the support of the (inverse) Fourier transform of R × Γ ∋ (t, x) → τ(t,x)(A). It is denoted SpAτ.

Wojciech Dybalski Scattering theory for spin systems

Arveson spectrum

Let (A, τ) be a C ∗-dynamical system. Definition The Arveson spectrum of A ∈ A is the support of the (inverse) Fourier transform of R × Γ ∋ (t, x) → τ(t,x)(A). It is denoted SpAτ. Fact 1: (Energy-momentum transfer relation)

1 Let τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. 2 Let 1

lU( · ) denote the spectral measure of U. Then A 1 lU(∆)H ⊂ 1 lU(∆ + SpAτ)H, A ∈ A.

Wojciech Dybalski Scattering theory for spin systems

Arveson spectrum

Let (A, τ) be a C ∗-dynamical system. Definition The Arveson spectrum of A ∈ A is the support of the (inverse) Fourier transform of R × Γ ∋ (t, x) → τ(t,x)(A). It is denoted SpAτ. Fact 1: (Energy-momentum transfer relation)

1 Let τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A. 2 Let 1

lU( · ) denote the spectral measure of U. Then A 1 lU(∆)H ⊂ 1 lU(∆ + SpAτ)H, A ∈ A. Fact 2: For any compact ∆ there are plenty almost-local operators A with SpAτ ⊂ ∆.

Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems

1 Γ - the abelian group of space translations (Rd or Zd). 2

Γ - Pontryagin dual of Γ (Rd or Sd

1 ).

3 (A, τ) - C ∗-dynamical system with R × Γ ∋ (t, x) → τ(t,x). 4 B⊂A -almost-local operators: [B1, τ(s,vs)(B2)] = O(|s|−∞). 5 A ⊂ B(H) and τ(t,x)(A) = U(t, x)AU(t, x)∗ for A ∈ A.

i

Sp U Ω H P Σ(p) p

∆ Β∗ i i

Sp U Ω H −π π Σ(p) p

∆ * Bi

Let B∗

1, . . . , B∗ n ∈ B be s.t. B∗ i Ω ∈ 1

lU(∆i)H are single-particle states.

Wojciech Dybalski Scattering theory for spin systems

Haag-Ruelle scattering theory

Theorem (Haag-Ruelle 62, Bachmann-Naaijkens-W.D.) The following limits exist and are called scattering states Ψout := lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω,

where B∗

t (gt) :=

• Γ dµ(x)τ(t,x)(B∗)gt(x), gt(x) :=
• Γ dp e−iΣ(p)t+ipx ˆ

g(p) and velocity supports V (gi) := { ∇Σ(p) | p ∈ supp ˆ gi } are disjoint.

i

Sp U Ω H P Σ(p) p

∆ Β∗ i i

Sp U Ω H −π π Σ(p) p

∆ * Bi

Wojciech Dybalski Scattering theory for spin systems

Haag-Ruelle scattering theory

Theorem (Haag-Ruelle 62, Bachmann-Naaijkens-W.D.) The following limits exist and are called scattering states Ψout := lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω,

where B∗

t (gt) :=

• Γ dµ(x)τ(t,x)(B∗)gt(x), gt(x) :=
• Γ dp e−iΣ(p)t+ipx ˆ

g(p) and velocity supports V (gi) := { ∇Σ(p) | p ∈ supp ˆ gi } are disjoint. Proof:

1 ∂t(B∗

t (gt))Ω = 0.

2 Let Ψt := B∗

1,t(g1,t)B∗ 2,t(g2,t)Ω.

Wojciech Dybalski Scattering theory for spin systems

Haag-Ruelle scattering theory

Theorem (Haag-Ruelle 62, Bachmann-Naaijkens-W.D.) The following limits exist and are called scattering states Ψout := lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω,

where B∗

t (gt) :=

• Γ dµ(x)τ(t,x)(B∗)gt(x), gt(x) :=
• Γ dp e−iΣ(p)t+ipx ˆ

g(p) and velocity supports V (gi) := { ∇Σ(p) | p ∈ supp ˆ gi } are disjoint. Proof:

1 ∂t(B∗

t (gt))Ω = 0.

2 Let Ψt := B∗

1,t(g1,t)B∗ 2,t(g2,t)Ω.

Wojciech Dybalski Scattering theory for spin systems

Haag-Ruelle scattering theory

Theorem (Haag-Ruelle 62, Bachmann-Naaijkens-W.D.) The following limits exist and are called scattering states Ψout := lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω,

where B∗

t (gt) :=

• Γ dµ(x)τ(t,x)(B∗)gt(x), gt(x) :=
• Γ dp e−iΣ(p)t+ipx ˆ

g(p) and velocity supports V (gi) := { ∇Σ(p) | p ∈ supp ˆ gi } are disjoint. Proof:

1 ∂t(B∗

t (gt))Ω = 0.

