classical and mean field limit of field-particle...

Marco FalconiCentre Henri Lebesgue et IRMAR;

Universite de Rennes1marco. falconi@ univ-rennes1. fr

Classical and mean field limit offield-particle systems

Roscoff, February 5th 2014

Marco Falconi (CHL, Univ. Rennes1) Classical and mean field limit of field-particle systems Roscoff, February 5th 2014 1 / 36

Outline

1 Overview

2 The Nelson model

3 Future developments

Outline

1 Overview

2 The Nelson model

Outline

1 Overview

2 The Nelson model

Overview

Overview The physical setting

Classical limit

A quantum system should behave as its classical counterpart when quantumeffects become negligible. That can be thought as the limit → 0. The easiestexample is, on L2(R):

H = −2∆/2m + V (x) ,

that should reduce classically to Newton equationsdξ

dt(t) =

mπ(t)

dt(t) = −∇V (ξ(t))

Classical limit

A quantum system should behave as its classical counterpart when quantumeffects become negligible.

That can be thought as the limit → 0. The easiestexample is, on L2(R):

H = −2∆/2m + V (x) ,

dt(t) =

mπ(t)

dt(t) = −∇V (ξ(t))

Classical limit

A quantum system should behave as its classical counterpart when quantumeffects become negligible. That can be thought as the limit → 0.

The easiestexample is, on L2(R):

H = −2∆/2m + V (x) ,

dt(t) =

mπ(t)

dt(t) = −∇V (ξ(t))

Classical limit

H = −2∆/2m + V (x) ,

dt(t) =

mπ(t)

dt(t) = −∇V (ξ(t))

Classical limit

H = −2∆/2m + V (x) ,

dt(t) =

mπ(t)

dt(t) = −∇V (ξ(t))

Mean field limit

A system consisting of a large number n of identical particles is very difficultto describe microscopically, due to the large number of system’s degrees offreedom. However it is expected that, when n is sufficiently large, eachparticle moves as it is subjected to the same external potential (the meanfield), generated by the totality of the particles.

Formally, we are looking to reduce a very big phase-space (the n particleone), to a single-particle phase space; in the limit n→∞.

Mean field limit

A system consisting of a large number n of identical particles is very difficultto describe microscopically, due to the large number of system’s degrees offreedom.

However it is expected that, when n is sufficiently large, eachparticle moves as it is subjected to the same external potential (the meanfield), generated by the totality of the particles.

Mean field limit

Mean field theory for many bosons

Consider the system of n non-relativistic bosons described by the followingHamiltonian of L2(Rnd ):

H =n∑

−∆xj +1

n∑i<j

V (xi − xj ) .

We expect that when n is very large the dynamics of each particle should bedictated by the mean field Hartree equation:

i∂tϕt = −∆ϕt + (V ∗ |ϕt |2)ϕt .

H =n∑

−∆xj +1

n∑i<j

V (xi − xj ) .

i∂tϕt = −∆ϕt + (V ∗ |ϕt |2)ϕt .

H =n∑

−∆xj +1

n∑i<j

V (xi − xj ) .

i∂tϕt = −∆ϕt + (V ∗ |ϕt |2)ϕt .

Particles interacting with fields

Nelson model. [Fs(L2(R3))⊗Fs(L2(R3))] Consider a system ofnon-relativistic bosons interacting with a relativistic scalar boson field,described by the following Hamiltonian:

∫(∇ψ)∗(x)∇ψ(x)dx+

∫ω(k)a∗(k)a(k)dk+λ

∫ϕ(x)ψ∗(x)ψ(x)dx

with ω(k) =√

k2 + µ2, M > 0, µ ≥ 0, coupling constant λ > 0 and

ϕ(x) =

∫χ(k)√ω

(a(k)e ikx + a∗(k)e−ikx

Nelson model. [Fs(L2(R3))⊗Fs(L2(R3))]

Consider a system ofnon-relativistic bosons interacting with a relativistic scalar boson field,described by the following Hamiltonian:

∫(∇ψ)∗(x)∇ψ(x)dx+

with ω(k) =√

ϕ(x) =

∫χ(k)√ω

∫(∇ψ)∗(x)∇ψ(x)dx+

with ω(k) =√

ϕ(x) =

∫χ(k)√ω

∫(∇ψ)∗(x)∇ψ(x)dx+

with ω(k) =√

ϕ(x) =

∫χ(k)√ω

In the mean field limit, we expect to obtain the following dynamics:

i∂t +1

2M∆)

u = (2π)−3/2(χ ∗ A)u

(∂2t −∆ + µ2)A = −(2π)−3/2χ ∗ |u|2

(The precise meaning of the mean field limit in this system will be explainedin detail.)

