Schrödinger-Klein-Gordon system as a classical limit of a scalar Quantum Field Theory.

Transcription

1 Schrödinger-Klein-Gordon system as a classical limit of a scalar Quantum Field Theory. (in collaboration with Marco Falconi) Zied Ammari Rennes University, IRMAR BECAM, 28 October 2014

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

3 The Yukawa Theory The Yukawa theory describes the nucleon-nucleon (NN) interaction. The strong nuclear force bounds nucleons together via the attractive Yukawa potential V (x) = e m x x.

4 The Yukawa Theory The Yukawa theory describes the nucleon-nucleon (NN) interaction. The strong nuclear force bounds nucleons together via the attractive Yukawa potential V (x) = e m x x. The interaction is mediated by the π-mesons (massive bosons) and it is described by a quantum field theory (QFT).

5 General point of view Quantum systems with a large number of particles obeying the Bose statistics have quite often an interesting collective behavior at certain scales. For instance, this is experimentally observed in condensate matter physics (Bose-Einstein condensate) and in quantum optics (coherent light).

6 General point of view Quantum systems with a large number of particles obeying the Bose statistics have quite often an interesting collective behavior at certain scales. For instance, this is experimentally observed in condensate matter physics (Bose-Einstein condensate) and in quantum optics (coherent light). The semiclassical analysis in finite dimension have turned the Bohr correspondence principle into a deep and powerful mathematical theory (h-pseudodifferential calculus, tunnelling effect, eigenvalues asymptotics, resonances, Weyl s law, propagation of singularities, quantum ergodicity and chaos...).

7 General point of view Quantum systems with a large number of particles obeying the Bose statistics have quite often an interesting collective behavior at certain scales. For instance, this is experimentally observed in condensate matter physics (Bose-Einstein condensate) and in quantum optics (coherent light). The semiclassical analysis in finite dimension have turned the Bohr correspondence principle into a deep and powerful mathematical theory (h-pseudodifferential calculus, tunnelling effect, eigenvalues asymptotics, resonances, Weyl s law, propagation of singularities, quantum ergodicity and chaos...). There exists an old attempt to extend microlocal analysis to infinite dimensional phase spaces (This was the main subject of Paul Krée seminar in the 70 s, Bernard Lascar,...). But there are several difficulties!

8 Aims Overcome the difficulties encountered in the 70 s and extend the semiclassical analysis to infinite dimensional phase spaces.

9 Aims Overcome the difficulties encountered in the 70 s and extend the semiclassical analysis to infinite dimensional phase spaces. Study of quantum systems with a large number of particles (bosons) using a general approach. In particular, addressing the questions of: Mean field theory Classical limit in the following frameworks:

10 Aims Overcome the difficulties encountered in the 70 s and extend the semiclassical analysis to infinite dimensional phase spaces. Study of quantum systems with a large number of particles (bosons) using a general approach. In particular, addressing the questions of: Mean field theory Classical limit in the following frameworks: Many-Body theory (N-body Schrödinger operator) Relativistic Quantum field theory ((ϕ) 4 2, P(ϕ) 2 models) Quantum electrodynamics (spin-boson, Nelson, Pauli-Fierz models)

11 Aims Overcome the difficulties encountered in the 70 s and extend the semiclassical analysis to infinite dimensional phase spaces. Study of quantum systems with a large number of particles (bosons) using a general approach. In particular, addressing the questions of: Mean field theory Classical limit in the following frameworks: Many-Body theory (N-body Schrödinger operator) Relativistic Quantum field theory ((ϕ) 4 2, P(ϕ) 2 models) Quantum electrodynamics (spin-boson, Nelson, Pauli-Fierz models) Results: semiclassical propagation theorems, eigenvalues asymptotics and more generally spectral and scattering properties of quantum dynamical systems (with infinite degrees of freedom) in the semiclassical regime.

12 Outline The Schrödinger Klein-Gordon (S-KG) system

13 Outline The Schrödinger Klein-Gordon (S-KG) system Mathematical foundation of QFT

14 Outline The Schrödinger Klein-Gordon (S-KG) system Mathematical foundation of QFT The classical limit ε 0

15 Outline The Schrödinger Klein-Gordon (S-KG) system Mathematical foundation of QFT The classical limit ε 0 Ground state energy limit

16 The S-KG system in dimension d

17 The S-KG system in dimension d i t u = u + Vu + (ϕ A)u 2M (S-KG) ( + m 2 )A = ϕ u 2

18 The S-KG system in dimension d i t u = u + Vu + (ϕ A)u 2M (S-KG) ( + m 2 )A = ϕ u 2 V is an external potential for the non-relativistic particle.

