Buchholz - Wightman - Strocchi - Morchio - Fröhlich ..... b k,λ. } Alessandro Pizzo /
UC Davis. 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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Infrared Problem in QED
I
Solvable models: I I
Bloch and Nordsieck Pauli and Fierz
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Infrared Problem in QED
I
Solvable models: I 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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Infrared Problem in QED
I
Solvable models: I I
I
S-Matrix, perturbative calculations: 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 - Fr¨ohlich
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Infrared Problem in QED
I
Solvable models: I I
I
S-Matrix, perturbative calculations: 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 - Fr¨ohlich
I
Nonrelativistic QED models
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
t
R1
R2
R3
(t,x(t))
(t=0,x(0)=0) xj
xi
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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)) )
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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(|~
Alessandro Pizzo / UC Davis
as |~y | → ∞
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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(|~
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 . ]]
Alessandro Pizzo / UC Davis
as |~y | → ∞
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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(|~
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 )
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Hilbert Space
I
One spinless nonrelativistic Electron coupled to the Quantized Radiation Field H := Hel ⊗ F ,
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Hilbert Space
I
One spinless nonrelativistic Electron coupled to the Quantized Radiation Field H := Hel ⊗ F , I
Hel := L2 (R3 ) ,
F :=
∞ M
F (N) ,
N=0
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Hilbert Space
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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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 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|
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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 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 ,
Alessandro Pizzo / UC Davis
~k · ~ε~ = 0 k,λ
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Dynamics I
Field Hamiltonian H
f
:=
XZ
∗ d 3 k |~k| a~k,λ a~k,λ
λ=±
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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
Alessandro Pizzo / UC Davis
2 + Hf
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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 ,
Alessandro Pizzo / UC Davis
α≈
1 137
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Translation Invariance I
Total Momentum ~ := ~p + P ~f P
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Translation Invariance I
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=±
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Translation Invariance I
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]
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Scattering in QFT, Haag Ruelle Scattering Theory I
~ Mass Spectrum and One-Particle States (for small |P|) ~ Ψ H Ψ = E (P)
~ = E (P)
Alessandro Pizzo / UC Davis
~2 P 2 ~ + o((|P|/M r) ) 2Mr
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Scattering in QFT, Haag Ruelle Scattering Theory I
~ Mass Spectrum and One-Particle States (for small |P|) ~ Ψ H Ψ = E (P)
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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Scattering in QFT, Haag Ruelle Scattering Theory I
~ Mass Spectrum and One-Particle States (for small |P|) ~ Ψ H Ψ = E (P)
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
Constructing Scattering States (Heisenberg picture) aout ∗ [f 1 ] aout ∗ [f 2 ] aout ∗ [f 3 ] Ψ
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Scattering in QFT, Haag Ruelle Scattering Theory I
~ Mass Spectrum and One-Particle States (for small |P|) ~ Ψ H Ψ = E (P)
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
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
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Fibration I
Fiber Space and Hamiltonian Z ⊕ H := HP~ d 3 P ,
Alessandro Pizzo / UC Davis
Z H =
⊕
HP~ d 3 P
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Fibration I
I
Fiber Space and Hamiltonian Z ⊕ H := HP~ d 3 P ,
Z H =
⊕
HP~ d 3 P
Fiber Hamiltonian
HP~ :=
~p z }| { ~ −P ~ f +α1/2 A ~ 2 P + Hf 2
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Model: Fibration I
I
Fiber Space and Hamiltonian Z ⊕ H := HP~ d 3 P ,
⊕
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|
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
E
E E
σ
P
σ
P
σ
P-k
- |k| |P-k|
Alessandro Pizzo / UC Davis
|P|
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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
If σ > 0 → unique groundstate ψ~σ
P
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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
If σ > 0 → unique groundstate ψ~σ P R ~ ψσ d 3P One-particle states: h(P) ~
I
P
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Fiber Hamiltonians: Existence of the Groundstate
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
Alessandro Pizzo / UC Davis
~ E σ ψ σ d 3 P = E σ (P) ~ h(P) ~ P ~ P
Z
~ ψσ d 3 P h(P) ~ P
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Fiber Hamiltonians: Lim σ → 0 I
~ = If P 6 0 → w − limσ→0 ψ~σ = 0 P
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Fiber Hamiltonians: Lim σ → 0 I
~ = 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
Alessandro Pizzo / UC Davis
k,λ
3 ~ ~) |~k| 2 (1 − kb · ∇E P
~ ~ · ~ε ∗ ∇E ~ P
k,λ
3 ~ ~) |~k| 2 (1 − kb · ∇E P
∗ =: B~k,λ
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Fiber Hamiltonians: Lim σ → 0 I
~ = 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
Alessandro Pizzo / UC Davis
k,λ
3 ~ ~) |~k| 2 (1 − kb · ∇E P
~ ~ · ~ε ∗ ∇E ~ P
k,λ
3 ~ ~) |~k| 2 (1 − kb · ∇E P
∗ =: B~k,λ
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Fiber Hamiltonians: Lim σ → 0 I
~ = 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 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
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Scattering Theory for Infraparticles
I
Fr¨ohlich’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
Alessandro Pizzo / UC Davis
d k
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1): Outline of Strategy I
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
× 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 Time-dependent Cell Partition of the P− space
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1): Outline of Strategy I
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
× 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 Time-dependent Cell Partition of the P− space I
R
d 3 P is transformed to a Riemann sum
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1): Outline of Strategy I
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
× 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 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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1): Outline of Strategy I
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
× 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 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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1) :Recipe I
Time-dependent Cell Partition G (t) I
(t)
One-Particle State for the cell Gj
Alessandro Pizzo / UC Davis
(t)
: ψj,σt :=
R
(t)
Gj
~ σt d 3 P h(P)ψ ~ P
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1) :Recipe I
Time-dependent Cell Partition G (t) I
I
(t)
One-Particle State for the cell Gj
(t)
: ψj,σt :=
R
(t)
Gj
~ σt d 3 P h(P)ψ ~ P
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
Alessandro Pizzo / UC Davis
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|
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1) :Recipe I
Time-dependent Cell Partition G (t) I
I
(t)
One-Particle State for the cell Gj
(t)
: ψj,σt :=
R
(t)
Gj
~ σt d 3 P h(P)ψ ~ P
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
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
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1) :Recipe I
Time-dependent Cell Partition G (t) I
I
(t)
One-Particle State for the cell Gj
(t)
: ψj,σt :=
R
(t)
Gj
~ σt d 3 P h(P)ψ ~ P
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
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
Alessandro Pizzo / UC Davis
P
σ P
,t) −iE~ t t
e
(t)
ψj,σt
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Approximating Vector (P ’05 and CFP1) :Recipe I
Time-dependent Cell Partition G (t) I
I
(t)
One-Particle State for the cell Gj
(t)
: ψj,σt :=
R
(t)
Gj
~ σt d 3 P h(P)ψ ~ P
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
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
Alessandro Pizzo / UC Davis
σ
,t) −iE~ t t (t) e P φj,σt
Solution of the Infrared Catastrophe Problem in Nonrelativistic
Times Scales
I
Fast infrared Cut-Off σt σt = t −β ,
β1
Alessandro Pizzo / UC Davis
(uncertainty principle)
Solution of the Infrared Catastrophe Problem in Nonrelativistic
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→±∞
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic
t
R1
R2
R3
(t,x(t))
(t=0,x(0)=0) xj
xi
Alessandro Pizzo / UC Davis
Solution of the Infrared Catastrophe Problem in Nonrelativistic