Derivation of the Maxwell-Schrodinger Equationsfrom the Pauli-Fierz Hamiltonian
Nikolai Leopold
Joint work with Peter Pickl(arXiv:1609.01545)
Oberwolfach, September 12, 2017
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Outline of the talk
Motivation
Pauli-Fierz Hamiltonian/Maxwell-Schrodinger equations
Main theorem
Idea of the proof
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Motivation
Question
Is it possible to derive Maxwell’s equations from Quantumelectrodynamics?
Physicists look at Heisenberg equations of the field operators.
More rigorous: find a physical situation which gives Maxwell’sequations in some limit.
|α0〉〈α0|N→∞←−−−− γ
(0,1)N,0 ←−−−− ΨN,0 −−−−→ γ
(1,0)N,0
N→∞−−−−→ |ϕ0〉〈ϕ0|
eff .
y y y y yeff .
|αt〉〈αt | ←−−−−N→∞
γ(0,1)N,t ←−−−− ΨN,t −−−−→ γ
(1,0)N,t −−−−→
N→∞|ϕt〉〈ϕt |
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Motivation
Question
Is it possible to derive Maxwell’s equations from Quantumelectrodynamics?
Physicists look at Heisenberg equations of the field operators.
More rigorous: find a physical situation which gives Maxwell’sequations in some limit.
|α0〉〈α0|N→∞←−−−− γ
(0,1)N,0 ←−−−− ΨN,0 −−−−→ γ
(1,0)N,0
N→∞−−−−→ |ϕ0〉〈ϕ0|
eff .
y y y y yeff .
|αt〉〈αt | ←−−−−N→∞
γ(0,1)N,t ←−−−− ΨN,t −−−−→ γ
(1,0)N,t −−−−→
N→∞|ϕt〉〈ϕt |
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Motivation
Question
Is it possible to derive Maxwell’s equations from Quantumelectrodynamics?
Physicists look at Heisenberg equations of the field operators.
More rigorous: find a physical situation which gives Maxwell’sequations in some limit.
|α0〉〈α0|N→∞←−−−− γ
(0,1)N,0 ←−−−− ΨN,0 −−−−→ γ
(1,0)N,0
N→∞−−−−→ |ϕ0〉〈ϕ0|
eff .
y y y y yeff .
|αt〉〈αt | ←−−−−N→∞
γ(0,1)N,t ←−−−− ΨN,t −−−−→ γ
(1,0)N,t −−−−→
N→∞|ϕt〉〈ϕt |
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Motivation
Question
Is it possible to derive Maxwell’s equations from Quantumelectrodynamics?
Physicists look at Heisenberg equations of the field operators.
More rigorous: find a physical situation which gives Maxwell’sequations in some limit.
|α0〉〈α0|N→∞←−−−− γ
(0,1)N,0 ←−−−− ΨN,0 −−−−→ γ
(1,0)N,0
N→∞−−−−→ |ϕ0〉〈ϕ0|
eff .
y y y y yeff .
|αt〉〈αt | ←−−−−N→∞
γ(0,1)N,t ←−−−− ΨN,t −−−−→ γ
(1,0)N,t −−−−→
N→∞|ϕt〉〈ϕt |
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Motivation
Question
Is it possible to derive Maxwell’s equations from Quantumelectrodynamics?
Physicists look at Heisenberg equations of the field operators.
More rigorous: find a physical situation which gives Maxwell’sequations in some limit.
|α0〉〈α0|N→∞←−−−− γ
(0,1)N,0 ←−−−− ΨN,0 −−−−→ γ
(1,0)N,0
N→∞−−−−→ |ϕ0〉〈ϕ0|
eff .
y y y y yeff .
|αt〉〈αt | ←−−−−N→∞
γ(0,1)N,t ←−−−− ΨN,t −−−−→ γ
(1,0)N,t −−−−→
N→∞|ϕt〉〈ϕt |
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Spinless Pauli-Fierz Hamiltonian
i∂tΨN,t =HNΨN,t
with
HN =N∑
j=1
(−i∇j −
Aκ(xj )√N
)2
+1
N
∑1≤j<k≤N
1
|xj − xk |+ Hf ,
Aκ(x) =∑λ=1,2
∫d3k
κ(k)√2|k |
ελ(k)(e ikxa(k , λ) + e−ikxa∗(k, λ)
).