19 The S-KG system in dimension d i t u = u + Vu + (ϕ A)u 2M (S-KG) ( + m 2 )A = ϕ u 2 V is an external potential for the non-relativistic particle. ϕ can be a (regularizing) function, or Dirac s delta distribution.

20 The S-KG system in dimension d i t u = u + Vu + (ϕ A)u 2M (S-KG) ( + m 2 )A = ϕ u 2 V is an external potential for the non-relativistic particle. ϕ can be a (regularizing) function, or Dirac s delta distribution. M > 0; m 0 (For simplicity, we suppose in this talk m > 0).

21 The S-KG system in dimension d i t u = u + Vu + (ϕ A)u 2M (S-KG) ( + m 2 )A = ϕ u 2 V is an external potential for the non-relativistic particle. ϕ can be a (regularizing) function, or Dirac s delta distribution. M > 0; m 0 (For simplicity, we suppose in this talk m > 0). It is a wave-particle system describing in some sense the nucleon-meson interaction.

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

23 The Yukawa interaction In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1 { i t u = u + Au ( + 1)A = u 2 (S-KG[δ])

24 The Yukawa interaction In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1 { i t u = u + Au ( + 1)A = u 2 (S-KG[δ]) u(t 0 ) = u 0, A(t 0 ) = A 0, t A(t 0 ) = A 1.

25 The Yukawa interaction In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1 { i t u = u + Au ( + 1)A = u 2 (S-KG[δ]) u(t 0 ) = u 0, A(t 0 ) = A 0, t A(t 0 ) = A 1. In the literature, global well-posedness of the above system has been extensively investigated (e.g. Fukuda and Tsutsumi [1975]; Baillon and Chadam [1978]; Bachelot [1984]; Ginibre and Velo [2002]; Colliander, Holmer and Tzirakis [2008]; Pecher [2012]...).

26 The Yukawa interaction In the case d = 3, V = 0, ϕ = δ, M = 1/2, and m = 1 { i t u = u + Au ( + 1)A = u 2 (S-KG[δ]) u(t 0 ) = u 0, A(t 0 ) = A 0, t A(t 0 ) = A 1. In the literature, global well-posedness of the above system has been extensively investigated (e.g. Fukuda and Tsutsumi [1975]; Baillon and Chadam [1978]; Bachelot [1984]; Ginibre and Velo [2002]; Colliander, Holmer and Tzirakis [2008]; Pecher [2012]...). Theorem (Pecher (2012)) Let 0 s σ s + 1 and u 0 H s (R 3 ), A 0 H σ (R 3 ), A 1 H σ 1 (R 3 ). Then (S-KG[δ]) is globally well-posed; i.e. there exists a unique solution u C 0 (R, H s (R 3 )), A C 0 (R, H σ (R 3 )) C 1 (R, H σ 1 (R 3 )).

27 The regularized system

28 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 case ϕ H 1/2 (R d ), m > 0, we have:

29 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 case ϕ H 1/2 (R d ), m > 0, we have: Proposition (d 2, (Falconi 2013)) Let V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) (m > 0); and u 0 L 2 (R d ), A 0 H 1 2 (R d ), A 1 H 1 2 (R d ). Then (S-KG) is globally well-posed; i.e. there exists a unique solution u C 0 (R, L 2 (R d )), A C 0 (R, H 1 2 (R d ).

30 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 case ϕ H 1/2 (R d ), m > 0, we have: Proposition (d 2, (Falconi 2013)) Let V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) (m > 0); and u 0 L 2 (R d ), A 0 H 1 2 (R d ), A 1 H 1 2 (R d ). Then (S-KG) is globally well-posed; i.e. there exists a unique solution u C 0 (R, L 2 (R d )), A C 0 (R, H 1 2 (R d ). In this regular setting the global well-posedness follows easily from standard contraction estimates and the conservation of mass u 2.