2 Let Ψt := B∗

1,t(g1,t)B∗ 2,t(g2,t)Ω.

Wojciech Dybalski Scattering theory for spin systems

Haag-Ruelle scattering theory

Theorem (Haag-Ruelle 62, Bachmann-Naaijkens-W.D.) The following limits exist and are called scattering states Ψout := lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω,

where B∗

t (gt) :=

• Γ dµ(x)τ(t,x)(B∗)gt(x), gt(x) :=
• Γ dp e−iΣ(p)t+ipx ˆ

g(p) and velocity supports V (gi) := { ∇Σ(p) | p ∈ supp ˆ gi } are disjoint. Proof:

1 ∂t(B∗

t (gt))Ω = 0.

2 Let Ψt := B∗

1,t(g1,t)B∗ 2,t(g2,t)Ω.

∂tΨt = ∂t(B∗

1,t(g1,t)) B∗ 2,t(g2,t)Ω + B∗ 1,t(g1,t) ∂t(B∗ 2,t(g2,t))Ω

• =0

Wojciech Dybalski Scattering theory for spin systems

Haag-Ruelle scattering theory

Theorem (Haag-Ruelle 62, Bachmann-Naaijkens-W.D.) The following limits exist and are called scattering states Ψout := lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω,

where B∗

t (gt) :=

• Γ dµ(x)τ(t,x)(B∗)gt(x), gt(x) :=
• Γ dp e−iΣ(p)t+ipx ˆ

g(p) and velocity supports V (gi) := { ∇Σ(p) | p ∈ supp ˆ gi } are disjoint. Proof:

1 ∂t(B∗

t (gt))Ω = 0.

2 Let Ψt := B∗

1,t(g1,t)B∗ 2,t(g2,t)Ω.

∂tΨt = ∂t(B∗

1,t(g1,t)) B∗ 2,t(g2,t)Ω + B∗ 1,t(g1,t) ∂t(B∗ 2,t(g2,t))Ω

• =0

= [∂t(B∗

1,t(g1,t)), B∗ 2,t(g2,t)]Ω = O(t−∞).

Wojciech Dybalski Scattering theory for spin systems

Wave-operators and S-matrix

1 H1 ⊂ H - single-particle subspace. 2 Γ(H1) - the symmetric Fock space over H1. 3 The outgoing wave-operator W out : Γ(H1) → H is defined by

W out(a∗(Ψ1) . . . a∗(Ψn)Ω) = lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω

for Ψi := B∗

i,t(gi,t)Ω.

Hout := Ran W out.

4 S := (W out)∗W in is the scattering matrix. 5 Def. If Hout = H the theory is asymptotically complete. Wojciech Dybalski Scattering theory for spin systems

Wave-operators and S-matrix

1 H1 ⊂ H - single-particle subspace. 2 Γ(H1) - the symmetric Fock space over H1. 3 The outgoing wave-operator W out : Γ(H1) → H is defined by

W out(a∗(Ψ1) . . . a∗(Ψn)Ω) = lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω

for Ψi := B∗

i,t(gi,t)Ω.

Hout := Ran W out.

4 S := (W out)∗W in is the scattering matrix. 5 Def. If Hout = H the theory is asymptotically complete. Wojciech Dybalski Scattering theory for spin systems

Wave-operators and S-matrix

1 H1 ⊂ H - single-particle subspace. 2 Γ(H1) - the symmetric Fock space over H1. 3 The outgoing wave-operator W out : Γ(H1) → H is defined by

W out(a∗(Ψ1) . . . a∗(Ψn)Ω) = lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω

for Ψi := B∗

i,t(gi,t)Ω.

Hout := Ran W out.

4 S := (W out)∗W in is the scattering matrix. 5 Def. If Hout = H the theory is asymptotically complete. Wojciech Dybalski Scattering theory for spin systems

Wave-operators and S-matrix

1 H1 ⊂ H - single-particle subspace. 2 Γ(H1) - the symmetric Fock space over H1. 3 The outgoing wave-operator W out : Γ(H1) → H is defined by

W out(a∗(Ψ1) . . . a∗(Ψn)Ω) = lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω

for Ψi := B∗

i,t(gi,t)Ω.

Hout := Ran W out.

4 S := (W out)∗W in is the scattering matrix. 5 Def. If Hout = H the theory is asymptotically complete. Wojciech Dybalski Scattering theory for spin systems

Wave-operators and S-matrix

1 H1 ⊂ H - single-particle subspace. 2 Γ(H1) - the symmetric Fock space over H1. 3 The outgoing wave-operator W out : Γ(H1) → H is defined by

W out(a∗(Ψ1) . . . a∗(Ψn)Ω) = lim

t→∞ B∗ 1,t(g1,t) . . . B∗ n,t(gn,t)Ω

for Ψi := B∗

i,t(gi,t)Ω.

