Schrodinger-Klein-Gordon system as a classical limit of ascalar Quantum Field Theory.

(in collaboration with Marco Falconi)

Zied Ammari

Rennes University, IRMAR

BECAM, 28 October 2014

The Yukawa Theory

I The Yukawa theory describes the nucleon-nucleon (NN) interaction.

I The strong nuclear force bounds nucleons together via the attractive

Yukawa potential V (x) = − e−m|x|

|x| .

I The interaction is mediated by the π-mesons (massive bosons) andit is described by a quantum field theory (QFT).

The Yukawa Theory

I The Yukawa theory describes the nucleon-nucleon (NN) interaction.

I The strong nuclear force bounds nucleons together via the attractive

Yukawa potential V (x) = − e−m|x|

|x| .

I The interaction is mediated by the π-mesons (massive bosons) andit is described by a quantum field theory (QFT).

The Yukawa Theory

I The Yukawa theory describes the nucleon-nucleon (NN) interaction.

I The strong nuclear force bounds nucleons together via the attractive

Yukawa potential V (x) = − e−m|x|

|x| .

I The interaction is mediated by the π-mesons (massive bosons) andit is described by a quantum field theory (QFT).

General point of viewI Quantum systems with a large number of particles obeying the Bose

statistics have quite often an interesting collective behavior atcertain scales. For instance, this is experimentally observed incondensate matter physics (Bose-Einstein condensate) and inquantum optics (coherent light).

I The semiclassical analysis in finite dimension have turned the Bohrcorrespondence principle into a deep and powerful mathematicaltheory (h-pseudodifferential calculus, tunnelling effect, eigenvaluesasymptotics, resonances, Weyl’s law, propagation of singularities,quantum ergodicity and chaos...).

I There exists an old attempt to extend microlocal analysis to infinitedimensional phase spaces (This was the main subject of Paul Kreeseminar in the 70’s, Bernard Lascar,...). But there are severaldifficulties!

General point of viewI Quantum systems with a large number of particles obeying the Bose

statistics have quite often an interesting collective behavior atcertain scales. For instance, this is experimentally observed incondensate matter physics (Bose-Einstein condensate) and inquantum optics (coherent light).

I The semiclassical analysis in finite dimension have turned the Bohrcorrespondence principle into a deep and powerful mathematicaltheory (h-pseudodifferential calculus, tunnelling effect, eigenvaluesasymptotics, resonances, Weyl’s law, propagation of singularities,quantum ergodicity and chaos...).

I There exists an old attempt to extend microlocal analysis to infinitedimensional phase spaces (This was the main subject of Paul Kreeseminar in the 70’s, Bernard Lascar,...). But there are severaldifficulties!

General point of viewI Quantum systems with a large number of particles obeying the Bose

statistics have quite often an interesting collective behavior atcertain scales. For instance, this is experimentally observed incondensate matter physics (Bose-Einstein condensate) and inquantum optics (coherent light).

I The semiclassical analysis in finite dimension have turned the Bohrcorrespondence principle into a deep and powerful mathematicaltheory (h-pseudodifferential calculus, tunnelling effect, eigenvaluesasymptotics, resonances, Weyl’s law, propagation of singularities,quantum ergodicity and chaos...).

I There exists an old attempt to extend microlocal analysis to infinitedimensional phase spaces (This was the main subject of Paul Kreeseminar in the 70’s, Bernard Lascar,...). But there are severaldifficulties!

AimsI Overcome the difficulties encountered in the 70’s and extend the

semiclassical analysis to infinite dimensional phase spaces.

I Study of quantum systems with a large number of particles (bosons)using a general approach. In particular, addressing the questions of:

I Mean field theoryI Classical limit

in the following frameworks:

I Many-Body theory (N-body Schrodinger operator)I Relativistic Quantum field theory ((ϕ)4

2,P(ϕ)2 models)I Quantum electrodynamics (spin-boson, Nelson, Pauli-Fierz models)

I Results: semiclassical propagation theorems, eigenvalues asymptoticsand more generally spectral and scattering properties of quantumdynamical systems (with infinite degrees of freedom) in thesemiclassical regime.

AimsI Overcome the difficulties encountered in the 70’s and extend the

semiclassical analysis to infinite dimensional phase spaces.

I Study of quantum systems with a large number of particles (bosons)using a general approach. In particular, addressing the questions of:

I Mean field theoryI Classical limit

in the following frameworks:

I Many-Body theory (N-body Schrodinger operator)I Relativistic Quantum field theory ((ϕ)4

2,P(ϕ)2 models)I Quantum electrodynamics (spin-boson, Nelson, Pauli-Fierz models)

I Results: semiclassical propagation theorems, eigenvalues asymptoticsand more generally spectral and scattering properties of quantumdynamical systems (with infinite degrees of freedom) in thesemiclassical regime.

AimsI Overcome the difficulties encountered in the 70’s and extend the

semiclassical analysis to infinite dimensional phase spaces.

I Study of quantum systems with a large number of particles (bosons)using a general approach. In particular, addressing the questions of:

I Mean field theoryI Classical limit

in the following frameworks:

I Many-Body theory (N-body Schrodinger operator)I Relativistic Quantum field theory ((ϕ)4

2,P(ϕ)2 models)I Quantum electrodynamics (spin-boson, Nelson, Pauli-Fierz models)

I Results: semiclassical propagation theorems, eigenvalues asymptoticsand more generally spectral and scattering properties of quantumdynamical systems (with infinite degrees of freedom) in thesemiclassical regime.

AimsI Overcome the difficulties encountered in the 70’s and extend the

semiclassical analysis to infinite dimensional phase spaces.

I Study of quantum systems with a large number of particles (bosons)using a general approach. In particular, addressing the questions of:

I Mean field theoryI Classical limit

in the following frameworks:

I Many-Body theory (N-body Schrodinger operator)I Relativistic Quantum field theory ((ϕ)4

2,P(ϕ)2 models)I Quantum electrodynamics (spin-boson, Nelson, Pauli-Fierz models)

I Results: semiclassical propagation theorems, eigenvalues asymptoticsand more generally spectral and scattering properties of quantumdynamical systems (with infinite degrees of freedom) in thesemiclassical regime.

Outline

The Schrodinger Klein-Gordon (S-KG) system

Mathematical foundation of QFT

The classical limit ε→ 0

Ground state energy limit

Outline

The Schrodinger Klein-Gordon (S-KG) system

Mathematical foundation of QFT

The classical limit ε→ 0

Ground state energy limit

Outline

The Schrodinger Klein-Gordon (S-KG) system

Mathematical foundation of QFT

The classical limit ε→ 0

Ground state energy limit

Outline

The Schrodinger Klein-Gordon (S-KG) system

Mathematical foundation of QFT

The classical limit ε→ 0

Ground state energy limit

The S-KG system in dimension d

i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

(� + m2)A = −ϕ ∗ |u|2(S-KG)

I V is an external potential for the non-relativistic particle.

I ϕ can be a (regularizing) function, or Dirac’s delta distribution.

I M > 0; m ≥ 0 (For simplicity, we suppose in this talk m > 0).

I It is a wave-particle system describing in some sense thenucleon-meson interaction.

The S-KG system in dimension d

i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

(� + m2)A = −ϕ ∗ |u|2(S-KG)

I V is an external potential for the non-relativistic particle.

I ϕ can be a (regularizing) function, or Dirac’s delta distribution.

I M > 0; m ≥ 0 (For simplicity, we suppose in this talk m > 0).

I It is a wave-particle system describing in some sense thenucleon-meson interaction.

The S-KG system in dimension d

i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

(� + m2)A = −ϕ ∗ |u|2(S-KG)

I V is an external potential for the non-relativistic particle.

I ϕ can be a (regularizing) function, or Dirac’s delta distribution.

I M > 0; m ≥ 0 (For simplicity, we suppose in this talk m > 0).

I It is a wave-particle system describing in some sense thenucleon-meson interaction.

The S-KG system in dimension d

i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

(� + m2)A = −ϕ ∗ |u|2(S-KG)

I V is an external potential for the non-relativistic particle.