31 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 case ϕ H 1/2 (R d ), m > 0, we have: Proposition (d 2, (Falconi 2013)) Let V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) (m > 0); and u 0 L 2 (R d ), A 0 H 1 2 (R d ), A 1 H 1 2 (R d ). Then (S-KG) is globally well-posed; i.e. there exists a unique solution u C 0 (R, L 2 (R d )), A C 0 (R, H 1 2 (R d ). In this regular setting the global well-posedness follows easily from standard contraction estimates and the conservation of mass u 2. We have a well defined continuous global flow on the phase-space.

32 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 case ϕ H 1/2 (R d ), m > 0, we have: Proposition (d 2, (Falconi 2013)) Let V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) (m > 0); and u 0 L 2 (R d ), A 0 H 1 2 (R d ), A 1 H 1 2 (R d ). Then (S-KG) is globally well-posed; i.e. there exists a unique solution u C 0 (R, L 2 (R d )), A C 0 (R, H 1 2 (R d ). In this regular setting the global well-posedness follows easily from standard contraction estimates and the conservation of mass u 2. We have a well defined continuous global flow on the phase-space. From now on, we work in this setting.

33 The complex fields QFT uses often complex fields rather than real fields. It is therefore convient to switch to this representation even at the classical level. So, we replace (A, t A) by the complex fields (α, ᾱ) defined by:

34 The complex fields QFT uses often complex fields rather than real fields. It is therefore convient to switch to this representation even at the classical level. So, we replace (A, t A) by the complex fields (α, ᾱ) defined by: A(x) = 1 (2π) d 2 2ω(k) (ᾱ(k)e ik x +α(k)e ik x) dk, ω(k) = k 2 + m 2. R d 1

35 The complex fields QFT uses often complex fields rather than real fields. It is therefore convient to switch to this representation even at the classical level. So, we replace (A, t A) by the complex fields (α, ᾱ) defined by: A(x) = 1 (2π) d 2 2ω(k) (ᾱ(k)e ik x +α(k)e ik x) dk, ω(k) = k 2 + m 2. R d 1 Then, with χ = (2π) d 2 ˆϕ, (S-KG) is equivalent to the following system with the unknown (u, α): i t u = u + Vu + (ϕ A)u 2M i t α = ωα + χ (S-KG[χ]) (ūu) 2ω

36 The complex fields QFT uses often complex fields rather than real fields. It is therefore convient to switch to this representation even at the classical level. So, we replace (A, t A) by the complex fields (α, ᾱ) defined by: A(x) = 1 (2π) d 2 2ω(k) (ᾱ(k)e ik x +α(k)e ik x) dk, ω(k) = k 2 + m 2. R d 1 Then, with χ = (2π) d 2 ˆϕ, (S-KG) is equivalent to the following system with the unknown (u, α): i t u = u + Vu + (ϕ A)u 2M i t α = ωα + χ (S-KG[χ]) (ūu) 2ω Corollary (d 2) Let V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) (m > 0) and u 0 L 2 (R d ), α 0 L 2 (R d ). Then (S-KG[χ]) is globally well-posed; i.e. there exists a unique solution u C 0 (R, L 2 (R d )), α C 0 (R, L 2 (R d )).

37 The classical Hamiltonian

38 The classical Hamiltonian The Schödinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian system.

39 The classical Hamiltonian The Schödinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian system. Its energy functional is densely defined on L 2 (R d ) L 2 (R d ): u(x)dx + ( h(u α) = ū(x) R d + 1 (2π) d 2 x 2M + V (x) ) R d ᾱ(k)ω(k)α(k)dk χ(k) ū(x) (ᾱ(k)e ik x + α(k)e ik x) u(x)dxdk R 2d 2ω(k)

40 The classical Hamiltonian The Schödinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian system. Its energy functional is densely defined on L 2 (R d ) L 2 (R d ): u(x)dx + ( h(u α) = ū(x) R d + 1 (2π) d 2 x 2M + V (x) ) R d ᾱ(k)ω(k)α(k)dk χ(k) ū(x) (ᾱ(k)e ik x + α(k)e ik x) u(x)dxdk R 2d 2ω(k) So that, the equation (S-KG[χ]) takes the more compact form: ( ) δh u δū i t = α δh δᾱ (S-KG[χ])

41 The classical Hamiltonian The Schödinger-Klein-Gordon equation S-KG[χ] is a Hamiltonian system. Its energy functional is densely defined on L 2 (R d ) L 2 (R d ): u(x)dx + ( h(u α) = ū(x) R d + 1 (2π) d 2 x 2M + V (x) ) R d ᾱ(k)ω(k)α(k)dk χ(k) ū(x) (ᾱ(k)e ik x + α(k)e ik x) u(x)dxdk R 2d 2ω(k) So that, the equation (S-KG[χ]) takes the more compact form: ( ) δh u δū i t = α δh δᾱ (S-KG[χ]) Let (Φ h ) t t 0 denotes the well defined flow of (S-KG[χ]) on the phase-space L 2 (R d ) L 2 (R d ).