In the mean field limit, we expect to obtain the following dynamics:(

i∂t +1

2M∆)

u = (2π)−3/2(χ ∗ A)u

(∂2t −∆ + µ2)A = −(2π)−3/2χ ∗ |u|2

In the mean field limit, we expect to obtain the following dynamics:(

i∂t +1

2M∆)

u = (2π)−3/2(χ ∗ A)u

(∂2t −∆ + µ2)A = −(2π)−3/2χ ∗ |u|2

Particle QED.[L2(R3)⊗Fs(L2(R3)⊗ C2)]

A rigid charge interacting withits own electromagnetic field can be described by:

(p − eA(q)

+ ∑λ=1,2

∫ω(k)a∗(k, λ)a(k, λ)dk ,

where p = −i√∇, q =

√x , ω(k) = c |k | and

A(x) =∑λ=1,2

∫ √

ω(k)eλ(k)χ(k)

(a(k, λ)e ik·x + a∗(k , λ)e−ik·x)dk .

Particle QED.[L2(R3)⊗Fs(L2(R3)⊗ C2)] A rigid charge interacting withits own electromagnetic field can be described by:

(p − eA(q)

+ ∑λ=1,2

∫ω(k)a∗(k , λ)a(k, λ)dk ,

√x , ω(k) = c |k | and

A(x) =∑λ=1,2

∫ √

ω(k)eλ(k)χ(k)

(a(k, λ)e ik·x + a∗(k , λ)e−ik·x)dk .

Particle QED.[L2(R3)⊗Fs(L2(R3)⊗ C2)] A rigid charge interacting withits own electromagnetic field can be described by:

(p − eA(q)

+ ∑λ=1,2

∫ω(k)a∗(k , λ)a(k, λ)dk ,

√x , ω(k) = c |k | and

A(x) =∑λ=1,2

∫ √

ω(k)eλ(k)χ(k)

(a(k , λ)e ik·x + a∗(k , λ)e−ik·x)dk .

Classically (→ 0), the same system is described by the Abraham model:

∂tB +∇× E = 0

∂tE −∇× B = −j

∇·E = ρ

∇·B = 0ξ = v

v = e[(ϕ ∗ E )(ξ) + v × (ϕ ∗ B)(ξ)]

j = evϕ(ξ − x) , ρ = eϕ(ξ − x) .

Classically (→ 0), the same system is described by the Abraham model:∂tB +∇× E = 0

∂tE −∇× B = −j

∇·E = ρ

∇·B = 0ξ = v

v = e[(ϕ ∗ E )(ξ) + v × (ϕ ∗ B)(ξ)]

j = evϕ(ξ − x) , ρ = eϕ(ξ − x) .

Classically (→ 0), the same system is described by the Abraham model:∂tB +∇× E = 0

∂tE −∇× B = −j

∇·E = ρ

∇·B = 0ξ = v

v = e[(ϕ ∗ E )(ξ) + v × (ϕ ∗ B)(ξ)]

j = evϕ(ξ − x) , ρ = eϕ(ξ − x) .

Overview Mathematical results

Mean field theory for many particles

BBGKY method. Hierarchy (in the number n of particles) of equations canbe written for the reduced density matrices; they follow from the Schrodingerequation of the system. This hierarchy is proved to converge in the limitn→∞, and the unique limit solution is written explicitly. This approach hasbeen introduced by Spohn [1980], and then developed by many authors [e.g.Bardos et al., 2000; Erdos and Yau, 2001; Erdos et al., 2010, . . . ]

PROs: Applicability in a large number of systems (Hartree, Gross-Pitaevskii,Hartree-Fock, . . . ).

CONs: Very specific initial states has to be considered (factorized states).No information on rate of convergence (apart for small times, in somesystems); nor on fluctuations around the mean field solution.

BBGKY method.

Hierarchy (in the number n of particles) of equations canbe written for the reduced density matrices; they follow from the Schrodingerequation of the system. This hierarchy is proved to converge in the limitn→∞, and the unique limit solution is written explicitly. This approach hasbeen introduced by Spohn [1980], and then developed by many authors [e.g.Bardos et al., 2000; Erdos and Yau, 2001; Erdos et al., 2010, . . . ]

BBGKY method. Hierarchy (in the number n of particles) of equations canbe written for the reduced density matrices; they follow from the Schrodingerequation of the system.

