Solution of the Infrared Catastrophe Problem in Nonrelativistic QED ...

Solution of the Infrared Catastrophe Problem in Nonrelativistic QED joint work with T. Chen and J. Fr¨ ohlich refs: CFP1 in CMP-2010, CFP2 in JMP-2009; related works: C ’06, CF ’06, F ’73, F ’74, P ’03, P ’05, FP ’08

Alessandro Pizzo / UC Davis

ESI-Vienna, 26-09-2011


Solution of the Infrared Catastrophe Problem in Nonrelativistic

Infrared Problem in QED

I

Solvable models: I I

Bloch and Nordsieck Pauli and Fierz




I


I

Bloch and Nordsieck Pauli and Fierz

S-Matrix, perturbative calculations: I I

Jauch and Rohrlich - Yennie, Frautschi, and Suura Chung - Kibble - Zwanziger - Faddeev and Kulish - Stapp Steinmann




I


I


I

Bloch and Nordsieck Pauli and Fierz Jauch and Rohrlich - Yennie, Frautschi, and Suura Chung - Kibble - Zwanziger - Faddeev and Kulish - Stapp Steinmann

General Quantum Field Theory: Buchholz - Wightman - Strocchi - Morchio - Fröhlich




I


I


I

Bloch and Nordsieck Pauli and Fierz Jauch and Rohrlich - Yennie, Frautschi, and Suura Chung - Kibble - Zwanziger - Faddeev and Kulish - Stapp Steinmann

General Quantum Field Theory: Buchholz - Wightman - Strocchi - Morchio - Fröhlich

I

Nonrelativistic QED models



t

R1

R2

R3

(t,x(t))

(t=0,x(0)=0) xj

xi



Classical Pointlike Charged Particle I

Maxwell equations ∂µ F µν (t, ~y ) = J ν (t, ~y ) := −q ( δ (3) (~y −~x (t)) , ~x˙ (t) δ (3) (~y −~x (t)) )





I

Class of solutions C~vL.W . µν ~ ~ |F[~µν y |−2 ) vL.W . ] (t, y ) − F~x ,~vL.W . (t, y )| = o(|~


as |~y | → ∞




I


I

as |~y | → ∞

For t > ¯t µν ~ ~ φµν y ) := F[~µν out (t, ~ vL.W . ] (t, y ) − F~x∗ ,~vout (t, y )

~ out (t, ~y ) ' α 12 A

XZ λ

∗ o d 3 k n (~vout · ~ε~k,λ )~ε~k,λ −i(~k·~y −|~k|t) q e +c.c. ~ 3 ˆ |~k| |k| 2 (1 − ~vout · k)

−[[~vL.W . ]]


as |~y | → ∞




I


I

as |~y | → ∞

For t > ¯t µν ~ ~ φµν y ) := F[~µν out (t, ~ vL.W . ] (t, y ) − F~x∗ ,~vout (t, y )

~ out (t, ~y ) ' α 12 A

XZ λ

∗ o d 3 k n (~vout · ~ε~k,λ )~ε~k,λ −i(~k·~y −|~k|t) q e +c.c. ~ 3 ˆ |~k| |k| 2 (1 − ~vout · k)

−[[~vL.W . ]] I

For t ¯t and |~y − ~x (t)| = o(t)

as |~y | → ∞

µν ~ ~ ⇒ F[~µν vL.W . ] (t, y ) ' F~x∗ ,~vout (t, y )



Model: Hilbert Space

I

One spinless nonrelativistic Electron coupled to the Quantized Radiation Field H := Hel ⊗ F ,




I

One spinless nonrelativistic Electron coupled to the Quantized Radiation Field H := Hel ⊗ F , I

Hel := L2 (R3 ) ,

F :=

∞ M

F (N) ,

N=0




I

One spinless nonrelativistic Electron coupled to the Quantized Radiation Field H := Hel ⊗ F , I