I ϕ can be a (regularizing) function, or Dirac’s delta distribution.

I M > 0; m ≥ 0 (For simplicity, we suppose in this talk m > 0).

I It is a wave-particle system describing in some sense thenucleon-meson interaction.

The S-KG system in dimension d

i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

(� + m2)A = −ϕ ∗ |u|2(S-KG)

I V is an external potential for the non-relativistic particle.

I ϕ can be a (regularizing) function, or Dirac’s delta distribution.

I M > 0; m ≥ 0 (For simplicity, we suppose in this talk m > 0).

I It is a wave-particle system describing in some sense thenucleon-meson interaction.

The S-KG system in dimension d

i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

(� + m2)A = −ϕ ∗ |u|2(S-KG)

I V is an external potential for the non-relativistic particle.

I ϕ can be a (regularizing) function, or Dirac’s delta distribution.

I M > 0; m ≥ 0 (For simplicity, we suppose in this talk m > 0).

I It is a wave-particle system describing in some sense thenucleon-meson interaction.

The Yukawa interaction

In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1

{i∂tu = −∆u + Au

(� + 1)A = −|u|2(S-KG[δ])

u(t0) = u0 , A(t0) = A0 , ∂tA(t0) = A1 .

In the literature, global well-posedness of the above system has beenextensively investigated (e.g. Fukuda and Tsutsumi [1975]; Baillon andChadam [1978]; Bachelot [1984]; Ginibre and Velo [2002]; Colliander,Holmer and Tzirakis [2008]; Pecher [2012]...).

Theorem (Pecher (2012))

Let 0 ≤ s ≤ σ ≤ s + 1 and u0 ∈ Hs(R3), A0 ∈ Hσ(R3), A1 ∈ Hσ−1(R3).Then (S-KG[δ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C0(R,Hs(R3)), A ∈ C0(R,Hσ(R3)) ∩ C1(R,Hσ−1(R3)).

The Yukawa interaction

In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1{i∂tu = −∆u + Au

(� + 1)A = −|u|2(S-KG[δ])

u(t0) = u0 , A(t0) = A0 , ∂tA(t0) = A1 .

In the literature, global well-posedness of the above system has beenextensively investigated (e.g. Fukuda and Tsutsumi [1975]; Baillon andChadam [1978]; Bachelot [1984]; Ginibre and Velo [2002]; Colliander,Holmer and Tzirakis [2008]; Pecher [2012]...).

Theorem (Pecher (2012))

Let 0 ≤ s ≤ σ ≤ s + 1 and u0 ∈ Hs(R3), A0 ∈ Hσ(R3), A1 ∈ Hσ−1(R3).Then (S-KG[δ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C0(R,Hs(R3)), A ∈ C0(R,Hσ(R3)) ∩ C1(R,Hσ−1(R3)).

The Yukawa interaction

In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1{i∂tu = −∆u + Au

(� + 1)A = −|u|2(S-KG[δ])

u(t0) = u0 , A(t0) = A0 , ∂tA(t0) = A1 .

In the literature, global well-posedness of the above system has beenextensively investigated (e.g. Fukuda and Tsutsumi [1975]; Baillon andChadam [1978]; Bachelot [1984]; Ginibre and Velo [2002]; Colliander,Holmer and Tzirakis [2008]; Pecher [2012]...).

Theorem (Pecher (2012))

Let 0 ≤ s ≤ σ ≤ s + 1 and u0 ∈ Hs(R3), A0 ∈ Hσ(R3), A1 ∈ Hσ−1(R3).Then (S-KG[δ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C0(R,Hs(R3)), A ∈ C0(R,Hσ(R3)) ∩ C1(R,Hσ−1(R3)).

The Yukawa interaction

In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1{i∂tu = −∆u + Au

(� + 1)A = −|u|2(S-KG[δ])

u(t0) = u0 , A(t0) = A0 , ∂tA(t0) = A1 .

In the literature, global well-posedness of the above system has beenextensively investigated (e.g. Fukuda and Tsutsumi [1975]; Baillon andChadam [1978]; Bachelot [1984]; Ginibre and Velo [2002]; Colliander,Holmer and Tzirakis [2008]; Pecher [2012]...).

Theorem (Pecher (2012))

Let 0 ≤ s ≤ σ ≤ s + 1 and u0 ∈ Hs(R3), A0 ∈ Hσ(R3), A1 ∈ Hσ−1(R3).Then (S-KG[δ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C0(R,Hs(R3)), A ∈ C0(R,Hσ(R3)) ∩ C1(R,Hσ−1(R3)).

The Yukawa interaction

In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1{i∂tu = −∆u + Au

(� + 1)A = −|u|2(S-KG[δ])

u(t0) = u0 , A(t0) = A0 , ∂tA(t0) = A1 .

In the literature, global well-posedness of the above system has beenextensively investigated (e.g. Fukuda and Tsutsumi [1975]; Baillon andChadam [1978]; Bachelot [1984]; Ginibre and Velo [2002]; Colliander,Holmer and Tzirakis [2008]; Pecher [2012]...).

Theorem (Pecher (2012))

Let 0 ≤ s ≤ σ ≤ s + 1 and u0 ∈ Hs(R3), A0 ∈ Hσ(R3), A1 ∈ Hσ−1(R3).Then (S-KG[δ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C0(R,Hs(R3)), A ∈ C0(R,Hσ(R3)) ∩ C1(R,Hσ−1(R3)).

The regularized system

If ϕ is sufficiently regular, we can study global well-posedness of (S-KG)with less restrictions on d , m and V . For instance, in the caseϕ ∈ H−1/2(Rd), m > 0, we have:

Proposition (d ≥ 2, (Falconi 2013))

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0); and u0 ∈ L2(Rd),

A0 ∈ H12 (Rd), A1 ∈ H−

12 (Rd).

Then (S-KG) is globally well-posed; i.e. there exists a unique solution

u ∈ C 0(R, L2(Rd)), A ∈ C 0(R,H12 (Rd).

I In this regular setting the global well-posedness follows easily fromstandard contraction estimates and the conservation of mass ‖u‖2.

I We have a well defined continuous global flow on the phase-space.

I From now on, we work in this setting.

The regularized system

If ϕ is sufficiently regular, we can study global well-posedness of (S-KG)with less restrictions on d , m and V . For instance, in the caseϕ ∈ H−1/2(Rd), m > 0, we have:

Proposition (d ≥ 2, (Falconi 2013))

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0); and u0 ∈ L2(Rd),

A0 ∈ H12 (Rd), A1 ∈ H−

12 (Rd).

Then (S-KG) is globally well-posed; i.e. there exists a unique solution

u ∈ C 0(R, L2(Rd)), A ∈ C 0(R,H12 (Rd).

I In this regular setting the global well-posedness follows easily fromstandard contraction estimates and the conservation of mass ‖u‖2.

I We have a well defined continuous global flow on the phase-space.

I From now on, we work in this setting.

The regularized system

If ϕ is sufficiently regular, we can study global well-posedness of (S-KG)with less restrictions on d , m and V . For instance, in the caseϕ ∈ H−1/2(Rd), m > 0, we have:

Proposition (d ≥ 2, (Falconi 2013))

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0); and u0 ∈ L2(Rd),

A0 ∈ H12 (Rd), A1 ∈ H−

12 (Rd).

Then (S-KG) is globally well-posed; i.e. there exists a unique solution

u ∈ C 0(R, L2(Rd)), A ∈ C 0(R,H12 (Rd).

I In this regular setting the global well-posedness follows easily fromstandard contraction estimates and the conservation of mass ‖u‖2.

I We have a well defined continuous global flow on the phase-space.

I From now on, we work in this setting.

The regularized system

If ϕ is sufficiently regular, we can study global well-posedness of (S-KG)with less restrictions on d , m and V . For instance, in the caseϕ ∈ H−1/2(Rd), m > 0, we have:

Proposition (d ≥ 2, (Falconi 2013))

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0); and u0 ∈ L2(Rd),

A0 ∈ H12 (Rd), A1 ∈ H−

12 (Rd).