This hierarchy is proved to converge in the limitn→∞, and the unique limit solution is written explicitly. This approach hasbeen introduced by Spohn [1980], and then developed by many authors [e.g.Bardos et al., 2000; Erdos and Yau, 2001; Erdos et al., 2010, . . . ]

BBGKY method. Hierarchy (in the number n of particles) of equations canbe written for the reduced density matrices; they follow from the Schrodingerequation of the system. This hierarchy is proved to converge in the limitn→∞, and the unique limit solution is written explicitly.

This approach hasbeen introduced by Spohn [1980], and then developed by many authors [e.g.Bardos et al., 2000; Erdos and Yau, 2001; Erdos et al., 2010, . . . ]

CONs: Very specific initial states has to be considered (factorized states).

No information on rate of convergence (apart for small times, in somesystems); nor on fluctuations around the mean field solution.

Hepp method.

Hepp [1974] proved that the dynamics of quantum variables

−i√∇ and

√x , averaged on suitable -dependent coherent states, reduces

to Newton dynamics in the limit → 0. This result was extended by Ginibreand Velo [1979, 1980] to systems of non-relativistic bosons with infinitedegrees of freedom. Recently, these ideas has been applied to the mean fieldlimit, in order to obtain a rate of convergence for reduced density matrices[see Rodnianski and Schlein, 2009; Chen and Lee, 2011; Chen et al., 2011;Benedikter et al., 2013, . . . ] in various systems of bosons (and alsofermions).

PROs: Informations on the evolution of fluctuations (between coherent states).Bound on the rate of convergence of reduced density matrices.

CONs: Specific initial states has to be considered, namely factorized and coherentones (also partially factorized and linear combinations of the above [F., 1305]).Even if, due to symmetries, the natural setting of the system is L2(Rnd ), thewhole Fock space F (L2(Rd )) has to be considered to prove convergence(since the method relies on the Weyl operators of the whole Fock space).

Hepp method. Hepp [1974] proved that the dynamics of quantum variables

−i√∇ and

to Newton dynamics in the limit → 0.

This result was extended by Ginibreand Velo [1979, 1980] to systems of non-relativistic bosons with infinitedegrees of freedom. Recently, these ideas has been applied to the mean fieldlimit, in order to obtain a rate of convergence for reduced density matrices[see Rodnianski and Schlein, 2009; Chen and Lee, 2011; Chen et al., 2011;Benedikter et al., 2013, . . . ] in various systems of bosons (and alsofermions).

−i√∇ and

to Newton dynamics in the limit → 0. This result was extended by Ginibreand Velo [1979, 1980] to systems of non-relativistic bosons with infinitedegrees of freedom.

Recently, these ideas has been applied to the mean fieldlimit, in order to obtain a rate of convergence for reduced density matrices[see Rodnianski and Schlein, 2009; Chen and Lee, 2011; Chen et al., 2011;Benedikter et al., 2013, . . . ] in various systems of bosons (and alsofermions).

−i√∇ and

PROs: Informations on the evolution of fluctuations (between coherent states).

Bound on the rate of convergence of reduced density matrices.CONs: Specific initial states has to be considered, namely factorized and coherent

ones (also partially factorized and linear combinations of the above [F., 1305]).Even if, due to symmetries, the natural setting of the system is L2(Rnd ), thewhole Fock space F (L2(Rd )) has to be considered to prove convergence(since the method relies on the Weyl operators of the whole Fock space).

−i√∇ and

CONs: Specific initial states has to be considered, namely factorized and coherentones (also partially factorized and linear combinations of the above [F., 1305]).

Even if, due to symmetries, the natural setting of the system is L2(Rnd ), thewhole Fock space F (L2(Rd )) has to be considered to prove convergence(since the method relies on the Weyl operators of the whole Fock space).

−i√∇ and

Wigner measures approach.

Inspired by semiclassical analysis results onfinite dimensions, Ammari and Nier [2008, 2009, 2011a,b] studied infinitedimensional Wigner measures and their propagation in the mean field limit.The time evolved density matrices of general states are proved to converge,in the mean field limit, to the push forward by the classical flow (e.g. Hartreeflow) of a probability measure. This result implies the convergence of reduceddensity matrices as well.

PROs: On one hand the infinite dimensional setting makes the approach very suitablefor quantum field theories; on the other hand its relation with finitedimensional analysis makes it suitable also for systems with a finitedimensional phase space.The mean field limit is proved for a very general class of states.

