Curing the infrared problem in nonrelativistic QED Daniela Cadamuro - - PowerPoint PPT Presentation

Curing the infrared problem in nonrelativistic QED

Daniela Cadamuro

Leipzig joint work with Wojciech Dybalski York 1 July 2019

1/13

Problem:

System of nonrelativistic QED:

• ne “slow” spinless electron interacting with a cloud of photons

◮ Algebra of observables of the system electron + photons ◮ Coherent states 𝜕𝑄 𝑄 𝑄 (ground states of an Hamiltonian 𝐼𝑄 𝑄 𝑄 ,

𝑄 𝑄 𝑄 a total momentum of the system) ⇒ they induce inequivalent representations of the algebra

problem of velocity superselection

Consequences:

◮ states of single electrons with different momenta 𝑄

𝑄 𝑄 cannot be coherently superimposed

◮ electron is an infraparticle (no definite mass) ◮ scattering theory of many electrons seems problematic

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The model

◮ Hamiltonian:

𝐼 = 1 2(−𝑗∇𝑦

𝑦 𝑦 + ˜

𝛽1/2𝐵 𝐵 𝐵(𝑦 𝑦 𝑦))2 + 𝐼photon selfadjoint on dense domain in ℋ = ℋelectron ⊗ ℱphoton, 𝐵 𝐵 𝐵 in Coulomb gauge with UV cutoff.

◮ total momentum 𝑄

𝑄 𝑄 := −𝑗∇ ∇ ∇𝑦

𝑦 𝑦 + 𝑄

𝑄 𝑄 photon, [𝐼,𝑄 𝑄 𝑄] = 0 ⇒ 𝐼 = Π*(︁ ∫︂ ⊕ 𝐼𝑄

𝑄 𝑄 𝑒3𝑄

𝑄 𝑄 )︁ Π, Π a unitary identification

3/13

Ground states of the Hamiltonians H𝑄

𝑄 𝑄

◮ Absence of ground states:

◮ 𝐼P

P P do not have ground states (eigenvectors) for 𝑄

𝑄 𝑄 ̸= 0 at least for small ˜ 𝛽 and for 𝑄 𝑄 𝑄 ∈ 𝑇 = {𝑄 𝑄 𝑄 ∈ R3 : |𝑄 𝑄 𝑄| < 1

3}.

◮ This is a feature of the infraparticle problem

◮ Introduce an infrared cutoff:

𝐼𝑄

𝑄 𝑄,𝜏 := 1

2(𝑄 𝑄 𝑄 − 𝑄 𝑄 𝑄 photon + ˜ 𝛽1/2𝐵 𝐵 𝐵[𝜏,𝜆](0))2 + 𝐼photon selfadjoint on dense domain in Fock space ℱ over 𝑀2

tr(R3; C3);

denote creators/annihilators as 𝑏*

𝜇(𝑙

𝑙 𝑙), 𝑏𝜇(𝑙 𝑙 𝑙). 𝐵[𝜏,𝜆](𝑦 𝑦 𝑦) = ∑︂

𝜇=±

∫︂ 𝑒3𝑙 𝑙 𝑙 √︁ |𝑙 𝑙 𝑙| 𝜓[𝜏,𝜆](|𝑙 𝑙 𝑙|)𝜗 𝜗 𝜗𝜇(𝑙 𝑙 𝑙) (︁ 𝑓−𝑗𝑙

𝑙 𝑙·𝑦 𝑦 𝑦𝑏* 𝜇(𝑙

𝑙 𝑙)+𝑓𝑗𝑙

𝑙 𝑙·𝑦 𝑦 𝑦𝑏𝜇(𝑙

𝑙 𝑙) )︁ (𝜆: UV cutoff, 𝜏: IR cutoff)

4/13

Ground states with IR cutoff

◮ Fact: For any 𝜏 > 0, the operator 𝐼𝑄 𝑄 𝑄,𝜏 has a ground states

(eigenvector) Ψ𝑄

𝑄 𝑄,𝜏 ∈ ℱ with isolated eigenvalues 𝐹𝑄 𝑄 𝑄,𝜏.

◮ ΨP

P P ,σ tend weakly to zero as 𝜏 → 0.

◮ Hence ground states exist at fixed cutoff. ◮ However, we will need to remove the cutoff to describe the

physical system.

◮ This will be done by considering suitable states on a CCR

algebra.