Hel := L2 (R3 ) ,

F :=

∞ M

F (N) ,

N=0 I

F (0) = C ,

F (N) := SN

N O

h,

h := L2 (R3 × Z2 )

j=1



Model: Kinematics I

C.C.R. (~ = c = 1) [a~k,λ , a~k∗ 0 ,λ0 ] = δλλ0 δ(~k−~k 0 ), [a~# , a~#0 0 ] = 0 , k,λ

k ,λ

~k, ~k 0 ∈ R3 ,

a# = a , a∗

/

λ, λ0 ∈ Z2 ≡ {±} a~k,λ Ω = 0

[x j , p l ] = i δj,l



Model: Kinematics I


k ,λ

~k, ~k 0 ∈ R3 ,

a# = a , a∗

/

λ, λ0 ∈ Z2 ≡ {±} a~k,λ Ω = 0

[x j , p l ] = i δj,l I

Vector Potential ~ ~y ) := A(0,

XZ λ=±

d 3k ~ ~ ∗ ∗ q + ~ε~k,λ e i k·~y a~k,λ ~ε~k,λ e −i k·~y a~k,λ |~k|



Model: Kinematics I


k ,λ

~k, ~k 0 ∈ R3 ,

a# = a , a∗

/

λ, λ0 ∈ Z2 ≡ {±} a~k,λ Ω = 0

[x j , p l ] = i δj,l I

Vector Potential ~ ~y ) := A(0,

XZ λ=±

I

d 3k ~ ~ ∗ ∗ q + ~ε~k,λ e i k·~y a~k,λ ~ε~k,λ e −i k·~y a~k,λ |~k|

Coulomb Gauge ∗ ~ε~k,λ · ~ε~k,µ = δλµ ,


~k · ~ε~ = 0 k,λ


Model: Dynamics I

Field Hamiltonian H

f

:=

XZ

∗ d 3 k |~k| a~k,λ a~k,λ

λ=±



Model: Dynamics I

Field Hamiltonian H

f

:=

XZ

∗ d 3 k |~k| a~k,λ a~k,λ

λ=± I

Minimal Coupling H :=

~ x) ~ ~x + α1/2 A(~ − i∇ 2


2 + Hf


Model: Dynamics I

Field Hamiltonian H

f

:=

XZ

∗ d 3 k |~k| a~k,λ a~k,λ

λ=± I

I I

Minimal Coupling ~ x) 2 ~ ~x + α1/2 A(~ − i∇ H := + Hf 2 R d 3 k ∗ −i ~k·~x ~ ~ x) ≡ P Λ √ A(~ a~ e ~ε~k,λ + ~ε~∗ a~k,λ e i k·~x λ 0 ~ |k|

k,λ

k,λ

Fixed uv cutoff Λ and small coupling Λ ≈ mel = 1 ,


α≈

1 137


Model: Translation Invariance I

Total Momentum ~ := ~p + P ~f P




Total Momentum ~ := ~p + P ~f P I

Electron Momentum ~ ~x ~p = −i ∇

I

Field Momentum ~ f := P

XZ

∗ d 3 k ~k a~k,λ a~k,λ

λ=±




Total Momentum ~ := ~p + P ~f P I

Electron Momentum ~ ~x ~p = −i ∇

I

Field Momentum ~ f := P

XZ

∗ d 3 k ~k a~k,λ a~k,λ

λ=±

I

~ = 0 [H , P]



Scattering in QFT, Haag Ruelle Scattering Theory I

~ Mass Spectrum and One-Particle States (for small |P|) ~ Ψ H Ψ = E (P)

~ = E (P)


~2 P 2 ~ + o((|P|/M r) ) 2Mr




I

~ = E (P)

~2 P 2 ~ + o((|P|/M r) ) 2Mr

L.S.Z. asymptotic condition q Z X −it |~k|2 +mph 2 out ∗ iHt ∗ fλ (~k)d 3 k e −iHt a (f ) := lim e a~k,λ e t→+∞

|λ {z Z X f f ∗ e iHt e −iH t a~k,λ fλ (~k)d 3 k e iH t e −iHt

}

λ





I

~ = E (P)

~2 P 2 ~ + o((|P|/M r) ) 2Mr



}

λ I

Constructing Scattering States (Heisenberg picture) aout ∗ [f 1 ] aout ∗ [f 2 ] aout ∗ [f 3 ] Ψ