42 Constructive QFT The goal of constructive QFT is to provide a solution for nonlinear quantum field equations of the type ( + m)φ(t, x) + λφ 2n+1 (t, x) = 0, where φ(t; x) and π(t; x) = t φ(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) and π(t; x) 1 H = 2 [π2 (t, x) + φ(t, x) 2 + mφ(t, x) 2 ] + λ 2n + 2 φ(t, x)2n+2 dx. So that, the time variation of quantum fields is given by the equation of motion i t φ(t; x) = [φ(t; x); H] and i t π(t; x) = [π(t; x); H].

43 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 L 2 (R d, R) are self-adjoint operators on a given Hilbert space H, satisfying the Heisenberg 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: Satisfying: W (g, f ) = e i 2 f,g e iπ(g) e iφ(f ), (i) W (g 1, f 1 )W (g 2, f 2 ) = e i 2 σ[(g1,f1),(g2,f2)] W (g 1 + g 2, f 1 + f 2 ). (ii) W (g, f ) = W ( g, f ). Here σ is a canonical symplectic form.

44 Fock representation It is the most important representation of the Weyl commutation relations and it is build on the symmetric Fock space. Recall that the symmetric Fock space over Z = L 2 (R d, C) is Γ s (Z ) = n=0 n s Z = n=0l 2 s (R nd, C).

45 Fock representation It is the most important representation of the Weyl commutation relations and it is build on the symmetric Fock space. Recall that the symmetric Fock space over Z = L 2 (R d, C) is Γ s (Z ) = n=0 n s Z = n=0l 2 s (R nd, C). Annihilation : a (f )f 1 s s f n = n 1 f, f σ1 f σ2 f σn, n! σ S n

46 Fock representation It is the most important representation of the Weyl commutation relations and it is build on the symmetric Fock space. Recall that the symmetric Fock space over Z = L 2 (R d, C) is Γ s (Z ) = n=0 n s Z = n=0l 2 s (R nd, C). Annihilation : a (f )f 1 s s f n = n 1 f, f σ1 f σ2 f σn, n! σ S n Creation : a (f )f 1 s s f n = (n + 1) f s f 1 s f n. [a (f ), a (g)] = f, g Id.

47 Fock representation It is the most important representation of the Weyl commutation relations and it is build on the symmetric Fock space. Recall that the symmetric Fock space over Z = L 2 (R d, C) is Γ s (Z ) = n=0 n s Z = n=0l 2 s (R nd, C). Annihilation : a (f )f 1 s s f n = n 1 f, f σ1 f σ2 f σn, n! σ S n Creation : a (f )f 1 s s f n = (n + 1) f s f 1 s f n. [a (f ), a (g)] = f, g Id. Weyl operator : W (f ) = e i 2 (a (f )+a (f )). This determines φ(t, x) and π(t, x) in the case λ = 0.

48 Fock representation It is the most important representation of the Weyl commutation relations and it is build on the symmetric Fock space. Recall that the symmetric Fock space over Z = L 2 (R d, C) is Γ s (Z ) = n=0 n s Z = n=0l 2 s (R nd, C). Annihilation : a (f )f 1 s s f n = n 1 f, f σ1 f σ2 f σn, n! σ S n Creation : a (f )f 1 s s f n = (n + 1) f s f 1 s f n. [a (f ), a (g)] = f, g Id. Weyl operator : W (f ) = e i 2 (a (f )+a (f )). This determines φ(t, x) and π(t, x) in the case λ = 0.

49 Wick Quantization It is a mapping associating to a polynomial functional (classical Hamiltonian) h(ᾱ, α) an operator in the Fock space h(ᾱ, α) h(a, a) = H with all a in the left and all a in the right.