5/13

Velocity superselection

Algebra of observables of the system “electron + photon cloud”:

◮ Weyl (CCR) algebra A generated (up to closure in 𝐷*-norm) by

𝑋(𝑔 𝑔 𝑔), 𝑔 𝑔 𝑔 ∈ ℒ := ⋃︁

𝜗>0 𝑀2 tr,𝜗(R3; C3),

symplectic form 𝜏(·, ·) := Im⟨·, ·⟩.

◮ Vacuum representation: 𝜌vac(𝑋(𝑔

𝑔 𝑔)) = 𝑓𝑏*(𝑔

𝑔 𝑔)−𝑏(𝑔 𝑔 𝑔)

State: Given any 𝐵 ∈ A, define 𝜕𝑄

𝑄 𝑄 (𝐵) := lim 𝜏→0⟨Ψ𝑄 𝑄 𝑄,𝜏, 𝜌vac(𝐵)Ψ𝑄 𝑄 𝑄,𝜏⟩ ◮ state on A, describes plane-wave configurations of the electron

with velocity 𝑄 𝑄 𝑄 Representations: States 𝜕𝑄

𝑄 𝑄 have irreducible GNS representations 𝜌𝑄 𝑄 𝑄 . ◮ Fact: 𝜌𝑄 𝑄 𝑄 ̸≃ 𝜌𝑄 𝑄 𝑄 ′ for any 𝑄

𝑄 𝑄 ̸= 𝑄 𝑄 𝑄 ′ “velocity superselection”

6/13

Cause of the superselection problem

Analyze the phenomenon closely:

◮ introduce auxiliary vectors Φ𝑄 𝑄 𝑄,𝜏 = 𝑋(𝑤

𝑤 𝑤𝑄

𝑄 𝑄,𝜏)Ψ𝑄 𝑄 𝑄,𝜏, where

𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏(𝑙

𝑙 𝑙) := ˜ 𝛽1/2𝑄tr 𝜓[𝜏,𝜆](|𝑙 𝑙 𝑙|) |𝑙 𝑙 𝑙|3/2 ∇𝐹𝑄

𝑄 𝑄,𝜏

1 − ˆ 𝑙 𝑙 𝑙 · ∇𝐹𝑄

𝑄 𝑄,𝜏

. Fact: Φ𝑄

𝑄 𝑄 := lim𝜏→0 Φ𝑄 𝑄 𝑄,𝜏 exists in norm for suitable choice of

the phases of Ψ𝑄

𝑄 𝑄,𝜏.

𝜕𝑄

𝑄 𝑄 (𝑋(𝑔

𝑔 𝑔)) = lim

𝜏→0⟨Φ𝑄 𝑄 𝑄,𝜏, 𝜌vac

(︁ 𝑋(𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏)𝑋(𝑔

𝑔 𝑔)𝑋(𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏)*

⏟ ⏞

:=𝛽v

v vP P P ,σ (𝑋(𝑔

𝑔 𝑔))

)︁ Φ𝑄

𝑄 𝑄,𝜏⟩

𝛽𝑤

𝑤 𝑤P

P P ,σ(𝑋(𝑔

𝑔 𝑔)) = 𝑓−2𝑗 Im⟨𝑤

𝑤 𝑤P

P P ,σ,𝑔

𝑔 𝑔⟩𝑋(𝑔

𝑔 𝑔)

◮ For 𝜏 > 0, we have 𝜌𝑄 𝑄 𝑄,𝜏 ≃ 𝜌vac, but 𝜌𝑄 𝑄 𝑄 ̸≃ 𝜌vac

A possible solution: regularize the map 𝛽𝑤

𝑤 𝑤P

P P ,σ ⇒ Infravacuum state

7/13

Infravacuum state

Walter Kunhardt: DHR theory for the free massless scalar field

◮ automorphisms 𝛿 of the algebra of the free massless scalar field:

similar structure to 𝛽𝑤P

P P

◮ 𝛿 have poor localization property in front of the vacuum:

𝜌vac ∘ 𝛿 ⃒ ⃒

A(𝒫′) ̸≃ 𝜌vac

⃒ ⃒

A(𝒫′),

𝜌vac ∘ 𝛿 ⃒ ⃒

A(𝒟′) ̸≃ 𝜌vac

⃒ ⃒

A(𝒟′)

(𝒫 a double cone, 𝒟 a spacelike cone)