Hout := Ran W out.

4 S := (W out)∗W in is the scattering matrix. 5 Def. If Hout = H the theory is asymptotically complete. Wojciech Dybalski Scattering theory for spin systems

The problem of asymptotic completeness in QM

Definition A QM theory given by H = − 1

2∆ + V is asymptotically complete

if scattering states and bound states of H span the entire Hilbert space.

Wojciech Dybalski Scattering theory for spin systems

The problem of asymptotic completeness in QM

Definition A QM theory given by H = − 1

2∆ + V is asymptotically complete

if scattering states and bound states of H span the entire Hilbert space.

V x y v v Wojciech Dybalski Scattering theory for spin systems

The problem of asymptotic completeness in QM

Definition A QM theory given by H = − 1

2∆ + V is asymptotically complete

if scattering states and bound states of H span the entire Hilbert space.

V x y v v Wojciech Dybalski Scattering theory for spin systems

The problem of asymptotic completeness in QM

Definition A QM theory given by H = − 1

2∆ + V is asymptotically complete

if scattering states and bound states of H span the entire Hilbert space.

x y Wojciech Dybalski Scattering theory for spin systems

Proving asymptotic completeness in QM

Excluding ‘fuzzy’ configurations in which the particle cannot decide between a bound state and a scattering state.

V x y v v

Wojciech Dybalski Scattering theory for spin systems

Proving asymptotic completeness in QM

Excluding ‘fuzzy’ configurations in which the particle cannot decide between a bound state and a scattering state.

V x y v v

A proof of asymptotic completeness is available in N-body QM [Faddeev 63,..., Sigal-Soffer 87, Graf 90, Dereziński 93]

Wojciech Dybalski Scattering theory for spin systems

Asymptotic completeness in QFT and spin systems

1 In systems with infinitely many degrees of freedom

there is an additional difficulty:

The Stone-von Neumann uniqueness theorem may break down. That is, the algebra of observables A may have many inequivalent representations.

Wojciech Dybalski Scattering theory for spin systems

Asymptotic completeness in QFT and spin systems

1 In systems with infinitely many degrees of freedom

there is an additional difficulty:

The Stone-von Neumann uniqueness theorem may break down. That is, the algebra of observables A may have many inequivalent representations.

Wojciech Dybalski Scattering theory for spin systems

Asymptotic completeness in QFT and spin systems

1 In systems with infinitely many degrees of freedom

there is an additional difficulty:

The Stone-von Neumann uniqueness theorem may break down. That is, the algebra of observables A may have many inequivalent representations.

Wojciech Dybalski Scattering theory for spin systems

Asymptotic completeness in QFT and spin systems

1 In systems with infinitely many degrees of freedom

there is an additional difficulty:

The Stone-von Neumann uniqueness theorem may break down. That is, the algebra of observables A may have many inequivalent representations.

∆ m Η P Ω 2m Wojciech Dybalski Scattering theory for spin systems

Asymptotic completeness in QFT and spin systems

1 In systems with infinitely many degrees of freedom

there is an additional difficulty:

The Stone-von Neumann uniqueness theorem may break down. That is, the algebra of observables A may have many inequivalent representations.

∆ m Η P Ω 2m

2

p p

1

Wojciech Dybalski Scattering theory for spin systems

Asymptotic completeness in QFT and spin systems

1 In systems with infinitely many degrees of freedom

there is an additional difficulty:

The Stone-von Neumann uniqueness theorem may break down. That is, the algebra of observables A may have many inequivalent representations.

m’ Η P Ω 2m ∆ m Wojciech Dybalski Scattering theory for spin systems

Asymptotic completeness in QFT and spin systems

1 In systems with infinitely many degrees of freedom

there is an additional difficulty:

The Stone-von Neumann uniqueness theorem may break down. That is, the algebra of observables A may have many inequivalent representations.

p

Η P Ω 2m

2

p

1

∆ m m’ + − Wojciech Dybalski Scattering theory for spin systems

Generalized asymptotic completeness

1 Conventional asymptotic completeness:

• ut

Ψ = Ψ

Wojciech Dybalski Scattering theory for spin systems

Generalized asymptotic completeness

1 Conventional asymptotic completeness:

• ut

Ψ = Ψ

2 Generalized asymptotic completeness [C. Gérard-W.D. 16]:

• ut

Ψ Ψout Α

Wojciech Dybalski Scattering theory for spin systems

Generalized asymptotic completeness

1 Conventional asymptotic completeness:

• ut

Ψ = Ψ

2 Generalized asymptotic completeness [C. Gérard-W.D. 16]:

• ut

Ψ Ψout Α

3 Fact. Generalized asymptotic completeness holds under our

assumptions.