CONs: No information on the fluctuations, nor on the rate of convergence.

Wigner measures approach. Inspired by semiclassical analysis results onfinite dimensions, Ammari and Nier [2008, 2009, 2011a,b] studied infinitedimensional Wigner measures and their propagation in the mean field limit.

The time evolved density matrices of general states are proved to converge,in the mean field limit, to the push forward by the classical flow (e.g. Hartreeflow) of a probability measure. This result implies the convergence of reduceddensity matrices as well.

Wigner measures approach. Inspired by semiclassical analysis results onfinite dimensions, Ammari and Nier [2008, 2009, 2011a,b] studied infinitedimensional Wigner measures and their propagation in the mean field limit.The time evolved density matrices of general states are proved to converge,in the mean field limit, to the push forward by the classical flow (e.g. Hartreeflow) of a probability measure.

This result implies the convergence of reduceddensity matrices as well.

Wigner measures approach. Inspired by semiclassical analysis results onfinite dimensions, Ammari and Nier [2008, 2009, 2011a,b] studied infinitedimensional Wigner measures and their propagation in the mean field limit.The time evolved density matrices of general states are proved to converge,in the mean field limit, to the push forward by the classical flow (e.g. Hartreeflow) of a probability measure. This result implies the convergence of reduceddensity matrices as well.

PROs: On one hand the infinite dimensional setting makes the approach very suitablefor quantum field theories

; on the other hand its relation with finitedimensional analysis makes it suitable also for systems with a finitedimensional phase space.The mean field limit is proved for a very general class of states.

PROs: On one hand the infinite dimensional setting makes the approach very suitablefor quantum field theories; on the other hand its relation with finitedimensional analysis makes it suitable also for systems with a finitedimensional phase space.

The mean field limit is proved for a very general class of states.CONs: No information on the fluctuations, nor on the rate of convergence.

Other approaches.

Recently, other methods have been developed thatdeserve further mention. The first has been developed by Pickl [2011], and itis based on counting the particles that are not condensed at some time t;provided we started with an initial state completely factorized, he proves thatthis number goes to zero when n→∞, hence obtaining convergence for thereduced density matrix.

Another approach is due to Lewin et al. [2013]. They are able to describe theevolution of fluctuations around Hartree states, instead of coherent ones, inthe limit n→∞. Their result implies as well the convergence of reduceddensity matrices and gives a bound on the rate of convergence.Using the construction of a truncated Fock space (and of a map thatsubstitutes the Weyl operators) [see Lewin et al., 2012] , they restrict theanalysis to a space isomorphic to L2(Rnd ), instead of considering the wholeFock space as in the Hepp method.

Also, I mention a work of Frohlich et al. [2007], where they provideasymptotics for observables in the mean field and classical limit.

Other approaches. Recently, other methods have been developed thatdeserve further mention. The first has been developed by Pickl [2011], and itis based on counting the particles that are not condensed at some time t;provided we started with an initial state completely factorized, he proves thatthis number goes to zero when n→∞, hence obtaining convergence for thereduced density matrix.

Another approach is due to Lewin et al. [2013]. They are able to describe theevolution of fluctuations around Hartree states, instead of coherent ones, inthe limit n→∞.

Their result implies as well the convergence of reduceddensity matrices and gives a bound on the rate of convergence.Using the construction of a truncated Fock space (and of a map thatsubstitutes the Weyl operators) [see Lewin et al., 2012] , they restrict theanalysis to a space isomorphic to L2(Rnd ), instead of considering the wholeFock space as in the Hepp method.

Another approach is due to Lewin et al. [2013]. They are able to describe theevolution of fluctuations around Hartree states, instead of coherent ones, inthe limit n→∞. Their result implies as well the convergence of reduceddensity matrices and gives a bound on the rate of convergence.

Using the construction of a truncated Fock space (and of a map thatsubstitutes the Weyl operators) [see Lewin et al., 2012] , they restrict theanalysis to a space isomorphic to L2(Rnd ), instead of considering the wholeFock space as in the Hepp method.

Mean field limit of the Nelson model. Ginibre et al. [2006] studied thepartial limit of the Nelson model without cut off, using the Hepp method. In[F., 1301], I analyzed the complete mean field limit of the Nelson model, butwith cut off. I will describe the results in detail in the following section.

Classical limit of Particle QED. No result yet!

Mean field limit of the Nelson model.