Then (S-KG) is globally well-posed; i.e. there exists a unique solution

u ∈ C 0(R, L2(Rd)), A ∈ C 0(R,H12 (Rd).

I In this regular setting the global well-posedness follows easily fromstandard contraction estimates and the conservation of mass ‖u‖2.

I We have a well defined continuous global flow on the phase-space.

I From now on, we work in this setting.

The regularized system

If ϕ is sufficiently regular, we can study global well-posedness of (S-KG)with less restrictions on d , m and V . For instance, in the caseϕ ∈ H−1/2(Rd), m > 0, we have:

Proposition (d ≥ 2, (Falconi 2013))

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0); and u0 ∈ L2(Rd),

A0 ∈ H12 (Rd), A1 ∈ H−

12 (Rd).

Then (S-KG) is globally well-posed; i.e. there exists a unique solution

u ∈ C 0(R, L2(Rd)), A ∈ C 0(R,H12 (Rd).

I In this regular setting the global well-posedness follows easily fromstandard contraction estimates and the conservation of mass ‖u‖2.

I We have a well defined continuous global flow on the phase-space.

I From now on, we work in this setting.

The regularized system

If ϕ is sufficiently regular, we can study global well-posedness of (S-KG)with less restrictions on d , m and V . For instance, in the caseϕ ∈ H−1/2(Rd), m > 0, we have:

Proposition (d ≥ 2, (Falconi 2013))

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0); and u0 ∈ L2(Rd),

A0 ∈ H12 (Rd), A1 ∈ H−

12 (Rd).

Then (S-KG) is globally well-posed; i.e. there exists a unique solution

u ∈ C 0(R, L2(Rd)), A ∈ C 0(R,H12 (Rd).

I In this regular setting the global well-posedness follows easily fromstandard contraction estimates and the conservation of mass ‖u‖2.

I We have a well defined continuous global flow on the phase-space.

I From now on, we work in this setting.

The complex fields

QFT uses often complex fields rather than real fields. It is thereforeconvient to switch to this representation even at the classical level. So,we replace (A, ∂tA) by the complex fields (α, α) defined by:

A(x) =1

(2π)d2

∫Rd

1√2ω(k)

(α(k)e−ik·x+α(k)e ik·x)dk , ω(k) =

√k2 + m2 .

Then, with χ = (2π)d2 ϕ, (S-KG) is equivalent to the following system

with the unknown (u, α):i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

i∂tα = ωα +χ√2ω

(uu)(S-KG[χ])

Corollary (d ≥ 2)

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0) and u0 ∈ L2(Rd),

α0 ∈ L2(Rd).Then (S-KG[χ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C 0(R, L2(Rd)), α ∈ C 0(R, L2(Rd)).

The complex fields

QFT uses often complex fields rather than real fields. It is thereforeconvient to switch to this representation even at the classical level. So,we replace (A, ∂tA) by the complex fields (α, α) defined by:

A(x) =1

(2π)d2

∫Rd

1√2ω(k)

(α(k)e−ik·x+α(k)e ik·x)dk , ω(k) =

√k2 + m2 .

Then, with χ = (2π)d2 ϕ, (S-KG) is equivalent to the following system

with the unknown (u, α):i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

i∂tα = ωα +χ√2ω

(uu)(S-KG[χ])

Corollary (d ≥ 2)

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0) and u0 ∈ L2(Rd),

α0 ∈ L2(Rd).Then (S-KG[χ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C 0(R, L2(Rd)), α ∈ C 0(R, L2(Rd)).

The complex fields

QFT uses often complex fields rather than real fields. It is thereforeconvient to switch to this representation even at the classical level. So,we replace (A, ∂tA) by the complex fields (α, α) defined by:

A(x) =1

(2π)d2

∫Rd

1√2ω(k)

(α(k)e−ik·x+α(k)e ik·x)dk , ω(k) =

√k2 + m2 .

Then, with χ = (2π)d2 ϕ, (S-KG) is equivalent to the following system

with the unknown (u, α):i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

i∂tα = ωα +χ√2ω

(uu)(S-KG[χ])

Corollary (d ≥ 2)

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0) and u0 ∈ L2(Rd),

α0 ∈ L2(Rd).Then (S-KG[χ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C 0(R, L2(Rd)), α ∈ C 0(R, L2(Rd)).

The complex fields

QFT uses often complex fields rather than real fields. It is thereforeconvient to switch to this representation even at the classical level. So,we replace (A, ∂tA) by the complex fields (α, α) defined by:

A(x) =1

(2π)d2

∫Rd

1√2ω(k)

(α(k)e−ik·x+α(k)e ik·x)dk , ω(k) =

√k2 + m2 .

Then, with χ = (2π)d2 ϕ, (S-KG) is equivalent to the following system

with the unknown (u, α):i∂tu = − ∆

2Mu + Vu + (ϕ ∗ A)u

i∂tα = ωα +χ√2ω

(uu)(S-KG[χ])

Corollary (d ≥ 2)

Let V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) (m > 0) and u0 ∈ L2(Rd),

α0 ∈ L2(Rd).Then (S-KG[χ]) is globally well-posed; i.e. there exists a unique solutionu ∈ C 0(R, L2(Rd)), α ∈ C 0(R, L2(Rd)).

The classical Hamiltonian

I The Schodinger-Klein-Gordon equation S-KG[χ] is a Hamiltoniansystem.

I Its energy functional is densely defined on L2(Rd)⊕ L2(Rd):

h(u ⊕ α) =

∫Rd

u(x)(−∆x

2M+ V (x)

)u(x)dx +

∫Rd

α(k)ω(k)α(k)dk

+1

(2π)d2

∫R2d

u(x)χ(k)√2ω(k)

(α(k)e−ik·x + α(k)e ik·x

)u(x)dxdk

I So that, the equation (S-KG[χ]) takes the more compact form:

i∂t

(u

α

)=

δh

δu

δh

δα

(S-KG[χ])

Let (Φh)tt0denotes the well defined flow of (S-KG[χ]) on the phase-space

L2(Rd)⊕ L2(Rd).

The classical HamiltonianI The Schodinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian

system.

I Its energy functional is densely defined on L2(Rd)⊕ L2(Rd):

h(u ⊕ α) =

∫Rd

u(x)(−∆x

2M+ V (x)

)u(x)dx +

∫Rd

α(k)ω(k)α(k)dk

+1

(2π)d2

∫R2d

u(x)χ(k)√2ω(k)

(α(k)e−ik·x + α(k)e ik·x

)u(x)dxdk

I So that, the equation (S-KG[χ]) takes the more compact form:

i∂t

(u

α

)=

δh

δu

δh

δα

(S-KG[χ])

Let (Φh)tt0denotes the well defined flow of (S-KG[χ]) on the phase-space

L2(Rd)⊕ L2(Rd).

The classical HamiltonianI The Schodinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian

system.

I Its energy functional is densely defined on L2(Rd)⊕ L2(Rd):

h(u ⊕ α) =

∫Rd

u(x)(−∆x

2M+ V (x)

)u(x)dx +

∫Rd

α(k)ω(k)α(k)dk

+1

(2π)d2

∫R2d

u(x)χ(k)√2ω(k)

(α(k)e−ik·x + α(k)e ik·x

)u(x)dxdk

I So that, the equation (S-KG[χ]) takes the more compact form:

i∂t

(u

α

)=

δh

δu

δh

δα

(S-KG[χ])

Let (Φh)tt0denotes the well defined flow of (S-KG[χ]) on the phase-space

L2(Rd)⊕ L2(Rd).

The classical HamiltonianI The Schodinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian

system.

I Its energy functional is densely defined on L2(Rd)⊕ L2(Rd):

h(u ⊕ α) =

∫Rd

u(x)(−∆x

2M+ V (x)

)u(x)dx +

∫Rd

α(k)ω(k)α(k)dk

+1

(2π)d2

∫R2d

u(x)χ(k)√2ω(k)

(α(k)e−ik·x + α(k)e ik·x

)u(x)dxdk

I So that, the equation (S-KG[χ]) takes the more compact form:

i∂t

(u

α

)=

δh

δu

δh

δα

(S-KG[χ])

Let (Φh)tt0denotes the well defined flow of (S-KG[χ]) on the phase-space

L2(Rd)⊕ L2(Rd).

