Derivation of the Maxwell-Schrödinger Equations from the ...pub.ist.ac.at/~nleopold/Pauli_Fierz_Oberwolfach.pdf · Derivation of the Maxwell-Schr odinger Equations from the Pauli-Fierz

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


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 |


Motivation

Question




|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 −−−−→



Motivation

Question




|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 −−−−→



Motivation

Question




|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 −−−−→



Motivation

Question




|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 −−−−→



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.



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 |


).



H = L2(R3N)⊗ [⊕n≥0h⊗n

s ] with h = L2(R3)⊗ C2,




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 |


).



H = L2(R3N)⊗ [⊕n≥0h⊗n

s ] with h = L2(R3)⊗ C2,




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 |


).



H = L2(R3N)⊗ [⊕n≥0h⊗n

s ] with h = L2(R3)⊗ C2,




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 |


).



H = L2(R3N)⊗ [⊕n≥0h⊗n

s ] with h = L2(R3)⊗ C2,



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)) .



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),


)with initial datum

ϕ0,

A(x , 0) = (2π)−3/2∑λ=1,2

∫d3k 1√

2|k|ελ(k)


),

E (x , 0) = (2π)−3/2∑λ=1,2 d

3k√|k|2 ελ(k)i


).




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),


)with initial datum

ϕ0,

A(x , 0) = (2π)−3/2∑λ=1,2

∫d3k 1√

2|k|ελ(k)


),

E (x , 0) = (2π)−3/2∑λ=1,2 d

3k√|k|2 ελ(k)i


).



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)).


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.


Idea of the proof








soft photons.


Idea of the proof








soft photons.


Idea of the proof








soft photons.


Idea of the proof








soft photons.


Idea of the proof








soft photons.


Idea of the proof








soft photons.


Idea of the proof








soft photons.


Idea of the proof








soft photons.


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.


Remarks and Outline

Remarks:





Outlook:




Remarks and Outline

Remarks:





Outlook:




Remarks and Outline

Remarks:





Outlook:




Remarks and Outline

Remarks:





Outlook:




Remarks and Outline

Remarks:





Outlook:




Thank you for listening!


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.


βa


f (x1, . . . , xN) 7→ ϕt(xj )

∫d3xj ϕ

∗t (xj )f (x1, . . . , xN).



βa[ΨN,t , ϕt ] := N−1N∑

j=1


t1ΨN,t〉.



√βa(t),

ΨN,0 = ϕ⊗N0 ⊗W (

√Nα0)Ω⇒ βa[ΨN,0, ϕ0] = 0.


βa


f (x1, . . . , xN) 7→ ϕt(xj )

∫d3xj ϕ

∗t (xj )f (x1, . . . , xN).



βa[ΨN,t , ϕt ] := N−1N∑

j=1


t1ΨN,t〉.



√βa(t),

ΨN,0 = ϕ⊗N0 ⊗W (

√Nα0)Ω⇒ βa[ΨN,0, ϕ0] = 0.


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〉.


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〉.


Trh|γ(0,1)N,t − |ut〉〈ut || ≤ C ||ut ||h


ΨN,0 = ϕ⊗N0 ⊗W (

√Nα0)Ω⇒ βb[ΨN,0, α0] = 0.


√Nα0), then


√Nα0)ΨN〉.


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〉.


Trh|γ(0,1)N,t − |ut〉〈ut || ≤ C ||ut ||h


ΨN,0 = ϕ⊗N0 ⊗W (

√Nα0)Ω⇒ βb[ΨN,0, α0] = 0.


√Nα0), then


√Nα0)ΨN〉.


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.


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.


Documents

Derivation of the Maxwell-Schrödinger Equations from the ...pub.ist.ac.at/~nleopold/Pauli_Fierz_Oberwolfach.pdf · Derivation of the Maxwell-Schr odinger Equations from the Pauli-Fierz