Ginibre et al. [2006] studied thepartial limit of the Nelson model without cut off, using the Hepp method. In[F., 1301], I analyzed the complete mean field limit of the Nelson model, butwith cut off. I will describe the results in detail in the following section.

Mean field limit of the Nelson model. Ginibre et al. [2006] studied thepartial limit of the Nelson model without cut off, using the Hepp method.

In[F., 1301], I analyzed the complete mean field limit of the Nelson model, butwith cut off. I will describe the results in detail in the following section.

Classical limit of Particle QED.

No result yet!

The Nelson model

The mean field limit as λ→ 0

Recall the Nelson Hamiltonian:

∫(∇ψ)∗(x)∇ψ(x)dx+

∫ϕ(x)ψ∗(x)ψ(x)dx .

Consider a state φn1,n2 such that 〈φn1,n2 , (N1 + N2)φn1,n2〉 ∼ n1 + n2; we wouldlike to describe its dynamics in the limit n1, n2 →∞ as a mean field theory, withthe particles coupled as described above.

In order to do that n1 and n2 has to be related, and it turns out that they havealso to be related to the coupling constant λ, as n1 ∼ n2 ∼ λ−2. So the meanfield limit is also a weak coupling limit λ→ 0.

The Nelson model

∫(∇ψ)∗(x)∇ψ(x)dx+

The Nelson model

∫(∇ψ)∗(x)∇ψ(x)dx+

The Nelson model Hepp method

Quantum dynamics

Proposition

H is essentially self-adjoint on Fs(L2(R3))⊗Fs(L2(R3)).

Consider the subspace Hn1 = L2(R3n1 )⊗Fs(L2(R3)) of the whole Fock spacewith fixed number n1 of non-relativistic particles, H

∣∣n1

the restriction of H to that

subspace.Then ∀ε < 1, ∃C (ε, n1) such that ∀φ ∈ D(H0

∣∣n1

∣∣n1φ‖2 ≤ ε2‖H0

∣∣n1φ‖2 + C (ε, n1)‖φ‖2 .

Quantum dynamics

Proposition

∣∣n1

subspace.

Then ∀ε < 1, ∃C (ε, n1) such that ∀φ ∈ D(H0

∣∣n1

∣∣n1φ‖2 ≤ ε2‖H0

∣∣n1φ‖2 + C (ε, n1)‖φ‖2 .

Quantum dynamics

Proposition

∣∣n1

subspace.Then ∀ε < 1, ∃C (ε, n1) such that ∀φ ∈ D(H0

∣∣n1

∣∣n1φ‖2 ≤ ε2‖H0

∣∣n1φ‖2 + C (ε, n1)‖φ‖2 .

→ H∣∣n1

is self-adjoint with domain D(H0

∣∣n1

), ∀n1 ∈ N.

→ Since ε does not depend on n1, we can define H as the direct sum:

H =∞⊕

H∣∣n1.

→ H∣∣n1

is self-adjoint with domain D(H0

∣∣n1

), ∀n1 ∈ N.

→ Since ε does not depend on n1, we can define H as the direct sum:

H =∞⊕

H∣∣n1.

Classical dynamics

Recall the classical equation,(

i∂t +1

2M∆)

u = (2π)−3/2(χ ∗ A)u

(∂2t −∆ + µ2)A = −(2π)−3/2χ ∗ |u|2

A can be written as

A(x) =

∫1√ω

(α(k)e ikx + α(k)e−ikx

Classical dynamics

Recall the classical equation,(

i∂t +1

2M∆)

u = (2π)−3/2(χ ∗ A)u

(∂2t −∆ + µ2)A = −(2π)−3/2χ ∗ |u|2

A can be written as

A(x) =

∫1√ω

(α(k)e ikx + α(k)e−ikx

The system of equations for u and α then becomes: i∂tu = − 1

2M∆u + (2π)−3/2(χ ∗ A)u

i∂tα = ωα + (ω)−1/2χ(|u|2) .

Proposition

Let (u0, α0) in L2(R3)⊕ L2(R3). Then the equation above admits an uniqueglobal solution (u(t), α(t)) ∈ C 0(R, L2(R3)⊕ L2(R3)).

The system of equations for u and α then becomes: i∂tu = − 1

2M∆u + (2π)−3/2(χ ∗ A)u

i∂tα = ωα + (ω)−1/2χ(|u|2) .

Proposition

Let (u0, α0) in L2(R3)⊕ L2(R3). Then the equation above admits an uniqueglobal solution (u(t), α(t)) ∈ C 0(R, L2(R3)⊕ L2(R3)).

