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

Problem: System of nonrelativistic QED: one “slow” spinless electron interacting with a cloud of photons ◮ Algebra of observables of the system electron + photons ◮ Coherent states 𝜕 𝑄 𝑄 (ground states of an Hamiltonian 𝐼 𝑄 𝑄 , 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 cannot be 𝑄 coherently superimposed ◮ electron is an infraparticle (no definite mass) ◮ scattering theory of many electrons seems problematic 1/13

The model ◮ Hamiltonian: 𝐼 = 1 𝑦 )) 2 + 𝐼 photon 𝛽 1 / 2 𝐵 2( − 𝑗 ∇ 𝑦 𝑦 + ˜ 𝐵 ( 𝑦 𝐵 𝑦 𝑦 selfadjoint on dense domain in ℋ = ℋ electron ⊗ ℱ photon , 𝐵 𝐵 𝐵 in Coulomb gauge with UV cutoff. ◮ total momentum 𝑄 ∇ 𝑄 := − 𝑗 ∇ 𝑄 ∇ 𝑦 𝑦 + 𝑄 𝑄 𝑄 photon , [ 𝐼,𝑄 𝑄 𝑄 ] = 0 𝑦 𝐼 = Π * (︁ ∫︂ ⊕ )︁ 𝑄 𝑒 3 𝑄 ⇒ 𝐼 𝑄 𝑄 𝑄 Π , Π a unitary identification 𝑄 2/13

Ground states of the Hamiltonians H 𝑄 𝑄 𝑄 ◮ Absence of ground states: ◮ 𝐼 P P do not have ground states (eigenvectors) for 𝑄 𝑄 𝑄 ̸ = 0 P 𝑄 ∈ R 3 : | 𝑄 𝑄 | < 1 at least for small ˜ 𝛽 and for 𝑄 𝑄 𝑄 ∈ 𝑇 = { 𝑄 𝑄 𝑄 3 } . ◮ This is a feature of the infraparticle problem ◮ Introduce an infrared cutoff: 𝑄,𝜏 := 1 𝐵 [ 𝜏,𝜆 ] (0)) 2 + 𝐼 photon 𝛽 1 / 2 𝐵 𝐼 𝑄 2( 𝑄 𝑄 𝑄 − 𝑄 𝑄 𝑄 photon + ˜ 𝐵 𝑄 selfadjoint on dense domain in Fock space ℱ over 𝑀 2 tr ( R 3 ; C 3 ) ; denote creators/annihilators as 𝑏 * 𝜇 ( 𝑙 𝑙 ) , 𝑏 𝜇 ( 𝑙 𝑙 𝑙 ) . 𝑙 ∫︂ 𝑒 3 𝑙 𝑙 𝑙 ∑︂ (︁ )︁ 𝑓 − 𝑗𝑙 𝑙 𝑙 · 𝑦 𝑦 𝑦 𝑏 * 𝑙 )+ 𝑓 𝑗𝑙 𝑙 · 𝑦 𝑙 𝑦 𝑏 𝜇 ( 𝑙 𝑦 𝐵 [ 𝜏,𝜆 ] ( 𝑦 𝑦 ) = 𝑦 𝜓 [ 𝜏,𝜆 ] ( | 𝑙 𝑙 𝑙 | ) 𝜗 𝜗 𝜗 𝜇 ( 𝑙 𝑙 ) 𝑙 𝜇 ( 𝑙 𝑙 𝑙 𝑙 ) √︁ | 𝑙 𝑙 | 𝑙 𝜇 = ± ( 𝜆 : UV cutoff, 𝜏 : IR cutoff) 3/13

Ground states with IR cutoff ◮ Fact: For any 𝜏 > 0 , the operator 𝐼 𝑄 𝑄,𝜏 has a ground states 𝑄 (eigenvector) Ψ 𝑄 𝑄,𝜏 ∈ ℱ with isolated eigenvalues 𝐹 𝑄 𝑄,𝜏 . 𝑄 𝑄 ◮ Ψ P P ,σ tend weakly to zero as 𝜏 → 0 . P ◮ 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. 4/13