two types of particles: (non-relativistic) bosons and photons,
photons have two polarizations ε1(k), ε2(k) (∇ · Aκ = 0),
H = L2(R3N)⊗ [⊕n≥0h⊗n
s ] with h = L2(R3)⊗ C2,
scaling: kinetic and potential energy are of the same order.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Spinless Pauli-Fierz Hamiltonian
i∂tΨN,t =HNΨN,t
with
HN =N∑
j=1
(−i∇j −
Aκ(xj )√N
)2
+1
N
∑1≤j<k≤N
1
|xj − xk |+ Hf ,
Aκ(x) =∑λ=1,2
∫d3k
κ(k)√2|k |
ελ(k)(e ikxa(k , λ) + e−ikxa∗(k, λ)
).
two types of particles: (non-relativistic) bosons and photons,
photons have two polarizations ε1(k), ε2(k) (∇ · Aκ = 0),
H = L2(R3N)⊗ [⊕n≥0h⊗n
s ] with h = L2(R3)⊗ C2,
scaling: kinetic and potential energy are of the same order.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Spinless Pauli-Fierz Hamiltonian
i∂tΨN,t =HNΨN,t
with
HN =N∑
j=1
(−i∇j −
Aκ(xj )√N
)2
+1
N
∑1≤j<k≤N
1
|xj − xk |+ Hf ,
Aκ(x) =∑λ=1,2
∫d3k
κ(k)√2|k |
ελ(k)(e ikxa(k , λ) + e−ikxa∗(k, λ)
).
two types of particles: (non-relativistic) bosons and photons,
photons have two polarizations ε1(k), ε2(k) (∇ · Aκ = 0),
H = L2(R3N)⊗ [⊕n≥0h⊗n
s ] with h = L2(R3)⊗ C2,
scaling: kinetic and potential energy are of the same order.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Spinless Pauli-Fierz Hamiltonian
i∂tΨN,t =HNΨN,t
with
HN =N∑
j=1
(−i∇j −
Aκ(xj )√N
)2
+1
N
∑1≤j<k≤N
1
|xj − xk |+ Hf ,
Aκ(x) =∑λ=1,2
∫d3k
κ(k)√2|k |
ελ(k)(e ikxa(k , λ) + e−ikxa∗(k, λ)
).
two types of particles: (non-relativistic) bosons and photons,
photons have two polarizations ε1(k), ε2(k) (∇ · Aκ = 0),
H = L2(R3N)⊗ [⊕n≥0h⊗n
s ] with h = L2(R3)⊗ C2,
scaling: kinetic and potential energy are of the same order.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Spinless Pauli-Fierz Hamiltonian
i∂tΨN,t =HNΨN,t
with
HN =N∑
j=1
(−i∇j −
Aκ(xj )√N
)2
+1
N
∑1≤j<k≤N
1
|xj − xk |+ Hf ,
Aκ(x) =∑λ=1,2
∫d3k
κ(k)√2|k |
ελ(k)(e ikxa(k , λ) + e−ikxa∗(k, λ)
).
two types of particles: (non-relativistic) bosons and photons,
photons have two polarizations ε1(k), ε2(k) (∇ · Aκ = 0),
H = L2(R3N)⊗ [⊕n≥0h⊗n
s ] with h = L2(R3)⊗ C2,
scaling: kinetic and potential energy are of the same order.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Maxwell-Schrodinger system of equations
i∂tϕt(x) =(
(−i∇− (κ ∗ A)(x , t))2 + (| · |−1 ∗ |ϕt |2)(x))ϕt(x),
∇ · A(x , t) = 0,
∂tA(x , t) = −E (x , t),
∂tE (x , t) = (−∆A) (x , t)−(1−∇div∆−1
)(κ ∗ j t) (x),
j t(x) = 2(Im(ϕ∗t∇ϕt)(x)− |ϕt |2(x)(κ ∗ A)(x , t)
)with initial datum
ϕ0,
A(x , 0) = (2π)−3/2∑λ=1,2
∫d3k 1√
2|k|ελ(k)
(e ikxα0(k, λ) + e−ikxα∗0(k, λ)
),
E (x , 0) = (2π)−3/2∑λ=1,2 d
3k√|k|2 ελ(k)i
(e ikxα0(k, λ)− e−ikxα∗0(k, λ)
).
ut(k, λ) := |k |1/2αt(k , λ) =1√2ελ(k) · (|k |FT [A](k , t)− iFT [E ](k, t)) .