Evolution of fluctuations

Let u, α ∈ L2(R3). Define the Weyl operator:

C (u, α) = expψ∗(u)− ψ(u) ⊗ expa∗(α)− a(α)

(ψ(u) =∫

u(x)ψ(x)dx , ψ∗(u) = (ψ(u))∗, analogous for a#(α)).

Let u, α ∈ L2(R3).

Define the Weyl operator:

(ψ(u) =∫

Then we can define the quantum evolution between coherent states as:

W (t, s) = C∗(u(t)/λ, α(t)/λ) exp−i(t − s)HC (u(s)/λ, α(s)/λ)e iΛ(t,s) ;

where (u(t), α(t)) is the solution of the classical equation with initial state(u(s), α(s)) and

Λ(t, s) = −1

2(2π)−3/2λ−2

(χ ∗ A)(τ)u(τ)u(τ)dxdτ .

where (u(t), α(t)) is the solution of the classical equation with initial state(u(s), α(s))

Λ(t, s) = −1

2(2π)−3/2λ−2

where (u(t), α(t)) is the solution of the classical equation with initial state(u(s), α(s)) and

Λ(t, s) = −1

2(2π)−3/2λ−2

More difficult to define is its putative limit when λ→ 0: we call it U2(t, s).

U2(t, s) is formally generated by the time-dependent Hamiltonian:

H2(t) =1

∫(∇ψ)∗(x)∇ψ(x)dx +

∫ωa∗(k)a(k)dk +

[∫ (1

2(2π)−3/2

(χ ∗ A(t))ψ∗ψ + u(t)ϕψ∗)

dx + h.c.

U2(t, s) can be rigorously defined by means of a truncated Dyson series in theinteraction representation.

More difficult to define is its putative limit when λ→ 0: we call it U2(t, s).U2(t, s) is formally generated by the time-dependent Hamiltonian:

H2(t) =1

∫(∇ψ)∗(x)∇ψ(x)dx +

[∫ (1

2(2π)−3/2

(χ ∗ A(t))ψ∗ψ + u(t)ϕψ∗)

dx + h.c.

More difficult to define is its putative limit when λ→ 0: we call it U2(t, s).U2(t, s) is formally generated by the time-dependent Hamiltonian:

H2(t) =1

∫(∇ψ)∗(x)∇ψ(x)dx +

[∫ (1

2(2π)−3/2

(χ ∗ A(t))ψ∗ψ + u(t)ϕψ∗)

dx + h.c.

Theorem 1 ([F., 1301])

Let φ ∈H . Then

limλ→0

W (t, s)φ = U2(t, s)φ

in the strong topology of H ; uniformly in t and s on compact intervals.

U2(t, s) describes the evolution of quantum fluctuations operators ψ# − u(s)/λand a# − α(s)/λ in the limit λ→ 0.

Theorem 1 ([F., 1301])

Let φ ∈H . Then

limλ→0

W (t, s)φ = U2(t, s)φ

in the strong topology of H ; uniformly in t and s on compact intervals.

U2(t, s) describes the evolution of quantum fluctuations operators ψ# − u(s)/λand a# − α(s)/λ in the limit λ→ 0.

Mean field limit of quantum variables

The quantum variables of the system are the annihilation and creationoperators for the two species of particles: ψ#(x) and a#(k).

We expect that, in some sense, in the limit λ→ 0: λψ#t (x) ∼ u#(t, x),

λa#t (k) ∼ α#(t, k).

(the multiplicative factor λ is necessary, otherwise ψ# and a# would divergein the limit as

√n1 ∼ λ−1 and

√n2 ∼ λ−1 respectively)

We may consider the mean field limit for initial states that are coherent

(C (u0/λ, α0/λ)Ω) or factorized (u⊗n10 ⊗ α⊗n2

0 ) or factorized in the first

species and coherent in the second one (u⊗n10 ⊗ C (α0/λ)Ω2).

Each one of these states has the property that〈ψ, (N1 + N2)ψ〉 ∼ n1 + n2 ∼ λ−2.

λa#t (k) ∼ α#(t, k).

√n1 ∼ λ−1 and

λa#t (k) ∼ α#(t, k).

√n1 ∼ λ−1 and

λa#t (k) ∼ α#(t, k).

√n1 ∼ λ−1 and

(C (u0/λ, α0/λ)Ω)

or factorized (u⊗n10 ⊗ α⊗n2

λa#t (k) ∼ α#(t, k).