The classical HamiltonianI The Schodinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian

system.

I Its energy functional is densely defined on L2(Rd)⊕ L2(Rd):

h(u ⊕ α) =

∫Rd

u(x)(−∆x

2M+ V (x)

)u(x)dx +

∫Rd

α(k)ω(k)α(k)dk

+1

(2π)d2

∫R2d

u(x)χ(k)√2ω(k)

(α(k)e−ik·x + α(k)e ik·x

)u(x)dxdk

I So that, the equation (S-KG[χ]) takes the more compact form:

i∂t

(u

α

)=

δh

δu

δh

δα

(S-KG[χ])

Let (Φh)tt0denotes the well defined flow of (S-KG[χ]) on the phase-space

L2(Rd)⊕ L2(Rd).

Constructive QFT

The goal of constructive QFT is to provide a solution for nonlinearquantum field equations of the type

(� + m)φ(t, x) + λφ2n+1(t, x) = 0 ,

where φ(t; x) and π(t; x) = ∂tφ(t; x) satisfy the quantum condition

[φ(t, x), φ(t, y)] = [π(t, x), π(t, y)] = 0, [π(t, x), iφ(t, y)] = ~δ(x − y).

The Hamiltonian H is a function of the canonical variables φ(t; x) andπ(t; x)

H =

∫1

2[π2(t, x) + |∇φ(t, x)|2 + mφ(t, x)2] +

λ

2n + 2φ(t, x)2n+2 dx .

So that, the time variation of quantum fields is given by the equation ofmotion

i~∂tφ(t; x) = [φ(t; x); H] and i~∂tπ(t; x) = [π(t; x); H].

Canonical commutation relations

In quantum field theory the smeared canonical variables

φ(f ) =

∫φ(0, x)f (x)dx and π(f ) =

∫π(0, x)f (x)dx , f ∈ L2(Rd ,R)

are self-adjoint operators on a given Hilbert space H, satisfying theHeisenberg commutation relations:

[φ(f ), φ(g)] = [π(f ), π(g)] = 0, [π(g), iφ(f )] = ~〈f , g〉 I .

Usually the Weyl commutation relations are preferred:

W (g , f ) = e−i~2 〈f ,g〉 e iπ(g) e iφ(f ) ,

Satisfying:

(i) W (g1, f1)W (g2, f2) = ei~2 σ[(g1,f1),(g2,f2)] W (g1 + g2, f1 + f2).

(ii) W (g , f )∗ = W (−g ,−f ).

Here σ is a canonical symplectic form.

Fock representation

It is the most important representation of the Weyl commutationrelations and it is build on the symmetric Fock space. Recall that thesymmetric Fock space over Z = L2(Rd ,C) is

Γs(Z ) = ⊕∞n=0 ⊗ns Z = ⊕∞n=0L2

s (Rnd ,C) .

I Annihilation :

a

~

(f )f1 ⊗s · · · ⊗s fn =√

~

n1

n!

∑σ∈Sn

〈f , fσ1〉 fσ2 ⊗ · · · ⊗ fσn ,

I Creation :

a

~

∗(f )f1 ⊗s · · · ⊗s fn =√

~

(n + 1) f ⊗s f1 · · · ⊗s fn.

[a

~

(f ), a

~

∗(g)] =

~

〈f , g〉Id .

I Weyl operator :

W

~

(f ) = ei√2

(a

~

∗(f )+a

~

(f )).

This determines φ(t, x) and π(t, x) in the case λ = 0.

Fock representation

It is the most important representation of the Weyl commutationrelations and it is build on the symmetric Fock space. Recall that thesymmetric Fock space over Z = L2(Rd ,C) is

Γs(Z ) = ⊕∞n=0 ⊗ns Z = ⊕∞n=0L2

s (Rnd ,C) .

I Annihilation :

a

~

(f )f1 ⊗s · · · ⊗s fn =√

~

n1

n!

∑σ∈Sn

〈f , fσ1〉 fσ2 ⊗ · · · ⊗ fσn ,

I Creation :

a

~

∗(f )f1 ⊗s · · · ⊗s fn =√

~

(n + 1) f ⊗s f1 · · · ⊗s fn.

[a

~

(f ), a

~

∗(g)] =

~

〈f , g〉Id .

I Weyl operator :

W

~

(f ) = ei√2

(a

~

∗(f )+a

~

(f )).

This determines φ(t, x) and π(t, x) in the case λ = 0.

Fock representation

It is the most important representation of the Weyl commutationrelations and it is build on the symmetric Fock space. Recall that thesymmetric Fock space over Z = L2(Rd ,C) is

Γs(Z ) = ⊕∞n=0 ⊗ns Z = ⊕∞n=0L2

s (Rnd ,C) .

I Annihilation :

a

~

(f )f1 ⊗s · · · ⊗s fn =√

~

n1

n!

∑σ∈Sn

〈f , fσ1〉 fσ2 ⊗ · · · ⊗ fσn ,

I Creation :

a

~

∗(f )f1 ⊗s · · · ⊗s fn =√

~

(n + 1) f ⊗s f1 · · · ⊗s fn.

[a

~

(f ), a

~

∗(g)] =

~

〈f , g〉Id .

I Weyl operator :

W

~

(f ) = ei√2

(a

~

∗(f )+a

~

(f )).

This determines φ(t, x) and π(t, x) in the case λ = 0.

Fock representation

It is the most important representation of the Weyl commutationrelations and it is build on the symmetric Fock space. Recall that thesymmetric Fock space over Z = L2(Rd ,C) is

Γs(Z ) = ⊕∞n=0 ⊗ns Z = ⊕∞n=0L2

s (Rnd ,C) .

I Annihilation :

a

~

(f )f1 ⊗s · · · ⊗s fn =√

~

n1

n!

∑σ∈Sn

〈f , fσ1〉 fσ2 ⊗ · · · ⊗ fσn ,

I Creation :

a

~

∗(f )f1 ⊗s · · · ⊗s fn =√

~

(n + 1) f ⊗s f1 · · · ⊗s fn.

[a

~

(f ), a

~

∗(g)] =

~

〈f , g〉Id .

I Weyl operator :

W

~

(f ) = ei√2

(a

~

∗(f )+a

~

(f )).

This determines φ(t, x) and π(t, x) in the case λ = 0.

Fock representation

It is the most important representation of the Weyl commutationrelations and it is build on the symmetric Fock space. Recall that thesymmetric Fock space over Z = L2(Rd ,C) is

Γs(Z ) = ⊕∞n=0 ⊗ns Z = ⊕∞n=0L2

s (Rnd ,C) .

I Annihilation :

a~(f )f1 ⊗s · · · ⊗s fn =√~n

1

n!

∑σ∈Sn

〈f , fσ1〉 fσ2 ⊗ · · · ⊗ fσn ,

I Creation :

a~∗(f )f1 ⊗s · · · ⊗s fn =

√~(n + 1) f ⊗s f1 · · · ⊗s fn.

[a~(f ), a~∗(g)] = ~〈f , g〉Id .

I Weyl operator :

W~(f ) = ei√2

(a~∗(f )+a~(f ))

.

This determines φ(t, x) and π(t, x) in the case λ = 0.

Wick Quantization

It is a mapping associating to a polynomial functional (classicalHamiltonian) h(α, α) an operator in the Fock space

h(α, α) −→ h(a∗, a) = H

with all a∗ in the left and all a in the right.

I The Wick operator with polynomial ”symbol” is defined by

b

~

Wick|⊗n

s Z=

√(n − p + q)!n!

(n − p)!

~p+q

2

b ⊗s 1n−p

where b : ⊗ps Z → ⊗q

s Z is a given operator and the symbol isb(α, α) = 〈α⊗q, bα⊗p〉.