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Maxwell-Schrodinger system of equations
i∂tϕt(x) =(
(−i∇− (κ ∗ A)(x , t))2 + (| · |−1 ∗ |ϕt |2)(x))ϕt(x),
∇ · A(x , t) = 0,
∂tA(x , t) = −E (x , t),
∂tE (x , t) = (−∆A) (x , t)−(1−∇div∆−1
)(κ ∗ j t) (x),
j t(x) = 2(Im(ϕ∗t∇ϕt)(x)− |ϕt |2(x)(κ ∗ A)(x , t)
)with initial datum
ϕ0,
A(x , 0) = (2π)−3/2∑λ=1,2
∫d3k 1√
2|k|ελ(k)
(e ikxα0(k, λ) + e−ikxα∗0(k, λ)
),
E (x , 0) = (2π)−3/2∑λ=1,2 d
3k√|k|2 ελ(k)i
(e ikxα0(k, λ)− e−ikxα∗0(k, λ)
).
ut(k, λ) := |k |1/2αt(k , λ) =1√2ελ(k) · (|k |FT [A](k , t)− iFT [E ](k, t)) .
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Maxwell-Schrodinger system of equations
i∂tϕt(x) =(
(−i∇− (κ ∗ A)(x , t))2 + (| · |−1 ∗ |ϕt |2)(x))ϕt(x),
∇ · A(x , t) = 0,
∂tA(x , t) = −E (x , t),
∂tE (x , t) = (−∆A) (x , t)−(1−∇div∆−1
)(κ ∗ j t) (x),
j t(x) = 2(Im(ϕ∗t∇ϕt)(x)− |ϕt |2(x)(κ ∗ A)(x , t)
)with initial datum
ϕ0,
A(x , 0) = (2π)−3/2∑λ=1,2
∫d3k 1√
2|k|ελ(k)
(e ikxα0(k, λ) + e−ikxα∗0(k, λ)
),
E (x , 0) = (2π)−3/2∑λ=1,2 d
3k√|k|2 ελ(k)i
(e ikxα0(k, λ)− e−ikxα∗0(k, λ)
).
ut(k, λ) := |k |1/2αt(k , λ) =1√2ελ(k) · (|k |FT [A](k , t)− iFT [E ](k, t)) .
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Main theorem
One-particle reduced density matrices:
γ(1,0)N,t := Tr2,...,N ⊗ TrF |ΨN,t〉〈ΨN,t |,
γ(0,1)N,t (k , λ; k ′, λ′) := N−1|k |1/2|k ′|1/2〈ΨN , a
∗(k ′, λ′)a(k, λ)ΨN〉.
Theorem (L. and Pickl [2016])
Let ϕ0 ∈ L2 with ||ϕ0|| = 1, α0 ∈ h s.t. (A(0),E (0)) ∈ (H3 ⊕ H2)and ΨN,0 = ϕ⊗N
0 ⊗W (√Nα0)Ω. Moreover, assume that
supt∈[0,T ]||ϕt ||H3 + ||A(t)||H3 + ||E (t)||H2 <∞ for any T ∈ R+.Then, for any t ∈ R+ there exist two constants C1,C2 such that
TrL2 |γ(1,0)N,t − |ϕt〉〈ϕt || ≤ N−1/2Λ2C1e
Λ4C2(t)(1 + Λ2C1/22 (t)),
Trh|γ(0,1)N,t − |ut〉〈ut || ≤ N−1/2Λ2C1e
Λ4C2(t)(1 + Λ2C1/22 (t)).
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Idea of the proof
Introduce the functional: β(t) = βa(t) + βb(t) + βc (t).
βa measures the dirt in the condensate of the non-relativisticparticles,
βb measures if the photons are close to a coherent state,
βc quantifies the fluctuations in the energy per particle of themany-body system.
Initially: β(0) ≈ 0Show: dtβ(t) ≤ C (β(t) + o(1))Gronwall: β(t) ≤ eCtβ(0) + o(1)
Two major difficulties:
minimal coupling term,
soft photons.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Remarks and Outline
Remarks:
Theorem holds for a larger class of initial conditions andpotentials.