√n1 ∼ λ−1 and

or factorized in the first

λa#t (k) ∼ α#(t, k).

√n1 ∼ λ−1 and

λa#t (k) ∼ α#(t, k).

√n1 ∼ λ−1 and

Let ψ#t (x) = expitHψ#(x) exp−itH, a#

t (k) = expitHa#(k) exp−itH.

Also, let (u(t), α(t)) be the classical solution corresponding to initial datum(u0, α0) with ‖u0‖2 = ‖α0‖2 = 1.

Theorem 2

As functions of L2(R3), we have the following convergence:

limλ→0〈C (u0/λ, α0/λ)Ω, λψ#

t (x)C (u0/λ, α0/λ)Ω〉 = u#(t, x) ;

limλ→0〈C (u0/λ, α0/λ)Ω, λa#

t (k)C (u0/λ, α0/λ)Ω〉 = α#(t, k) .

t (k) = expitHa#(k) exp−itH.Also, let (u(t), α(t)) be the classical solution corresponding to initial datum(u0, α0) with ‖u0‖2 = ‖α0‖2 = 1.

Theorem 2

t (x)C (u0/λ, α0/λ)Ω〉 = u#(t, x) ;

t (k)C (u0/λ, α0/λ)Ω〉 = α#(t, k) .

t (k) = expitHa#(k) exp−itH.Also, let (u(t), α(t)) be the classical solution corresponding to initial datum(u0, α0) with ‖u0‖2 = ‖α0‖2 = 1.

Theorem 2

t (x)C (u0/λ, α0/λ)Ω〉 = u#(t, x) ;

t (k)C (u0/λ, α0/λ)Ω〉 = α#(t, k) .

The convergence result holds for arbitrary products of normal orderedannihilation and creation operators

, converging to products of the classicalsolutions.

The rate of convergence is of order λ2 (obtained using quantum fluctuations).

We obtain the same convergence and rate if we replace C (u0/λ, α0/λ)Ω with

u⊗n10 ⊗ C (α0/λ)Ω2; in this case the limit is different from zero only if we

consider a product with the same number of ψ∗ and ψ.

Up to a normalization factor, the result above (for products with the samenumber of creation and annihilation operators of each type) is equivalent tothe convergence of reduced density matrices in the Hilbert-Schmidt norm.

The convergence result holds for arbitrary products of normal orderedannihilation and creation operators, converging to products of the classicalsolutions.

The rate of convergence is of order λ2

(obtained using quantum fluctuations).

u⊗n10 ⊗ C (α0/λ)Ω2

; in this case the limit is different from zero only if weconsider a product with the same number of ψ∗ and ψ.

The result for factorized vectors u⊗n10 ⊗ α⊗n2

0 deserves special attention.

Let(uθ(t), αθ(t)) be the classical solution corresponding to initial datum(u0, exp−iθα0).

Theorem 3

As functions of L2(R6) and L2(R3) respectively, we have the followingconvergence:

limλ→0〈u⊗n1(λ)

0 ⊗α⊗n2(λ)

0 , λ2ψ∗t (x)ψt(y)u⊗n1(λ)

0 ⊗α⊗n2(λ)

0 〉 =

∫ 2π

uθ(t, x)uθ(t, y)dθ

0 ⊗ α⊗n2(λ)

0 , λa#t (k)u

⊗n1(λ)

0 ⊗ α⊗n2(λ)

0 〉 =

∫ 2π

α#θ (t, k)

0 deserves special attention. Let(uθ(t), αθ(t)) be the classical solution corresponding to initial datum(u0, exp−iθα0).

Theorem 3

0 ⊗α⊗n2(λ)

0 〉 =

∫ 2π

0 ⊗ α⊗n2(λ)

0 , λa#t (k)u

⊗n1(λ)

0 ⊗ α⊗n2(λ)

0 〉 =

∫ 2π

α#θ (t, k)

0 deserves special attention. Let(uθ(t), αθ(t)) be the classical solution corresponding to initial datum(u0, exp−iθα0).

Theorem 3

0 ⊗α⊗n2(λ)

0 〉 =

∫ 2π

0 ⊗ α⊗n2(λ)

0 , λa#t (k)u

⊗n1(λ)

0 ⊗ α⊗n2(λ)

0 〉 =

∫ 2π

α#θ (t, k)

The convergence result holds for arbitrary products of normal orderedannihilation and creation operators.