Wojciech Dybalski Scattering theory for spin systems

Araki-Haag detectors

Theorem (Araki-Haag 67, Buchholz 90) Let Ct :=

• Γ dµ(x)τ(t,x)(B∗B)h

x

t

• , h ∈ C ∞

0 (R3).

Wojciech Dybalski Scattering theory for spin systems

Araki-Haag detectors

Theorem (Araki-Haag 67, Buchholz 90) Let Ct :=

• Γ dµ(x)τ(t,x)(B∗B)h

x

t

• , h ∈ C ∞

0 (R3). Then

lim

t→∞Ψout, CtΨout

Wojciech Dybalski Scattering theory for spin systems

Araki-Haag detectors

Theorem (Araki-Haag 67, Buchholz 90) Let Ct :=

• Γ dµ(x)τ(t,x)(B∗B)h

x

t

• , h ∈ C ∞

0 (R3). Then

lim

t→∞Ψout, CtΨout

=

• Γ

dp p|B∗B|ph(∇Σ(p))

• sensitivity of the detector

Ψout, a∗

• ut(p)aout(p)Ψout
• particle density

.

Wojciech Dybalski Scattering theory for spin systems

Generalized asymptotic completeness

Theorem (Gérard-W.D. 14, W.D. 16) Let ∆1 + · · · + ∆n ⊂ ∆ and hi have disjoint supports. Then the following limits exist Aout := s- lim

t→∞ C1,t . . . Cn,t1

lU(∆), Ct :=

• Γ

dµ(x)τ(t,x)(B∗B)h x t

• .

Also, generalized AC holds, i.e. Hout = [AoutH] ⊕ CΩ.

∗

Sp U Ω H P

∆

∆

∆1

2 Β2 ∗ Β1

1

Sp U Ω H −π π

∆ ∆

∆

Β Β

∗ ∗ 1 2 2

Wojciech Dybalski Scattering theory for spin systems

Generalized asymptotic completeness

Theorem (Gérard-W.D. 14, W.D. 16) Let ∆1 + · · · + ∆n ⊂ ∆ and hi have disjoint supports. Then the following limits exist Aout := s- lim

t→∞ C1,t . . . Cn,t1

lU(∆), Ct :=

• Γ

dµ(x)τ(t,x)(B∗B)h x t

• .

Also, generalized AC holds, i.e. Hout = [AoutH] ⊕ CΩ.

1 Generalized asymptotic completeness:

• ut

Ψ Ψout Α

Wojciech Dybalski Scattering theory for spin systems

Conclusions and outlook

1 We developed Haag-Ruelle scattering theory for a class of

gapped quantum spin systems.

2 The construction relies on the Lieb-Robinson bounds and on

the existence of isolated mass-shells of (quasi-)particles.

3 Our work provides a basis for a systematic study of the

problem of asymptotic completeness for quantum spin systems.

4 A related future direction is scattering of charged particles.

(anyons, topological order, Kitaev-type models...)

Wojciech Dybalski Scattering theory for spin systems

Conclusions and outlook

1 We developed Haag-Ruelle scattering theory for a class of

gapped quantum spin systems.

2 The construction relies on the Lieb-Robinson bounds and on

the existence of isolated mass-shells of (quasi-)particles.

3 Our work provides a basis for a systematic study of the

problem of asymptotic completeness for quantum spin systems.

4 A related future direction is scattering of charged particles.

(anyons, topological order, Kitaev-type models...)

Wojciech Dybalski Scattering theory for spin systems

Conclusions and outlook

1 We developed Haag-Ruelle scattering theory for a class of

gapped quantum spin systems.

2 The construction relies on the Lieb-Robinson bounds and on

the existence of isolated mass-shells of (quasi-)particles.

3 Our work provides a basis for a systematic study of the

problem of asymptotic completeness for quantum spin systems.

4 A related future direction is scattering of charged particles.

(anyons, topological order, Kitaev-type models...)

Wojciech Dybalski Scattering theory for spin systems

Conclusions and outlook

1 We developed Haag-Ruelle scattering theory for a class of

gapped quantum spin systems.

2 The construction relies on the Lieb-Robinson bounds and on

the existence of isolated mass-shells of (quasi-)particles.

3 Our work provides a basis for a systematic study of the

problem of asymptotic completeness for quantum spin systems.

4 A related future direction is scattering of charged particles.

(anyons, topological order, Kitaev-type models...)

Wojciech Dybalski Scattering theory for spin systems