50 Wick Quantization It is a mapping associating to a polynomial functional (classical Hamiltonian) h(ᾱ, α) an operator in the Fock space h(ᾱ, α) h(a, a) = H with all a in the left and all a in the right. The Wick operator with polynomial symbol is defined by Wick (n p + q)!n! b n s Z = p+q 2 b s 1 n p (n p)! where b : p s Z q s Z is a given operator and the symbol is b(ᾱ, α) = α q, bα p.

51 Wick Quantization It is a mapping associating to a polynomial functional (classical Hamiltonian) h(ᾱ, α) an operator in the Fock space h(ᾱ, α) h(a, a) = H with all a in the left and all a in the right. The Wick operator with polynomial symbol is defined by Wick (n p + q)!n! b n s Z = p+q 2 b s 1 n p (n p)! where b : p s Z q s Z is a given operator and the symbol is b(ᾱ, α) = α q, bα p. This procedure provides a canonical quantization of classical systems.

52 Wick Quantization It is a mapping associating to a polynomial functional (classical Hamiltonian) h(ᾱ, α) an operator in the Fock space h(ᾱ, α) h(a, a) = H with all a in the left and all a in the right. The Wick operator with polynomial symbol is defined by Wick (n p + q)!n! b n s Z = p+q 2 b s 1 n p (n p)! where b : p s Z q s Z is a given operator and the symbol is b(ᾱ, α) = α q, bα p. This procedure provides a canonical quantization of classical systems. Example: The Klein-Gordon free energy ᾱ(k)ω(k)α(k)dk α, ω(k)α Wick = a (k)ω(k) a (k)dk. R d R d

53 Classical-Quantum correspondence

54 Classical-Quantum correspondence L 2 (R d ) L 2 (R d ) Classical phase space (infinite dim.) Γ s (L 2 (R d ) L 2 (R d )) Quantum Fock space

55 Classical-Quantum correspondence L 2 (R d ) L 2 (R d ) Classical phase space (infinite dim.) Γ s (L 2 (R d ) L 2 (R d )) Quantum Fock space u(x), ū(x) and α(k), ᾱ(k) Classical variables (scalar fields) ψ(x), ψ (x) and a(k), a (k) Quantum variables (op.valued distributions)

56 Classical-Quantum correspondence L 2 (R d ) L 2 (R d ) Classical phase space (infinite dim.) Γ s (L 2 (R d ) L 2 (R d )) Quantum Fock space u(x), ū(x) and α(k), ᾱ(k) Classical variables (scalar fields) ψ(x), ψ (x) and a(k), a (k) Quantum variables (op.valued distributions) f (u α) : D(f ) R Classical observables (functionals) F := f (u, α) Wick = f (ψ, a) Quantum observables (operators on Fock sp.)

57 Classical-Quantum correspondence L 2 (R d ) L 2 (R d ) Classical phase space (infinite dim.) Γ s (L 2 (R d ) L 2 (R d )) Quantum Fock space u(x), ū(x) and α(k), ᾱ(k) Classical variables (scalar fields) ψ(x), ψ (x) and a(k), a (k) Quantum variables (op.valued distributions) f (u α) : D(f ) R Classical observables (functionals) F := f (u, α) Wick = f (ψ, a) Quantum observables (operators on Fock sp.) (Φ h ) t t 0 Classical evolution (flow on phase sp.) e i (t t0)h, H = h(ψ, a) Quantum evolution (unitary group on Fock sp.)

58 Classical-Quantum correspondence L 2 (R d ) L 2 (R d ) Classical phase space (infinite dim.) Γ s (L 2 (R d ) L 2 (R d )) Quantum Fock space u(x), ū(x) and α(k), ᾱ(k) Classical variables (scalar fields) ψ(x), ψ (x) and a(k), a (k) Quantum variables (op.valued distributions) f (u α) : D(f ) R Classical observables (functionals) F := f (u, α) Wick = f (ψ, a) Quantum observables (operators on Fock sp.) (Φ h ) t t 0 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 and we replace by ε a small parameter in the quantum theory (ε 0).

59 General scheme Classical Hamiltonians Quantization Quantum field Hamiltonians Classical limit This scheme extends to various models: Spin-boson, Nelson, Pauli-Fierz models (QED); self-interacting models P(ϕ) 2 (QFT) and Many-body Hamiltonians (MFT).