The rate of convergence is of order λ2.

The quantum evolution does not preserve the number n2 of relativisticparticles; this affects the classical limit, for initial states that have a fixednumber of relativistic particles, in an unexpected fashion.

The quantum evolution does not preserve the number n2 of relativisticparticles

; this affects the classical limit, for initial states that have a fixednumber of relativistic particles, in an unexpected fashion.

The Nelson model Wigner measures

Mean field limit for a general class of states

In a joint work with Ammari (in preparation) we utilize the Wigner measuresapproach on the Nelson model. Let p1, p2 ∈ N, we call γp1,p2

λ the reduced densitymatrix corresponding to a state ρλ (p1 and p2 stands for the number of particlesof the non-relativistic and relativistic type respectively).

We say that a family of density matrices (ρλ)λ∈(0,λ) on a Fock space Fs(Z )converges to a measure µ on Z if a probability measure on Z exists such that forall ξ ∈ Z :

limλ→0

Tr[ρλ exp

i(a∗(ξ) + a(ξ)

e i√

2Re〈ξ,ζ〉Z dµ(ζ) .

In a joint work with Ammari (in preparation) we utilize the Wigner measuresapproach on the Nelson model.

Let p1, p2 ∈ N, we call γp1,p2

limλ→0

Tr[ρλ exp

i(a∗(ξ) + a(ξ)

e i√

limλ→0

Tr[ρλ exp

i(a∗(ξ) + a(ξ)

e i√

We say that a family of density matrices (ρλ)λ∈(0,λ) on a Fock space Fs(Z )converges to a measure µ on Z

if a probability measure on Z exists such that forall ξ ∈ Z :

limλ→0

Tr[ρλ exp

i(a∗(ξ) + a(ξ)

e i√

limλ→0

Tr[ρλ exp

i(a∗(ξ) + a(ξ)

e i√

Proposition

Let (ρλ)λ∈(0,λ) be a family of normal states on Fs(L2(R3))⊗Fs(L2(R3))(satisfying some regularity properties) that converges to a probability measure µ0

of Z := L2(R3)⊕ L2(R3) when λ→ 0.

Also, let Φ(t, s) be the classical flux, i.e.(u(t), α(t)) = Φ(t, s)(u(s), α(s)), and µt = Φ(t, 0)∗µ0. Then in theL 1(L2

s (R3p1 )⊗ L2s (R3p1 ))-norm, for all p1, p2 ∈ N:

limλ→0

γp1,p2

λ (t) =1∫

Z |z1|2p1 |z2|2p2 dµt(z)

|z⊗p11 ⊗ z

⊗p22 〉〈z⊗p1

1 ⊗ z⊗p22 |dµt(z) .

Proposition

of Z := L2(R3)⊕ L2(R3) when λ→ 0. Also, let Φ(t, s) be the classical flux, i.e.(u(t), α(t)) = Φ(t, s)(u(s), α(s)), and µt = Φ(t, 0)∗µ0.

Then in theL 1(L2

limλ→0

γp1,p2

λ (t) =1∫

Z |z1|2p1 |z2|2p2 dµt(z)

|z⊗p11 ⊗ z

⊗p22 〉〈z⊗p1

1 ⊗ z⊗p22 |dµt(z) .

Proposition

of Z := L2(R3)⊕ L2(R3) when λ→ 0. Also, let Φ(t, s) be the classical flux, i.e.(u(t), α(t)) = Φ(t, s)(u(s), α(s)), and µt = Φ(t, 0)∗µ0. Then in theL 1(L2

limλ→0

γp1,p2

λ (t) =1∫

Z |z1|2p1 |z2|2p2 dµt(z)

|z⊗p11 ⊗ z

⊗p22 〉〈z⊗p1

1 ⊗ z⊗p22 |dµt(z) .

Proposition

of Z := L2(R3)⊕ L2(R3) when λ→ 0. Also, let Φ(t, s) be the classical flux, i.e.(u(t), α(t)) = Φ(t, s)(u(s), α(s)), and µt = Φ(t, 0)∗µ0. Then in theL 1(L2

limλ→0

γp1,p2

λ (t) =1∫

Z |z1|2p1 |z2|2p2 dµt(z)

|z⊗p11 ⊗ z

⊗p22 〉〈z⊗p1

1 ⊗ z⊗p22 |dµt(z) .

Future developments

Classical limit of particle QED.

Mean field limit of the Nelson model without cut off.

Scattering in the mean field limit.

Future developments

References

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Thank you.

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