Method also works for the Nelson model.
UV-cutoff is essential, but can be chosen N-dependent.
Literature:[Knowles, 2009], [Falconi, Velo, 2012], [Ammari, Falconi, 2016].
Outlook:
Mean-field limits for fermions,
Renormalized Nelson model.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Remarks and Outline
Remarks:
Theorem holds for a larger class of initial conditions andpotentials.
Method also works for the Nelson model.
UV-cutoff is essential, but can be chosen N-dependent.
Literature:[Knowles, 2009], [Falconi, Velo, 2012], [Ammari, Falconi, 2016].
Outlook:
Mean-field limits for fermions,
Renormalized Nelson model.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Remarks and Outline
Remarks:
Theorem holds for a larger class of initial conditions andpotentials.
Method also works for the Nelson model.
UV-cutoff is essential, but can be chosen N-dependent.
Literature:[Knowles, 2009], [Falconi, Velo, 2012], [Ammari, Falconi, 2016].
Outlook:
Mean-field limits for fermions,
Renormalized Nelson model.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Remarks and Outline
Remarks:
Theorem holds for a larger class of initial conditions andpotentials.
Method also works for the Nelson model.
UV-cutoff is essential, but can be chosen N-dependent.
Literature:[Knowles, 2009], [Falconi, Velo, 2012], [Ammari, Falconi, 2016].
Outlook:
Mean-field limits for fermions,
Renormalized Nelson model.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Remarks and Outline
Remarks:
Theorem holds for a larger class of initial conditions andpotentials.
Method also works for the Nelson model.
UV-cutoff is essential, but can be chosen N-dependent.
Literature:[Knowles, 2009], [Falconi, Velo, 2012], [Ammari, Falconi, 2016].
Outlook:
Mean-field limits for fermions,
Renormalized Nelson model.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Remarks and Outline
Remarks:
Theorem holds for a larger class of initial conditions andpotentials.
Method also works for the Nelson model.
UV-cutoff is essential, but can be chosen N-dependent.
Literature:[Knowles, 2009], [Falconi, Velo, 2012], [Ammari, Falconi, 2016].
Outlook:
Mean-field limits for fermions,
Renormalized Nelson model.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
Thank you for listening!
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βa
Let ϕt ∈ L2(R3) and ptj : L2(R3N)→ L2(R3N) be given by
f (x1, . . . , xN) 7→ ϕt(xj )
∫d3xj ϕ
∗t (xj )f (x1, . . . , xN).
Define qtj := 1− pt
j and the functional
βa[ΨN,t , ϕt ] := N−1N∑
j=1
〈ΨN,t , qtj ⊗ 1FΨN,t〉 = 〈ΨN,t , q
t1ΨN,t〉.
βa measures the relative number of particles which are not in thestate ϕt :
βa(t) ≤ TrL2(R3)|γ(1,0)N,t − |ϕt〉〈ϕt || ≤ C
√βa(t),
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βa[ΨN,0, ϕ0] = 0.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βa
Let ϕt ∈ L2(R3) and ptj : L2(R3N)→ L2(R3N) be given by
f (x1, . . . , xN) 7→ ϕt(xj )
∫d3xj ϕ
∗t (xj )f (x1, . . . , xN).
Define qtj := 1− pt
j and the functional
βa[ΨN,t , ϕt ] := N−1N∑
j=1
〈ΨN,t , qtj ⊗ 1FΨN,t〉 = 〈ΨN,t , q
t1ΨN,t〉.
βa measures the relative number of particles which are not in thestate ϕt :
βa(t) ≤ TrL2(R3)|γ(1,0)N,t − |ϕt〉〈ϕt || ≤ C
√βa(t),
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βa[ΨN,0, ϕ0] = 0.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βa
Let ϕt ∈ L2(R3) and ptj : L2(R3N)→ L2(R3N) be given by
f (x1, . . . , xN) 7→ ϕt(xj )
∫d3xj ϕ
∗t (xj )f (x1, . . . , xN).
Define qtj := 1− pt
j and the functional
βa[ΨN,t , ϕt ] := N−1N∑
j=1
〈ΨN,t , qtj ⊗ 1FΨN,t〉 = 〈ΨN,t , q
t1ΨN,t〉.