◮ improve the localization property: infravacuum state

𝜕𝑈 (𝑋(𝑔)) = 𝑓− 1

4 ‖𝑈𝑔‖2

◮ Fact: 𝜌𝑈 ∘ 𝛿

⃒ ⃒

A(𝒟′) ≃ 𝜌𝑈

⃒ ⃒

A(𝒟′) ◮ automorphism of the algebra 𝛽𝑈 : 𝛽𝑈 (𝑋(𝑔)) = 𝑋(𝑈𝑔)

8/13

The symplectic map T

◮ Recall ℒ := ⋃︁ 𝜗>0 𝑀2 tr,𝜗(R3; C3) ◮ 𝑈 : ℒ → ℒ,

𝑈 = 𝑈1 1+𝐾

2

+ 𝑈2 1−𝐾

2

𝑈1 := 1 1 1+s- lim

𝑜→∞ 𝑜

∑︂

𝑗=1

(𝑐𝑗−1)𝑅 𝑅 𝑅𝑗, 𝑈2 := 1 1 1+s- lim

𝑜→∞ 𝑜

∑︂

𝑗=1

(︁ 1 𝑐𝑗 −1 )︁ 𝑅 𝑅 𝑅𝑗

◮ 𝑅

𝑅 𝑅i orthogonal projectors on 𝑀2

tr(R3; C3), ∑︁ i 𝑅

𝑅 𝑅i = 1

◮ 𝑗 large means “low energy” ◮ 𝑐i decay with 𝑗 large

◮ 𝑈 modify the low energy behaviour of wave functions in ℒ, and

in particular of 𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏, in such a way that

lim𝜏→0 𝑈𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏 ∈ 𝑀2 tr(R3; C3).

9/13

Infravacuum state

◮ Idea: Instead of 𝜕𝑄 𝑄 𝑄 , consider a modified state 𝜕𝑄 𝑄 𝑄,𝑈 defined by

𝜕𝑄

𝑄 𝑄,𝑈 (𝐵) := lim 𝜏→0⟨Φ𝑄 𝑄 𝑄,𝜏, 𝜌vac

(︁ 𝛽𝑈 (𝛽𝑤

𝑤 𝑤P

P P ,σ(𝐵))

)︁ Φ𝑄

𝑄 𝑄,𝜏⟩

= lim

𝜏→0⟨Φ𝑄 𝑄 𝑄,𝜏, 𝜌vac

(︁ 𝛽𝑈 (𝑋(𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏)𝐵𝑋(𝑤

𝑤 𝑤𝑄

𝑄 𝑄,𝜏)*)

)︁ Φ𝑄

𝑄 𝑄,𝜏⟩

= lim

𝜏→0⟨Φ𝑄 𝑄 𝑄,𝜏, 𝜌vac

(︁ 𝑋(𝑈𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏)𝛽𝑈 (𝐵)𝑋(𝑈𝑤

𝑤 𝑤𝑄

𝑄 𝑄,𝜏)*)︁

Φ𝑄

𝑄 𝑄,𝜏⟩ ◮ Fact: lim𝜏→0 𝑈𝑤

𝑤 𝑤𝑄

𝑄 𝑄,𝜏 := 𝑈𝑤

𝑤 𝑤𝑄

𝑄 𝑄 ∈ 𝑀2 tr(R3; C3) ◮ Result: 𝜌𝑄 𝑄 𝑄,𝑈 ≃ 𝜌𝑄 𝑄 𝑄 ′,𝑈 for any 𝑄

𝑄 𝑄 ̸= 𝑄 𝑄 𝑄 ′.

10/13

Restriction to the light cone

Alternative approach:

◮ Arbitrariness in the choice of the algebra A as long as it acts

irreducibly on ℱ and the states 𝜕𝑄

𝑄 𝑄 are well-defined. ◮ choose A to be the algebra of observables of the free

electromagnetic field → local and relativistic A(𝒫) := 𝐷*{𝑓𝑗(𝐹

𝐹 𝐹(𝑔 𝑔 𝑔e)+𝐶 𝐶 𝐶(𝑔 𝑔 𝑔b)) | supp𝑔

𝑔 𝑔𝑓, supp𝑔 𝑔 𝑔𝑐 ⊂ 𝒫,𝑔 𝑔 𝑔𝑓,𝑐 ∈ 𝒠(R4, R3)}

◮ Result: if A := ⋃︁ 𝒫⊂R4 A(𝒫) (quasi-local algebra),

then 𝜌𝑄

𝑄 𝑄 ̸≃ 𝜌𝑄 𝑄 𝑄 ′, but with A(𝑊+) := ⋃︁ 𝒫⊂𝑊+ A(𝒫),

𝜌𝑄

𝑄 𝑄

⃒ ⃒

A(𝑊+) ≃ 𝜌𝑄 𝑄 𝑄 ′

⃒ ⃒

A(𝑊+) for any 𝑄

𝑄 𝑄,𝑄 𝑄 𝑄 ′ ∈ 𝑇 (𝑊+: forward light cone)