I

~ = E (P)

~2 P 2 ~ + o((|P|/M r) ) 2Mr



}

λ I

Constructing Scattering States (Heisenberg picture) aout ∗ [f 1 ] aout ∗ [f 2 ] aout ∗ [f 3 ] Ψ

I

If mph = 0 ⇒ problems: Absence of a Proper Mass Shell (Schroer ’63), Decoupling (Buchholz ’77) Alessandro Pizzo / UC Davis


Model: Fibration I

Fiber Space and Hamiltonian Z ⊕ H := HP~ d 3 P ,


Z H =

⊕

HP~ d 3 P


Model: Fibration I

I


Z H =

⊕

HP~ d 3 P

Fiber Hamiltonian

HP~ :=

~p z }| { ~ −P ~ f +α1/2 A ~ 2 P + Hf 2



Model: Fibration I

I


⊕

HP~ d 3 P

Fiber Hamiltonian ~p z }| { ~ −P ~ f +α1/2 A ~ 2 P + Hf 2

HP~ := I

Z H =

Fiber Vector Potential ~

∗ e −i k·~x a~k,λ

~ ≡ A

XZ λ

0

Λ

d 3 k z}|{ ∗ q b~k,λ ~ε~k,λ + ~ε~∗k,λ b~k,λ |~k|



E

E E

σ

P

σ

P

σ

P-k

- |k| |P-k|


|P|


Fiber Hamiltonians: Existence of the Groundstate

I

Fiber Hamiltonian with infrared cut-off σ ~ −P ~ f + α1/2 A ~σ 2 P σ HP~ := + Hf 2




I


I

If σ > 0 → unique groundstate ψ~σ

P




I


I

If σ > 0 → unique groundstate ψ~σ P R ~ ψσ d 3P One-particle states: h(P) ~

I

P




I

Fiber Hamiltonian with infrared cut-off σ ~ −P ~ f + α1/2 A ~σ 2 P σ + Hf HP~ := 2