I This procedure provides a canonical quantization of classicalsystems.

Example: The Klein-Gordon free energy∫Rd

α(k)ω(k)α(k)dk −→ 〈α, ω(k)α〉Wick =

∫Rd

a∗~(k)ω(k) a~(k)dk .

Wick Quantization

It is a mapping associating to a polynomial functional (classicalHamiltonian) h(α, α) an operator in the Fock space

h(α, α) −→ h(a∗, a) = H

with all a∗ in the left and all a in the right.

I The Wick operator with polynomial ”symbol” is defined by

b~Wick|⊗n

s Z=

√(n − p + q)!n!

(n − p)!~

p+q2 b ⊗s 1n−p

where b : ⊗ps Z → ⊗q

s Z is a given operator and the symbol isb(α, α) = 〈α⊗q, bα⊗p〉.

I This procedure provides a canonical quantization of classicalsystems.

Example: The Klein-Gordon free energy∫Rd

α(k)ω(k)α(k)dk −→ 〈α, ω(k)α〉Wick =

∫Rd

a∗~(k)ω(k) a~(k)dk .

Wick Quantization

It is a mapping associating to a polynomial functional (classicalHamiltonian) h(α, α) an operator in the Fock space

h(α, α) −→ h(a∗, a) = H

with all a∗ in the left and all a in the right.

I The Wick operator with polynomial ”symbol” is defined by

b~Wick|⊗n

s Z=

√(n − p + q)!n!

(n − p)!~

p+q2 b ⊗s 1n−p

where b : ⊗ps Z → ⊗q

s Z is a given operator and the symbol isb(α, α) = 〈α⊗q, bα⊗p〉.

I This procedure provides a canonical quantization of classicalsystems.

Example: The Klein-Gordon free energy∫Rd

α(k)ω(k)α(k)dk −→ 〈α, ω(k)α〉Wick =

∫Rd

a∗~(k)ω(k) a~(k)dk .

Wick Quantization

It is a mapping associating to a polynomial functional (classicalHamiltonian) h(α, α) an operator in the Fock space

h(α, α) −→ h(a∗, a) = H

with all a∗ in the left and all a in the right.

I The Wick operator with polynomial ”symbol” is defined by

b~Wick|⊗n

s Z=

√(n − p + q)!n!

(n − p)!~

p+q2 b ⊗s 1n−p

where b : ⊗ps Z → ⊗q

s Z is a given operator and the symbol isb(α, α) = 〈α⊗q, bα⊗p〉.

I This procedure provides a canonical quantization of classicalsystems.

Example: The Klein-Gordon free energy∫Rd

α(k)ω(k)α(k)dk −→ 〈α, ω(k)α〉Wick =

∫Rd

a∗~(k)ω(k) a~(k)dk .

Classical-Quantum correspondence

I

L2(Rd)⊕ L2(Rd)Classical phase space (infinite dim.)

−→ Γs(L2(Rd)⊕ L2(Rd))Quantum Fock space

I

u(x), u(x) and α(k), α(k)Classical variables (scalar fields)

−→ ψ(x), ψ∗(x) and a(k), a∗(k)Quantum variables (op.valued distributions)

I

f (u ⊕ α) : D(f )→ RClassical observables (functionals)

−→ F := f (u, α)Wick = f (ψ, a)Quantum observables (operators on Fock sp.)

I

(Φh)tt0

Classical evolution (flow on phase sp.)

−→ e−i~ (t−t0)H , H = h(ψ, a)

Quantum evolution (unitary group on Fock sp.)

From now on, the dependence on ~ at the quantum level is implicite andwe replace ~ by ε a small parameter in the quantum theory (ε→ 0).

Classical-Quantum correspondenceI

L2(Rd)⊕ L2(Rd)Classical phase space (infinite dim.)

−→ Γs(L2(Rd)⊕ L2(Rd))Quantum Fock space

I

u(x), u(x) and α(k), α(k)Classical variables (scalar fields)

−→ ψ(x), ψ∗(x) and a(k), a∗(k)Quantum variables (op.valued distributions)

I

f (u ⊕ α) : D(f )→ RClassical observables (functionals)

−→ F := f (u, α)Wick = f (ψ, a)Quantum observables (operators on Fock sp.)

I

(Φh)tt0

Classical evolution (flow on phase sp.)

−→ e−i~ (t−t0)H , H = h(ψ, a)

Quantum evolution (unitary group on Fock sp.)

From now on, the dependence on ~ at the quantum level is implicite andwe replace ~ by ε a small parameter in the quantum theory (ε→ 0).

Classical-Quantum correspondenceI

L2(Rd)⊕ L2(Rd)Classical phase space (infinite dim.)

−→ Γs(L2(Rd)⊕ L2(Rd))Quantum Fock space

I

u(x), u(x) and α(k), α(k)Classical variables (scalar fields)

−→ ψ(x), ψ∗(x) and a(k), a∗(k)Quantum variables (op.valued distributions)

I

f (u ⊕ α) : D(f )→ RClassical observables (functionals)

−→ F := f (u, α)Wick = f (ψ, a)Quantum observables (operators on Fock sp.)

I

(Φh)tt0

Classical evolution (flow on phase sp.)

−→ e−i~ (t−t0)H , H = h(ψ, a)

Quantum evolution (unitary group on Fock sp.)

From now on, the dependence on ~ at the quantum level is implicite andwe replace ~ by ε a small parameter in the quantum theory (ε→ 0).

Classical-Quantum correspondenceI

L2(Rd)⊕ L2(Rd)Classical phase space (infinite dim.)

−→ Γs(L2(Rd)⊕ L2(Rd))Quantum Fock space

I

u(x), u(x) and α(k), α(k)Classical variables (scalar fields)

−→ ψ(x), ψ∗(x) and a(k), a∗(k)Quantum variables (op.valued distributions)

I

f (u ⊕ α) : D(f )→ RClassical observables (functionals)

−→ F := f (u, α)Wick = f (ψ, a)Quantum observables (operators on Fock sp.)

I

(Φh)tt0

Classical evolution (flow on phase sp.)

−→ e−i~ (t−t0)H , H = h(ψ, a)

Quantum evolution (unitary group on Fock sp.)

From now on, the dependence on ~ at the quantum level is implicite andwe replace ~ by ε a small parameter in the quantum theory (ε→ 0).

Classical-Quantum correspondenceI

L2(Rd)⊕ L2(Rd)Classical phase space (infinite dim.)

−→ Γs(L2(Rd)⊕ L2(Rd))Quantum Fock space

I

u(x), u(x) and α(k), α(k)Classical variables (scalar fields)

−→ ψ(x), ψ∗(x) and a(k), a∗(k)Quantum variables (op.valued distributions)

I

f (u ⊕ α) : D(f )→ RClassical observables (functionals)

−→ F := f (u, α)Wick = f (ψ, a)Quantum observables (operators on Fock sp.)

I

(Φh)tt0

Classical evolution (flow on phase sp.)

−→ e−i~ (t−t0)H , H = h(ψ, a)

Quantum evolution (unitary group on Fock sp.)

From now on, the dependence on ~ at the quantum level is implicite andwe replace ~ by ε a small parameter in the quantum theory (ε→ 0).

Classical-Quantum correspondenceI

L2(Rd)⊕ L2(Rd)Classical phase space (infinite dim.)

−→ Γs(L2(Rd)⊕ L2(Rd))Quantum Fock space

I

u(x), u(x) and α(k), α(k)Classical variables (scalar fields)

−→ ψ(x), ψ∗(x) and a(k), a∗(k)Quantum variables (op.valued distributions)

I

f (u ⊕ α) : D(f )→ RClassical observables (functionals)

−→ F := f (u, α)Wick = f (ψ, a)Quantum observables (operators on Fock sp.)

I

(Φh)tt0

Classical evolution (flow on phase sp.)

−→ e−i~ (t−t0)H , H = h(ψ, a)

Quantum evolution (unitary group on Fock sp.)

From now on, the dependence on ~ at the quantum level is implicite andwe replace ~ by ε a small parameter in the quantum theory (ε→ 0).