60 General scheme Classical Hamiltonians Quantization Quantum field Hamiltonians Classical limit This scheme extends to various models: Spin-boson, Nelson, Pauli-Fierz models (QED); self-interacting models P(ϕ) 2 (QFT) and Many-body Hamiltonians (MFT). The (formal) quantization of the (S-KG[χ]) energy gives a QFT Hamiltonian H = (h(u α)) Wick, known as the Nelson Hamiltonian.

61 General scheme Classical Hamiltonians Quantization Quantum field Hamiltonians Classical limit This scheme extends to various models: Spin-boson, Nelson, Pauli-Fierz models (QED); self-interacting models P(ϕ) 2 (QFT) and Many-body Hamiltonians (MFT). The (formal) quantization of the (S-KG[χ]) energy gives a QFT Hamiltonian H = (h(u α)) Wick, known as the Nelson Hamiltonian. This model has been extensively investigated in the past two decades (e.g. Fröhlich, Spohn, Gérard-Derezinski, Bruneau-Derezinski, Georgescu-Gérard-Møller, Arai-Hiroshima, Pizzo, Bach-Fröhlich-Segal, Barbaroux-Faupin-Guillot, Griesemer-Lieb-Loss...).

62 The Nelson model. The Hamiltonian ( H = ψ (x) R d + 1 (2π) d 2 x 2M + V (x) ) ψ(x)dx + a (k)ω(k)a(k)dk R d ψ χ(k) (x) (a (k)e ik x + a(k)e ik x) ψ(x)dxdk R 2d 2ω(k) (1)

63 The Nelson model. The Hamiltonian ( H = ψ (x) R d + 1 (2π) d 2 x 2M + V (x) ) ψ(x)dx + a (k)ω(k)a(k)dk R d ψ χ(k) (x) (a (k)e ik x + a(k)e ik x) ψ(x)dxdk R 2d 2ω(k) (1) With the CCR s: [a(x), a (y)] = εδ(x y) ; [ψ(x), ψ (y)] = εδ(x y).

64 The Nelson model. The Hamiltonian ( H = ψ (x) R d + 1 (2π) d 2 x 2M + V (x) ) ψ(x)dx + a (k)ω(k)a(k)dk R d ψ χ(k) (x) (a (k)e ik x + a(k)e ik x) ψ(x)dxdk R 2d 2ω(k) (1) With the CCR s: [a(x), a (y)] = εδ(x y) ; [ψ(x), ψ (y)] = εδ(x y). For any ω 1/2 χ L 2 (R d ), H is a bounded from below self-adjoint operator on Γ s (L 2 (R d )) Γ s (L 2 (R d )) Γ s (L 2 (R d ) L 2 (R d )).

65 The Nelson model. The Hamiltonian ( H = ψ (x) R d + 1 (2π) d 2 x 2M + V (x) ) ψ(x)dx + a (k)ω(k)a(k)dk R d ψ χ(k) (x) (a (k)e ik x + a(k)e ik x) ψ(x)dxdk R 2d 2ω(k) (1) With the CCR s: [a(x), a (y)] = εδ(x y) ; [ψ(x), ψ (y)] = εδ(x y). For any ω 1/2 χ L 2 (R d ), H is a bounded from below self-adjoint operator on Γ s (L 2 (R d )) Γ s (L 2 (R d )) Γ s (L 2 (R d ) L 2 (R d )). If χ = 1, H is ill-defined. A renormalization procedure is necessary ( Nelson [1964]).

66 The classical limit

67 The classical limit The probabilistic interpretation of quantum systems suggests that quantum states should converge in some sense to a probability distribution on the classical phase space (Wigner measures).

68 The classical limit The probabilistic interpretation of quantum systems suggests that quantum states should converge in some sense to a probability distribution on the classical phase space (Wigner measures). In finite dimensional phase spaces, Wigner measures have been extensively studied (e.g. Schleirmann, Colin de Verdière, Helffer-Martinez-Robert; Tartar; P. Gérard; Lions-Paul...)

69 The classical limit The probabilistic interpretation of quantum systems suggests that quantum states should converge in some sense to a probability distribution on the classical phase space (Wigner measures). In finite dimensional phase spaces, Wigner measures have been extensively studied (e.g. Schleirmann, Colin de Verdière, Helffer-Martinez-Robert; Tartar; P. Gérard; Lions-Paul...) The concept has been extended to infinite dimensional phase spaces in a series of works by Ammari-Nier.