I

If σ > 0 → unique groundstate ψ~σ P R σ ~ One-particle states: h(P) ψ~ d 3 P ∈ H

I

P

Hσ

Z

~ ψσ d 3 P = h(P) ~ P

Z


~ E σ ψ σ d 3 P = E σ (P) ~ h(P) ~ P ~ P

Z

~ ψσ d 3 P h(P) ~ P


Fiber Hamiltonians: Lim σ → 0 I

~ = If P 6 0 → w − limσ→0 ψ~σ = 0 P




~ = If P 6 0 → w − limσ→0 ψ~σ = 0

I

Weyl Operators, Intertwiners

P

~ ~ ) := W (∇E P = exp α

1 2

XZ

Λ 3

d k

0

~ ~ · {~ε~ b ∗ − ~ε ∗ b~ } ∇E ~ P k,λ ~ k,λ k,λ

~ ~ ) = b ∗ −α 21 ~ ~ ) b ∗ W ∗ (∇E W (∇E ~k,λ ~k,λ P P


k,λ

3 ~ ~) |~k| 2 (1 − kb · ∇E P

~ ~ · ~ε ∗ ∇E ~ P

k,λ

3 ~ ~) |~k| 2 (1 − kb · ∇E P

∗ =: B~k,λ



~ = If P 6 0 → w − limσ→0 ψ~σ = 0

I


P

~ ~ ) := W (∇E P = exp α

1 2

XZ

Λ 3

d k

0

~ ~ · {~ε~ b ∗ − ~ε ∗ b~ } ∇E ~ P k,λ ~ k,λ k,λ

~ ~ ) = b ∗ −α 21 ~ ~ ) b ∗ W ∗ (∇E W (∇E ~k,λ ~k,λ P P


k,λ

3 ~ ~) |~k| 2 (1 − kb · ∇E P

~ ~ · ~ε ∗ ∇E ~ P

k,λ

3 ~ ~) |~k| 2 (1 − kb · ∇E P

∗ =: B~k,λ



~ = If P 6 0 → w − limσ→0 ψ~σ = 0

I


P

~ ~ ) := W (∇E P = exp α

1 2

XZ

Λ 3

d k

0

~ ~ · {~ε~ b ∗ − ~ε ∗ b~ } ∇E ~ P k,λ ~ k,λ k,λ

~ ~ ) = b ∗ −α 21 ~ ~ ) b ∗ W ∗ (∇E W (∇E ~k,λ ~k,λ P P I

k,λ

3 ~ ~) |~k| 2 (1 − kb · ∇E P

~ ~ · ~ε ∗ ∇E ~ P

k,λ

3 ~ ~) |~k| 2 (1 − kb · ∇E P

∗ =: B~k,λ

Transformed Hamiltonians ~ ~ ) H~ W ∗ (∇E ~ ~) HP~w := W (∇E P P P → groundstate φP~ Alessandro Pizzo / UC Davis


Scattering Theory for Infraparticles

I

Fröhlich’s Proposal (Fr¨ ohlich ’73), take the limit σ → 0, t→∞ e

iHt

Z

1 2

XZ

Λ1

3

~ σ · {~ε~ a∗ e −i|~k|t − h.c.} ∇E ~ k,λ ~ P

k,λ

× 3 ~ σ) σ |~k| 2 (1 − kb · ∇E λ ~ P (n) ~ x −i P ~ f ·~x ~ −iE~σ t i P·~ σ 3 ×e P e e h(P)ΨP~ d P (k1 , ..., kn ; λ1 , ...., λn ) exp α


d k


Approximating Vector (P ’05 and CFP1): Outline of Strategy I

e

I

iHt

Z

X

1 2

Z

Λ1

3

~ σ · {~ε~ a∗ e −i|~k|t − h.c.} ∇E ~ k,λ ~ P

k,λ

× 3 ~ σ) σ |~k| 2 (1 − kb · ∇E λ ~ P (n) ~ x −i P ~ f ·~x ~ −iE~σ t i P·~ ×e P e e h(P)ΨσP~ d 3 P (k1 , ..., kn ; λ1 , ...., λn ) exp

α

d k

Diagonal limit: σt →t→∞ 0




e

I I

iHt

Z

X

1 2

Z

Λ1

3

~ σ · {~ε~ a∗ e −i|~k|t − h.c.} ∇E ~ k,λ ~ P

k,λ


α

d k

Diagonal limit: σt →t→∞ 0 Time-dependent Cell Partition of the P− space




e

I I

iHt

Z

X

1 2

Z

Λ1

3

~ σ · {~ε~ a∗ e −i|~k|t − h.c.} ∇E ~ k,λ ~ P

k,λ


α

d k

Diagonal limit: σt →t→∞ 0 Time-dependent Cell Partition of the P− space I

R

d 3 P is transformed to a Riemann sum




e

I I

iHt

Z

X

1 2

Z

Λ1

3

~ σ · {~ε~ a∗ e −i|~k|t − h.c.} ∇E ~ k,λ ~ P

k,λ


α

d k

Diagonal limit: σt →t→∞ 0 Time-dependent Cell Partition of the P− space I I

d 3 P is transformed to a Riemann sum ~ → ~vj := ∇E ~ ∗) ~ σt (P) ~ σt ( P Replacement ∇E j R




e

I I

Z

X

1 2

Z

Λ1

3

~ σ · {~ε~ a∗ e −i|~k|t − h.c.} ∇E ~ k,λ ~ P

k,λ


α

d k

Diagonal limit: σt →t→∞ 0 Time-dependent Cell Partition of the P− space I I

I

iHt

d 3 P is transformed to a Riemann sum ~ → ~vj := ∇E ~ ∗) ~ σt (P) ~ σt ( P Replacement ∇E j R

Phase factor



Approximating Vector (P ’05 and CFP1) :Recipe I

Time-dependent Cell Partition G (t) I

(t)

One-Particle State for the cell Gj


(t)

: ψj,σt :=

R

(t)

Gj

~ σt d 3 P h(P)ψ ~ P




I

(t)