General scheme

Classical HamiltoniansQuantization−−−−−−−−−−⇀↽−−−−−−−−−−Classical limit

Quantum field Hamiltonians

I This scheme extends to various models: Spin-boson, Nelson,Pauli-Fierz models (QED); self-interacting models P(ϕ)2 (QFT) andMany-body Hamiltonians (MFT).

I The (formal) quantization of the (S-KG[χ]) energy gives a QFTHamiltonian

H = (h(u ⊕ α))Wick,

known as the Nelson Hamiltonian.

I This model has been extensively investigated in the past twodecades (e.g. Frohlich, Spohn, Gerard-Derezinski,Bruneau-Derezinski, Georgescu-Gerard-Møller, Arai-Hiroshima,Pizzo, Bach-Frohlich-Segal, Barbaroux-Faupin-Guillot,Griesemer-Lieb-Loss...).

General scheme

Classical HamiltoniansQuantization−−−−−−−−−−⇀↽−−−−−−−−−−Classical limit

Quantum field Hamiltonians

I This scheme extends to various models: Spin-boson, Nelson,Pauli-Fierz models (QED); self-interacting models P(ϕ)2 (QFT) andMany-body Hamiltonians (MFT).

I The (formal) quantization of the (S-KG[χ]) energy gives a QFTHamiltonian

H = (h(u ⊕ α))Wick,

known as the Nelson Hamiltonian.

I This model has been extensively investigated in the past twodecades (e.g. Frohlich, Spohn, Gerard-Derezinski,Bruneau-Derezinski, Georgescu-Gerard-Møller, Arai-Hiroshima,Pizzo, Bach-Frohlich-Segal, Barbaroux-Faupin-Guillot,Griesemer-Lieb-Loss...).

General scheme

Classical HamiltoniansQuantization−−−−−−−−−−⇀↽−−−−−−−−−−Classical limit

Quantum field Hamiltonians

I This scheme extends to various models: Spin-boson, Nelson,Pauli-Fierz models (QED); self-interacting models P(ϕ)2 (QFT) andMany-body Hamiltonians (MFT).

I The (formal) quantization of the (S-KG[χ]) energy gives a QFTHamiltonian

H = (h(u ⊕ α))Wick,

known as the Nelson Hamiltonian.

I This model has been extensively investigated in the past twodecades (e.g. Frohlich, Spohn, Gerard-Derezinski,Bruneau-Derezinski, Georgescu-Gerard-Møller, Arai-Hiroshima,Pizzo, Bach-Frohlich-Segal, Barbaroux-Faupin-Guillot,Griesemer-Lieb-Loss...).

The Nelson model.

The Hamiltonian

H =

∫Rd

ψ∗(x)(−∆x

2M+ V (x)

)ψ(x)dx +

∫Rd

a∗(k)ω(k)a(k)dk

+1

(2π)d2

∫R2d

ψ∗(x)χ(k)√2ω(k)

(a∗(k)e−ik·x + a(k)e ik·x

)ψ(x)dxdk

(1)

With the CCR’s:

[a(x), a∗(y)] = εδ(x − y) ; [ψ(x), ψ∗(y)] = εδ(x − y) .

I For any ω−1/2χ ∈ L2(Rd), H is a bounded from below self-adjointoperator on Γs(L2(Rd))⊗ Γs(L2(Rd)) ' Γs(L2(Rd)⊕ L2(Rd)).

I If χ = 1, H is ill-defined. A renormalization procedure is necessary (Nelson [1964]).

The Nelson model.

The Hamiltonian

H =

∫Rd

ψ∗(x)(−∆x

2M+ V (x)

)ψ(x)dx +

∫Rd

a∗(k)ω(k)a(k)dk

+1

(2π)d2

∫R2d

ψ∗(x)χ(k)√2ω(k)

(a∗(k)e−ik·x + a(k)e ik·x

)ψ(x)dxdk

(1)

With the CCR’s:

[a(x), a∗(y)] = εδ(x − y) ; [ψ(x), ψ∗(y)] = εδ(x − y) .

I For any ω−1/2χ ∈ L2(Rd), H is a bounded from below self-adjointoperator on Γs(L2(Rd))⊗ Γs(L2(Rd)) ' Γs(L2(Rd)⊕ L2(Rd)).

I If χ = 1, H is ill-defined. A renormalization procedure is necessary (Nelson [1964]).

The Nelson model.

The Hamiltonian

H =

∫Rd

ψ∗(x)(−∆x

2M+ V (x)

)ψ(x)dx +

∫Rd

a∗(k)ω(k)a(k)dk

+1

(2π)d2

∫R2d

ψ∗(x)χ(k)√2ω(k)

(a∗(k)e−ik·x + a(k)e ik·x

)ψ(x)dxdk

(1)

With the CCR’s:

[a(x), a∗(y)] = εδ(x − y) ; [ψ(x), ψ∗(y)] = εδ(x − y) .

I For any ω−1/2χ ∈ L2(Rd), H is a bounded from below self-adjointoperator on Γs(L2(Rd))⊗ Γs(L2(Rd)) ' Γs(L2(Rd)⊕ L2(Rd)).

I If χ = 1, H is ill-defined. A renormalization procedure is necessary (Nelson [1964]).

The Nelson model.

The Hamiltonian

H =

∫Rd

ψ∗(x)(−∆x

2M+ V (x)

)ψ(x)dx +

∫Rd

a∗(k)ω(k)a(k)dk

+1

(2π)d2

∫R2d

ψ∗(x)χ(k)√2ω(k)

(a∗(k)e−ik·x + a(k)e ik·x

)ψ(x)dxdk

(1)

With the CCR’s:

[a(x), a∗(y)] = εδ(x − y) ; [ψ(x), ψ∗(y)] = εδ(x − y) .

I For any ω−1/2χ ∈ L2(Rd), H is a bounded from below self-adjointoperator on Γs(L2(Rd))⊗ Γs(L2(Rd)) ' Γs(L2(Rd)⊕ L2(Rd)).

I If χ = 1, H is ill-defined. A renormalization procedure is necessary (Nelson [1964]).

The classical limit

I The probabilistic interpretation of quantum systems suggests thatquantum states should converge in some sense to a probabilitydistribution on the classical phase space (Wigner measures).

I In finite dimensional phase spaces, Wigner measures have beenextensively studied (e.g. Schleirmann, Colin de Verdiere,Helffer-Martinez-Robert; Tartar; P. Gerard; Lions-Paul...)

I The concept has been extended to infinite dimensional phase spacesin a series of works by Ammari-Nier.

I So convergence of quantum dynamics in the classical limit reducesto the study of

limε→0

Tr[ρε e i t

εH Oε e−itεH]

=

∫L2⊕L2

O(u, α)dµt .

where ρε is a sequence of states with Wigner measure µ0 and Oε isa quantum observable and µt = (Φh)t0 # µ0.

The classical limitI The probabilistic interpretation of quantum systems suggests that

quantum states should converge in some sense to a probabilitydistribution on the classical phase space (Wigner measures).

I In finite dimensional phase spaces, Wigner measures have beenextensively studied (e.g. Schleirmann, Colin de Verdiere,Helffer-Martinez-Robert; Tartar; P. Gerard; Lions-Paul...)

I The concept has been extended to infinite dimensional phase spacesin a series of works by Ammari-Nier.

I So convergence of quantum dynamics in the classical limit reducesto the study of

limε→0

Tr[ρε e i t

εH Oε e−itεH]

=

∫L2⊕L2

O(u, α)dµt .

where ρε is a sequence of states with Wigner measure µ0 and Oε isa quantum observable and µt = (Φh)t0 # µ0.

The classical limitI The probabilistic interpretation of quantum systems suggests that

quantum states should converge in some sense to a probabilitydistribution on the classical phase space (Wigner measures).

I In finite dimensional phase spaces, Wigner measures have beenextensively studied (e.g. Schleirmann, Colin de Verdiere,Helffer-Martinez-Robert; Tartar; P. Gerard; Lions-Paul...)