70 The classical limit The probabilistic interpretation of quantum systems suggests that quantum states should converge in some sense to a probability distribution on the classical phase space (Wigner measures). In finite dimensional phase spaces, Wigner measures have been extensively studied (e.g. Schleirmann, Colin de Verdière, Helffer-Martinez-Robert; Tartar; P. Gérard; Lions-Paul...) The concept has been extended to infinite dimensional phase spaces in a series of works by Ammari-Nier. So convergence of quantum dynamics in the classical limit reduces to the study of [ lim Tr ρ ε e i t ε H O ε e i t H] ε = O(u, α)dµ t. ε 0 L 2 L 2 where ρ ε is a sequence of states with Wigner measure µ 0 and O ε is a quantum observable and µ t = (Φ h ) t 0 # µ 0.

71 Wigner measures

72 Wigner measures The Wigner measures µ are probability measures on Z := L 2 (R d ) L 2 (R d ), i.e. they satisfy (z Z, z = z 1 z 2 ): µ(z ) = dµ(z) = 1 Z

73 Wigner measures The Wigner measures µ are probability measures on Z := L 2 (R d ) L 2 (R d ), i.e. they satisfy (z Z, z = z 1 z 2 ): µ(z ) = dµ(z) = 1 Let Z ξ = ξ 1 ξ 2. Recall that the unitary Weyl operator W (ξ) on Γ s (Z ) is: Z W (ξ) = e i 2 (ψ(ξ 1) +ψ(ξ 1)) e i 2 (ψ(ξ 2) +ψ(ξ 2)).

74 Wigner measures The Wigner measures µ are probability measures on Z := L 2 (R d ) L 2 (R d ), i.e. they satisfy (z Z, z = z 1 z 2 ): µ(z ) = dµ(z) = 1 Let Z ξ = ξ 1 ξ 2. Recall that the unitary Weyl operator W (ξ) on Γ s (Z ) is: Z W (ξ) = e i 2 (ψ(ξ 1) +ψ(ξ 1)) e i 2 (ψ(ξ 2) +ψ(ξ 2)). A probability µ is a Wigner measure of a family of states ( ) ρ ε ε (0, ε) if there exists a subinterval E (0, ε) (with 0 E ) such that for any ξ Z : lim Tr[ρ εw (ξ)] = e 2i(R ξ 1,z 1 2+R ξ 2,z 2 2) dµ(z) ε 0,ε E Z

75 Wigner measures The Wigner measures µ are probability measures on Z := L 2 (R d ) L 2 (R d ), i.e. they satisfy (z Z, z = z 1 z 2 ): µ(z ) = dµ(z) = 1 Let Z ξ = ξ 1 ξ 2. Recall that the unitary Weyl operator W (ξ) on Γ s (Z ) is: Z W (ξ) = e i 2 (ψ(ξ 1) +ψ(ξ 1)) e i 2 (ψ(ξ 2) +ψ(ξ 2)). A probability µ is a Wigner measure of a family of states ( ) ρ ε ε (0, ε) if there exists a subinterval E (0, ε) (with 0 E ) such that for any ξ Z : lim Tr[ρ εw (ξ)] = e 2i(R ξ 1,z 1 2+R ξ 2,z 2 2) dµ(z) ε 0,ε E M (ρ ε, ε (0, ε)) denotes the set of all Wigner measures of ( ρ ε )ε (0, ε). Z

76 Under suitable assumptions the set of Wigner measures M (ρ ε, ε (0, ε)) is not empty.

77 Under suitable assumptions the set of Wigner measures M (ρ ε, ε (0, ε)) is not empty. We shall consider, without loss of generality, only families of states with a single associated Wigner measure.