βa measures the relative number of particles which are not in thestate ϕt :
βa(t) ≤ TrL2(R3)|γ(1,0)N,t − |ϕt〉〈ϕt || ≤ C
√βa(t),
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βa[ΨN,0, ϕ0] = 0.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βb
βb[ΨN,t , αt ] :=∑λ=1,2
∫d3k |k |〈ΨN,t ,
(a∗(k , λ)√
N− α∗t (k , λ)
)(a(k , λ)√
N− αt(k, λ)
)ΨN,t〉.
βb measures the fluctuations of the quantized field modes aroundthe classical mode function αt :
Trh|γ(0,1)N,t − |ut〉〈ut || ≤ C ||ut ||h
√βb[ΨN,t , αt ],
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βb[ΨN,0, α0] = 0.
Let UN (t) = W−1(√Nαt)e−iHN tW (
√Nα0), then
βb[ΨN,t , αt ] = N−1〈UN (t)W−1(√Nα0)ΨN ,Hf UN (t)W−1(
√Nα0)ΨN〉.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βb
βb[ΨN,t , αt ] :=∑λ=1,2
∫d3k |k |〈ΨN,t ,
(a∗(k , λ)√
N− α∗t (k , λ)
)(a(k , λ)√
N− αt(k, λ)
)ΨN,t〉.
βb measures the fluctuations of the quantized field modes aroundthe classical mode function αt :
Trh|γ(0,1)N,t − |ut〉〈ut || ≤ C ||ut ||h
√βb[ΨN,t , αt ],
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βb[ΨN,0, α0] = 0.
Let UN (t) = W−1(√Nαt)e−iHN tW (
√Nα0), then
βb[ΨN,t , αt ] = N−1〈UN (t)W−1(√Nα0)ΨN ,Hf UN (t)W−1(
√Nα0)ΨN〉.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βb
βb[ΨN,t , αt ] :=∑λ=1,2
∫d3k |k |〈ΨN,t ,
(a∗(k , λ)√
N− α∗t (k , λ)
)(a(k , λ)√
N− αt(k, λ)
)ΨN,t〉.
βb measures the fluctuations of the quantized field modes aroundthe classical mode function αt :
Trh|γ(0,1)N,t − |ut〉〈ut || ≤ C ||ut ||h
√βb[ΨN,t , αt ],
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βb[ΨN,0, α0] = 0.
Let UN (t) = W−1(√Nαt)e−iHN tW (
√Nα0), then
βb[ΨN,t , αt ] = N−1〈UN (t)W−1(√Nα0)ΨN ,Hf UN (t)W−1(
√Nα0)ΨN〉.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βc
βc [ΨN,t , ϕt , αt ] :=
∣∣∣∣∣∣∣∣(HN
N− EMS [ϕt , αt ]
)ΨN,t
∣∣∣∣∣∣∣∣2H,
with
EMS [ϕt , αt ] := ||(−i∇− Aκ(t))ϕt ||2 + 1/2〈ϕt ,(| · |−1 ∗ |ϕt |2
)ϕt〉
+ 1/2∑λ=1,2
∫d3k |k ||αt(k , λ)|2.
βc restricts our consideration to many-body states, whose energyper particle only fluctuates little around the energy of the effectivesystem:
dtβc (t) = 0,
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βc [ΨN,0, ϕ0, α0] ≤ CΛ4N−1.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
βc
βc [ΨN,t , ϕt , αt ] :=
∣∣∣∣∣∣∣∣(HN
N− EMS [ϕt , αt ]
)ΨN,t
∣∣∣∣∣∣∣∣2H,
with
EMS [ϕt , αt ] := ||(−i∇− Aκ(t))ϕt ||2 + 1/2〈ϕt ,(| · |−1 ∗ |ϕt |2
)ϕt〉
+ 1/2∑λ=1,2
∫d3k |k ||αt(k , λ)|2.
βc restricts our consideration to many-body states, whose energyper particle only fluctuates little around the energy of the effectivesystem:
dtβc (t) = 0,
ΨN,0 = ϕ⊗N0 ⊗W (
√Nα0)Ω⇒ βc [ΨN,0, ϕ0, α0] ≤ CΛ4N−1.
Nikolai Leopold, Peter Pickl Derivation of the Maxwell-Schrodinger Equations from the Pauli-Fierz Hamiltonian
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