I The concept has been extended to infinite dimensional phase spacesin a series of works by Ammari-Nier.

I So convergence of quantum dynamics in the classical limit reducesto the study of

limε→0

Tr[ρε e i t

εH Oε e−itεH]

=

∫L2⊕L2

O(u, α)dµt .

where ρε is a sequence of states with Wigner measure µ0 and Oε isa quantum observable and µt = (Φh)t0 # µ0.

The classical limitI The probabilistic interpretation of quantum systems suggests that

quantum states should converge in some sense to a probabilitydistribution on the classical phase space (Wigner measures).

I In finite dimensional phase spaces, Wigner measures have beenextensively studied (e.g. Schleirmann, Colin de Verdiere,Helffer-Martinez-Robert; Tartar; P. Gerard; Lions-Paul...)

I The concept has been extended to infinite dimensional phase spacesin a series of works by Ammari-Nier.

I So convergence of quantum dynamics in the classical limit reducesto the study of

limε→0

Tr[ρε e i t

εH Oε e−itεH]

=

∫L2⊕L2

O(u, α)dµt .

where ρε is a sequence of states with Wigner measure µ0 and Oε isa quantum observable and µt = (Φh)t0 # µ0.

The classical limitI The probabilistic interpretation of quantum systems suggests that

quantum states should converge in some sense to a probabilitydistribution on the classical phase space (Wigner measures).

I In finite dimensional phase spaces, Wigner measures have beenextensively studied (e.g. Schleirmann, Colin de Verdiere,Helffer-Martinez-Robert; Tartar; P. Gerard; Lions-Paul...)

I The concept has been extended to infinite dimensional phase spacesin a series of works by Ammari-Nier.

I So convergence of quantum dynamics in the classical limit reducesto the study of

limε→0

Tr[ρε e i t

εH Oε e−itεH]

=

∫L2⊕L2

O(u, α)dµt .

where ρε is a sequence of states with Wigner measure µ0 and Oε isa quantum observable and µt = (Φh)t0 # µ0.

Wigner measures

I The Wigner measures µ are probability measures onZ := L2(Rd)⊕ L2(Rd), i.e. they satisfy (z ∈ Z , z = z1 ⊕ z2):

µ(Z ) =

∫Z

dµ(z) = 1

I Let Z 3 ξ = ξ1 ⊕ ξ2. Recall that the unitary Weyl operator W (ξ)on Γs(Z ) is:

W (ξ) = ei√2

(ψ(ξ1)∗+ψ(ξ1)) ⊗ ei√2

(ψ(ξ2)∗+ψ(ξ2)).

I A probability µ is a Wigner measure of a family of states(ρε)ε∈(0,ε)

if there exists a subinterval E ⊂ (0, ε) (with 0 ∈ E ) such that forany ξ ∈ Z :

limε→0,ε∈E

Tr[ρεW (ξ)] =

∫Z

e√

2i(<〈ξ1,z1〉2+<〈ξ2,z2〉2)dµ(z)

M (ρε, ε ∈ (0, ε)) denotes the set of all Wigner measures of(ρε)ε∈(0,ε)

.

Wigner measuresI The Wigner measures µ are probability measures on

Z := L2(Rd)⊕ L2(Rd), i.e. they satisfy (z ∈ Z , z = z1 ⊕ z2):

µ(Z ) =

∫Z

dµ(z) = 1

I Let Z 3 ξ = ξ1 ⊕ ξ2. Recall that the unitary Weyl operator W (ξ)on Γs(Z ) is:

W (ξ) = ei√2

(ψ(ξ1)∗+ψ(ξ1)) ⊗ ei√2

(ψ(ξ2)∗+ψ(ξ2)).

I A probability µ is a Wigner measure of a family of states(ρε)ε∈(0,ε)

if there exists a subinterval E ⊂ (0, ε) (with 0 ∈ E ) such that forany ξ ∈ Z :

limε→0,ε∈E

Tr[ρεW (ξ)] =

∫Z

e√

2i(<〈ξ1,z1〉2+<〈ξ2,z2〉2)dµ(z)

M (ρε, ε ∈ (0, ε)) denotes the set of all Wigner measures of(ρε)ε∈(0,ε)

.

Wigner measuresI The Wigner measures µ are probability measures on

Z := L2(Rd)⊕ L2(Rd), i.e. they satisfy (z ∈ Z , z = z1 ⊕ z2):

µ(Z ) =

∫Z

dµ(z) = 1

I Let Z 3 ξ = ξ1 ⊕ ξ2. Recall that the unitary Weyl operator W (ξ)on Γs(Z ) is:

W (ξ) = ei√2

(ψ(ξ1)∗+ψ(ξ1)) ⊗ ei√2

(ψ(ξ2)∗+ψ(ξ2)).

I A probability µ is a Wigner measure of a family of states(ρε)ε∈(0,ε)

if there exists a subinterval E ⊂ (0, ε) (with 0 ∈ E ) such that forany ξ ∈ Z :

limε→0,ε∈E

Tr[ρεW (ξ)] =

∫Z

e√

2i(<〈ξ1,z1〉2+<〈ξ2,z2〉2)dµ(z)

M (ρε, ε ∈ (0, ε)) denotes the set of all Wigner measures of(ρε)ε∈(0,ε)

.

Wigner measuresI The Wigner measures µ are probability measures on

Z := L2(Rd)⊕ L2(Rd), i.e. they satisfy (z ∈ Z , z = z1 ⊕ z2):

µ(Z ) =

∫Z

dµ(z) = 1

I Let Z 3 ξ = ξ1 ⊕ ξ2. Recall that the unitary Weyl operator W (ξ)on Γs(Z ) is:

W (ξ) = ei√2

(ψ(ξ1)∗+ψ(ξ1)) ⊗ ei√2

(ψ(ξ2)∗+ψ(ξ2)).

I A probability µ is a Wigner measure of a family of states(ρε)ε∈(0,ε)

if there exists a subinterval E ⊂ (0, ε) (with 0 ∈ E ) such that forany ξ ∈ Z :

limε→0,ε∈E

Tr[ρεW (ξ)] =

∫Z

e√

2i(<〈ξ1,z1〉2+<〈ξ2,z2〉2)dµ(z)

M (ρε, ε ∈ (0, ε)) denotes the set of all Wigner measures of(ρε)ε∈(0,ε)

.

Wigner measuresI The Wigner measures µ are probability measures on

Z := L2(Rd)⊕ L2(Rd), i.e. they satisfy (z ∈ Z , z = z1 ⊕ z2):

µ(Z ) =

∫Z

dµ(z) = 1

I Let Z 3 ξ = ξ1 ⊕ ξ2. Recall that the unitary Weyl operator W (ξ)on Γs(Z ) is:

W (ξ) = ei√2

(ψ(ξ1)∗+ψ(ξ1)) ⊗ ei√2

(ψ(ξ2)∗+ψ(ξ2)).

I A probability µ is a Wigner measure of a family of states(ρε)ε∈(0,ε)

if there exists a subinterval E ⊂ (0, ε) (with 0 ∈ E ) such that forany ξ ∈ Z :

limε→0,ε∈E

Tr[ρεW (ξ)] =

∫Z

e√

2i(<〈ξ1,z1〉2+<〈ξ2,z2〉2)dµ(z)

M (ρε, ε ∈ (0, ε)) denotes the set of all Wigner measures of(ρε)ε∈(0,ε)

.

I Under suitable assumptions the set of Wigner measuresM (ρε, ε ∈ (0, ε)) is not empty.

I We shall consider, without loss of generality, only families of stateswith a single associated Wigner measure.

I Can we determine the Wigner measures µt0 (t) of(ρε(t − t0)

)ε∈(0,ε)

where

ρε(t − t0) = e−iε (t−t0)Hρεe

iε (t−t0)H

I The answer is affirmative:

µt0 (t) = (Φh)tt0 # µt0

µt0 is the initial measure, associated to(ρε)ε∈(0,ε)

and (Φh)tt0is the

flow of the Schrodinger Klein-Gordon equation (S-KG[χ]).