11/13

Restriction to the light cone

Recall that 𝜕𝑄

𝑄 𝑄 (𝐵) = lim 𝜏→0⟨Φ𝑄 𝑄 𝑄,𝜏, 𝜌vac

(︁ 𝑋(𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝜏)𝐵𝑋(𝑤

𝑤 𝑤𝑄

𝑄 𝑄,𝜏)*)︁

Φ𝑄

𝑄 𝑄,𝜏⟩

Idea of proof:

◮ Use Huygens principle: A(𝑊−) ⊂ A(𝑊+)′ ◮ Approximate 𝑤

𝑤 𝑤𝑄

𝑄 𝑄,𝜏 with functions in the symplectic space of the

backward light cone 𝑊−.

◮ Then 𝑋(𝑤

𝑤 𝑤𝑄

𝑄 𝑄,𝜏) and 𝐵 ∈ A(𝑊+) approximately commute, hence

𝜕𝑄

𝑄 𝑄 lives in the vacuum representation. ◮ Hence 𝜌𝑄 𝑄 𝑄

⃒ ⃒

A(𝑊+) ≃ 𝜌vac ≃ 𝜌𝑄 𝑄 𝑄 ′

⃒ ⃒

A(𝑊+) for any 𝑄

𝑄 𝑄,𝑄 𝑄 𝑄 ′ ∈ 𝑇.

12/13

Restriction to the light cone

Local approximation of 𝑤 𝑤 𝑤𝑄

𝑄 𝑄 : ◮ The symplectic space for a double cone 𝑃𝑠 + 𝜐𝑓0 is:

𝑓𝑗|𝑙

𝑙 𝑙|𝜐ℒ𝐶𝐾(𝑃𝑠) :=

𝑓𝑗|𝑙

𝑙 𝑙|𝜐(1+𝐾)|𝑙

𝑙 𝑙|−1/2(𝑗𝑙 𝑙 𝑙 × ˜ 𝒠(𝑃𝑠; R3))+(1−𝐾)|𝑙 𝑙 𝑙|1/2𝑄tr ˜ 𝒠(𝑃𝑠; R3)

◮ Local approximant: Let 𝑕 : R3 → R be smooth and compactly

supported, ˜ 𝑕(0 0) = 1, consider ˆ 𝑤 𝑤 𝑤𝑄

𝑄 𝑄 (𝑙

𝑙 𝑙) := ˜ 𝛽1/2𝑄tr ˜ 𝑕(𝑙 𝑙 𝑙)𝑓−𝑗𝑣|𝑙

𝑙 𝑙|∇𝐹𝑄 𝑄 𝑄

|𝑙 𝑙 𝑙|3/2(1 − ∇𝐹𝑄

𝑄 𝑄 · ˆ

𝑙 𝑙 𝑙)

◮ Hence, local approximant for 𝑋(−𝑗𝑤

𝑤 𝑤𝑄

𝑄 𝑄 ) is

𝑋(−𝑗ˆ 𝑤 𝑤 𝑤𝑄

𝑄 𝑄,𝑈 ) =

exp (︂ −𝑗˜ 𝛽1/2 ∫︂ 𝑈 𝑒𝑢 ∫︂ 𝑈

𝑢

𝑒𝜐 1 (2𝜌)3/2 ∇𝐹𝑄

𝑄 𝑄 ·𝐹

𝐹 𝐹(𝑕)(−𝑣−𝜐, −∇𝐹𝑄

𝑄 𝑄 𝑢)

)︂

13/13

Conclusions and Outlook

◮ We have investigated the problem of velocity superselection in a

non-relativistic QED model.

◮ Our resolution of this problem rely on two possible methods:

◮ infravacuum state ◮ restriction of the algebra to the forward light-cone

◮ It would be interesting to investigate the problem of scattering

theory of many electrons in these two approaches.

◮ A method like Haag-Ruelle scattering theory could be applied:

◮ In the first approach one needs to make sense of Haag-Ruelle

scattering theory with respect to the infravacuum background state

◮ In the second approach only either an outgoing or an incoming

particle is available at the same time