78 Under suitable assumptions the set of Wigner measures M (ρ ε, ε (0, ε)) is not empty. We shall consider, without loss of generality, only families of states with a single associated Wigner measure. Can we determine the Wigner measures µ t0 (t) of ( ρ ε (t t 0 ) ) ε (0, ε) where ρ ε (t t 0 ) = e i ε (t t0)h ρ ε e i ε (t t0)h

79 Under suitable assumptions the set of Wigner measures M (ρ ε, ε (0, ε)) is not empty. We shall consider, without loss of generality, only families of states with a single associated Wigner measure. Can we determine the Wigner measures µ t0 (t) of ( ρ ε (t t 0 ) ) ε (0, ε) where ρ ε (t t 0 ) = e i ε (t t0)h ρ ε e i ε (t t0)h The answer is affirmative: µ t0 (t) = (Φ h ) t t 0 # µ t0

80 Under suitable assumptions the set of Wigner measures M (ρ ε, ε (0, ε)) is not empty. We shall consider, without loss of generality, only families of states with a single associated Wigner measure. Can we determine the Wigner measures µ t0 (t) of ( ρ ε (t t 0 ) ) ε (0, ε) where ρ ε (t t 0 ) = e i ε (t t0)h ρ ε e i ε (t t0)h The answer is affirmative: µ t0 (t) = (Φ h ) t t 0 # µ t0 µ t0 is the initial measure, associated to ( ρ ε )ε (0, ε) and (Φ h) t t 0 is the flow of the Schrödinger Klein-Gordon equation (S-KG[χ]).

81 Propagation result Let d 2, V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) and m > 0. Then the Nelson Hamiltonian (1) is self-adjoint on H = Γ s (L 2 (R d )) Γ s (L 2 (R d )) and the Schrödinger Klein-Gordon equation (S-KG[χ]) admits a well defined continuous flow on Z = L 2 (R d ) L 2 (R d ).

82 Propagation result Let d 2, V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) and m > 0. Then the Nelson Hamiltonian (1) is self-adjoint on H = Γ s (L 2 (R d )) Γ s (L 2 (R d )) and the Schrödinger Klein-Gordon equation (S-KG[χ]) admits a well defined continuous flow on Z = L 2 (R d ) L 2 (R d ). Theorem (Am.-Falconi 14) Let (ϱ ε ) ε (0, ε) be a family of normal states on the Hilbert space H verifying the assumption: Then for any t R δ > 0, C > 0, ε (0, ε) Tr[ϱ ε ˆNδ ] < C. M(e i t ε H ϱ ε e i t ε H, ε (0, ε)) = {(Φ h ) t 0 # µ 0, µ 0 M(ϱ ε, ε (0, ε))}, where (Φ h ) t 0 denotes the classical flow of the coupled Klein-Gordon Schrödinger equation well defined on the phase space Z = L 2 (R d ) L 2 (R d ).

83 Propagation result Let d 2, V L 2 loc (Rd, R + ), ϕ H 1/2 (R d ) and m > 0. Then the Nelson Hamiltonian (1) is self-adjoint on H = Γ s (L 2 (R d )) Γ s (L 2 (R d )) and the Schrödinger Klein-Gordon equation (S-KG[χ]) admits a well defined continuous flow on Z = L 2 (R d ) L 2 (R d ). Theorem (Am.-Falconi 14) Let (ϱ ε ) ε (0, ε) be a family of normal states on the Hilbert space H verifying the assumption: Then for any t R δ > 0, C > 0, ε (0, ε) Tr[ϱ ε ˆNδ ] < C. M(e i t ε H ϱ ε e i t ε H, ε (0, ε)) = {(Φ h ) t 0 # µ 0, µ 0 M(ϱ ε, ε (0, ε))}, where (Φ h ) t 0 denotes the classical flow of the coupled Klein-Gordon Schrödinger equation well defined on the phase space Z = L 2 (R d ) L 2 (R d ). Remark: The theorem holds also for massless Nelson model.

84 Ground state energy limit Consider again the the Nelson model H with a (smooth) ultraviolet cutoff. 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 restricted Nelson Hamiltonian has the following limit, for any λ > 0, lim inf σ(h ε 0,nε=λ 2 L 2 s (Rdn ) Γ s (L 2 (R ))) = inf h(u α), (2) d u L 2 (R d ) =λ where the infimum on the right hand side is taken over all u D( + V ) and α D(ω 1/2 ) with the constraint u L 2 (R d ) = λ.

85 Further developments 1- Removal of the ultraviolet cutoff and the effect of renormalization on the classical limit (Work in progress Am.-Falconi).

86 Further developments 1- Removal of the ultraviolet cutoff and the effect of renormalization on 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.

87 Further developments 1- Removal of the ultraviolet cutoff and the effect of renormalization on 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.

88 Further developments 1- Removal of the ultraviolet cutoff and the effect of renormalization on 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 phase transition.