I Under suitable assumptions the set of Wigner measuresM (ρε, ε ∈ (0, ε)) is not empty.

I We shall consider, without loss of generality, only families of stateswith a single associated Wigner measure.

I Can we determine the Wigner measures µt0 (t) of(ρε(t − t0)

)ε∈(0,ε)

where

ρε(t − t0) = e−iε (t−t0)Hρεe

iε (t−t0)H

I The answer is affirmative:

µt0 (t) = (Φh)tt0 # µt0

µt0 is the initial measure, associated to(ρε)ε∈(0,ε)

and (Φh)tt0is the

flow of the Schrodinger Klein-Gordon equation (S-KG[χ]).

I Under suitable assumptions the set of Wigner measuresM (ρε, ε ∈ (0, ε)) is not empty.

I We shall consider, without loss of generality, only families of stateswith a single associated Wigner measure.

I Can we determine the Wigner measures µt0 (t) of(ρε(t − t0)

)ε∈(0,ε)

where

ρε(t − t0) = e−iε (t−t0)Hρεe

iε (t−t0)H

I The answer is affirmative:

µt0 (t) = (Φh)tt0 # µt0

µt0 is the initial measure, associated to(ρε)ε∈(0,ε)

and (Φh)tt0is the

flow of the Schrodinger Klein-Gordon equation (S-KG[χ]).

I Under suitable assumptions the set of Wigner measuresM (ρε, ε ∈ (0, ε)) is not empty.

I We shall consider, without loss of generality, only families of stateswith a single associated Wigner measure.

I Can we determine the Wigner measures µt0 (t) of(ρε(t − t0)

)ε∈(0,ε)

where

ρε(t − t0) = e−iε (t−t0)Hρεe

iε (t−t0)H

I The answer is affirmative:

µt0 (t) = (Φh)tt0 # µt0

µt0 is the initial measure, associated to(ρε)ε∈(0,ε)

and (Φh)tt0is the

flow of the Schrodinger Klein-Gordon equation (S-KG[χ]).

I Under suitable assumptions the set of Wigner measuresM (ρε, ε ∈ (0, ε)) is not empty.

I We shall consider, without loss of generality, only families of stateswith a single associated Wigner measure.

I Can we determine the Wigner measures µt0 (t) of(ρε(t − t0)

)ε∈(0,ε)

where

ρε(t − t0) = e−iε (t−t0)Hρεe

iε (t−t0)H

I The answer is affirmative:

µt0 (t) = (Φh)tt0 # µt0

µt0 is the initial measure, associated to(ρε)ε∈(0,ε)

and (Φh)tt0is the

flow of the Schrodinger Klein-Gordon equation (S-KG[χ]).

Propagation result

Let d ≥ 2, V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) and m > 0. Then the

Nelson Hamiltonian (1) is self-adjoint on H = Γs(L2(Rd))⊗ Γs(L2(Rd))and the Schrodinger Klein-Gordon equation (S-KG[χ]) admits a welldefined continuous flow on Z = L2(Rd)⊕ L2(Rd).

Theorem (Am.-Falconi’14)

Let (%ε)ε∈(0,ε) be a family of normal states on the Hilbert space Hverifying the assumption:

∃δ > 0,∃C > 0,∀ε ∈ (0, ε) Tr[%εNδ] < C .

Then for any t ∈ R

M(e−itεH%εe

i tεH , ε ∈ (0, ε)) = {(Φh)t0 # µ0, µ0 ∈M(%ε, ε ∈ (0, ε))} ,

where (Φh)t0 denotes the classical flow of the coupled Klein-GordonSchrodinger equation well defined on the phase spaceZ = L2(Rd)⊕ L2(Rd).

I Remark: The theorem holds also for massless Nelson model.

Propagation result

Let d ≥ 2, V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) and m > 0. Then the

Nelson Hamiltonian (1) is self-adjoint on H = Γs(L2(Rd))⊗ Γs(L2(Rd))and the Schrodinger Klein-Gordon equation (S-KG[χ]) admits a welldefined continuous flow on Z = L2(Rd)⊕ L2(Rd).

Theorem (Am.-Falconi’14)

Let (%ε)ε∈(0,ε) be a family of normal states on the Hilbert space Hverifying the assumption:

∃δ > 0,∃C > 0,∀ε ∈ (0, ε) Tr[%εNδ] < C .

Then for any t ∈ R

M(e−itεH%εe

i tεH , ε ∈ (0, ε)) = {(Φh)t0 # µ0, µ0 ∈M(%ε, ε ∈ (0, ε))} ,

where (Φh)t0 denotes the classical flow of the coupled Klein-GordonSchrodinger equation well defined on the phase spaceZ = L2(Rd)⊕ L2(Rd).

I Remark: The theorem holds also for massless Nelson model.

Propagation result

Let d ≥ 2, V ∈ L2loc(Rd ,R+), ϕ ∈ H−1/2(Rd) and m > 0. Then the

Nelson Hamiltonian (1) is self-adjoint on H = Γs(L2(Rd))⊗ Γs(L2(Rd))and the Schrodinger Klein-Gordon equation (S-KG[χ]) admits a welldefined continuous flow on Z = L2(Rd)⊕ L2(Rd).

Theorem (Am.-Falconi’14)

Let (%ε)ε∈(0,ε) be a family of normal states on the Hilbert space Hverifying the assumption:

∃δ > 0,∃C > 0,∀ε ∈ (0, ε) Tr[%εNδ] < C .

Then for any t ∈ R

M(e−itεH%εe

i tεH , ε ∈ (0, ε)) = {(Φh)t0 # µ0, µ0 ∈M(%ε, ε ∈ (0, ε))} ,

where (Φh)t0 denotes the classical flow of the coupled Klein-GordonSchrodinger equation well defined on the phase spaceZ = L2(Rd)⊕ L2(Rd).

I Remark: The theorem holds also for massless Nelson model.

Ground state energy limit

Consider again the the Nelson model H with a (smooth) ultravioletcutoff.

Theorem (Am.-Falconi’14)

Assume that d ≥ 2, m > 0 and V is a confining potential, i.e.:lim|x|→∞ V (x) = +∞. Then the ground state energy of the restrictedNelson Hamiltonian has the following limit, for any λ > 0,

limε→0,nε=λ2

inf σ(H|L2s (Rdn)⊗Γs (L2(Rd ))) = inf

||u||L2(Rd )

=λh(u ⊕ α) , (2)

where the infimum on the right hand side is taken over allu ∈ D(

√−∆ + V ) and α ∈ D(ω1/2) with the constraint ||u||L2(Rd ) = λ.

Further developments

1- Removal of the ultraviolet cutoff and the effect of renormalizationon the classical limit (Work in progress Am.-Falconi).

2- Dispersive properties, long time asymptotics and scattering theory of(S-KG[χ]) and the Nelson model.

3- Ground state energy limit in the translation invariant case.

4- Bose-Hubbard model and superfluid to Mott insulator phasetransition.

Further developments

1- Removal of the ultraviolet cutoff and the effect of renormalizationon the classical limit (Work in progress Am.-Falconi).

2- Dispersive properties, long time asymptotics and scattering theory of(S-KG[χ]) and the Nelson model.

3- Ground state energy limit in the translation invariant case.

4- Bose-Hubbard model and superfluid to Mott insulator phasetransition.

Further developments

1- Removal of the ultraviolet cutoff and the effect of renormalizationon the classical limit (Work in progress Am.-Falconi).

2- Dispersive properties, long time asymptotics and scattering theory of(S-KG[χ]) and the Nelson model.

3- Ground state energy limit in the translation invariant case.

4- Bose-Hubbard model and superfluid to Mott insulator phasetransition.

Further developments

1- Removal of the ultraviolet cutoff and the effect of renormalizationon the classical limit (Work in progress Am.-Falconi).

2- Dispersive properties, long time asymptotics and scattering theory of(S-KG[χ]) and the Nelson model.

3- Ground state energy limit in the translation invariant case.

4- Bose-Hubbard model and superfluid to Mott insulator phasetransition.