(t)

: ψj,σt :=

R

(t)

Gj


Soft-Photon Cloud e iHt Wσt ,Λ1 (~vj , t)e −iH

σt t

,

~ ∗) ~ σt (P ~vj ≡ ∇E j

Wσt ,Λ1 (~vj , t) := =e

−iH f t

exp α

1 2

XZ λ


Λ1

σt

∗ ∗ d 3 k ~vj · {~ε~k,λ a~k,λ − ~ε~k,λ a~k,λ } iH f t q e ~k|(1 − kb · ~vj ) | ~ |k|




I

(t)


(t)

: ψj,σt :=

R

(t)

Gj



σt t

,

~ ∗) ~ σt (P ~vj ≡ ∇E j

Wσt ,Λ1 (~vj , t) := =e

−iH f t

exp α

1 2

XZ λ

I

Λ1

σt

∗ ∗ d 3 k ~vj · {~ε~k,λ a~k,λ − ~ε~k,λ a~k,λ } iH f t q e ~k|(1 − kb · ~vj ) | ~ |k|

Putting all together N(t)

ψh,Λ1 (t) := e

iHt

X

Wσt ,Λ1 (~vj , t)e

σ P

σ P

t ,t) −iE t t ~ iγσt (~vj ,∇E ~ ~

e

(t)

ψj,σt

j=1





I

(t)


(t)

: ψj,σt :=

R

(t)

Gj



σt t

,

~ ∗) ~ σt (P ~vj ≡ ∇E j

Wσt ,Λ1 (~vj , t) := =e

−iH f t

exp α

1 2

XZ

σt

λ I

Λ1

∗ ∗ d 3 k ~vj · {~ε~k,λ a~k,λ − ~ε~k,λ a~k,λ } iH f t q e |~k|(1 − kb · ~vj ) |~k|

Putting all together σt

~ ~ σt )Wσt (∇E ~ σt )e iγσt (~vj ,∇E ~ P e iHt Wσt ,Λ1 (~vj , t)Wσ∗t (∇E ~ ~ P


P

σ P

,t) −iE~ t t

e

(t)

ψj,σt




I

(t)


(t)

: ψj,σt :=

R

(t)

Gj



σt t

,

~ ∗) ~ σt (P ~vj ≡ ∇E j

Wσt ,Λ1 (~vj , t) := =e

−iH f t

exp α

1 2

XZ λ

I

Λ1

σt

∗ ∗ d 3 k ~vj · {~ε~k,λ a~k,λ − ~ε~k,λ a~k,λ } iH f t q e |~k|(1 − kb · ~vj ) |~k|

Putting all together σt

~ ~ σt )e iγσt (~vj ,∇E ~ P e iHt Wσt ,Λ1 (~vj , t)Wσ∗t (∇E ~ P


σ

,t) −iE~ t t (t) e P φj,σt


Times Scales

I

Fast infrared Cut-Off σt σt = t −β ,

β1


(uncertainty principle)


Times Scales

I

Fast infrared Cut-Off σt σt = t −β ,

I

β1

(uncertainty principle)

Slow infrared Cut-Off σtS σtS = t −θ ,

0 1) large enough, (0 0) (out) small enough, limt→+∞ ψh,Λ1 (t) =: ψh,Λ1 exists. I Asymptotic Velocity of the Electron ~x (out/in) (out/in) (out/in) =ψ ~ =: f (~v out/in ) ψh ,Λ1 lim e iHt f ( ) e −iHt ψh,Λ1 h f (∇E ),Λ1 t→±∞ t I

Asymptotic Fields: Coherent Factors out/in k,λ

a~ I

v~k→0

1

a~k,λ + α 2

~v out/in · ~ε~k,λ 3 |~k| 2 (1 − kb · ~v out/in )

Lienard Wiechert Fields for large |~d| Z (out/in) ~ (~x + ~d) e −iHt ψ (out/in) ) lim (ψh,Λ1 , e iHt E h,Λ1 t→±∞



t

R1

R2

R3

(t,x(t))

(t=0,x(0)=0) xj

xi