Velocity superselection Algebra of observables of the system “electron + photon cloud”: ◮ Weyl (CCR) algebra A generated (up to closure in 𝐷 * -norm) by 𝑔 ∈ ℒ := ⋃︁ 𝜗> 0 𝑀 2 tr ,𝜗 ( R 3 ; C 3 ) , 𝑋 ( 𝑔 𝑔 𝑔 ) , 𝑔 𝑔 symplectic form 𝜏 ( · , · ) := Im ⟨· , ·⟩ . 𝑔 )) = 𝑓 𝑏 * ( 𝑔 𝑔 𝑔 ) − 𝑏 ( 𝑔 𝑔 ) 𝑔 ◮ Vacuum representation: 𝜌 vac ( 𝑋 ( 𝑔 𝑔 State: Given any 𝐵 ∈ A , define 𝜕 𝑄 𝑄 ( 𝐵 ) := lim 𝜏 → 0 ⟨ Ψ 𝑄 𝑄,𝜏 , 𝜌 vac ( 𝐵 )Ψ 𝑄 𝑄,𝜏 ⟩ 𝑄 𝑄 𝑄 ◮ state on A , describes plane-wave configurations of the electron with velocity 𝑄 𝑄 𝑄 Representations: States 𝜕 𝑄 𝑄 have irreducible GNS representations 𝜌 𝑄 𝑄 . 𝑄 𝑄 𝑄 ′ ◮ Fact: 𝜌 𝑄 𝑄 ̸≃ 𝜌 𝑄 𝑄 ′ for any 𝑄 𝑄 𝑄 ̸ = 𝑄 𝑄 “velocity superselection” 𝑄 𝑄 5/13

Cause of the superselection problem Analyze the phenomenon closely: ◮ introduce auxiliary vectors Φ 𝑄 𝑄,𝜏 = 𝑋 ( 𝑤 𝑤 𝑄,𝜏 )Ψ 𝑄 𝑄,𝜏 , where 𝑤 𝑄 𝑄 𝑄 𝑄 𝜓 [ 𝜏,𝜆 ] ( | 𝑙 𝑙 𝑙 | ) ∇ 𝐹 𝑄 𝑄 𝑄,𝜏 𝛽 1 / 2 𝑄 tr 𝑤 𝑄,𝜏 ( 𝑙 𝑙 𝑙 ) := ˜ 𝑤 𝑤 𝑄 . 𝑄 𝑙 | 3 / 2 1 − ˆ | 𝑙 𝑙 𝑙 𝑙 𝑙 · ∇ 𝐹 𝑄 𝑄 𝑄,𝜏 Fact: Φ 𝑄 𝑄 := lim 𝜏 → 0 Φ 𝑄 𝑄,𝜏 exists in norm for suitable choice of 𝑄 𝑄 the phases of Ψ 𝑄 𝑄,𝜏 . 𝑄 (︁ )︁ 𝑄,𝜏 ) * 𝑄 ( 𝑋 ( 𝑔 𝑔 𝑔 )) = lim 𝜏 → 0 ⟨ Φ 𝑄 𝑋 ( 𝑤 𝑤 𝑄,𝜏 ) 𝑋 ( 𝑔 𝑔 𝑔 ) 𝑋 ( 𝑤 𝑤 Φ 𝑄 𝑄,𝜏 ⟩ 𝜕 𝑄 𝑄,𝜏 , 𝜌 vac 𝑤 𝑄 𝑤 𝑄 𝑄 𝑄 𝑄 𝑄 𝑄 ⏟ ⏞ := 𝛽 v P ,σ ( 𝑋 ( 𝑔 𝑔 𝑔 )) v vP P 𝑤 𝑔 𝑔 )) = 𝑓 − 2 𝑗 Im ⟨ 𝑤 𝑤 P P ,σ ,𝑔 𝑔 ⟩ 𝑋 ( 𝑔 𝛽 𝑤 P ,σ ( 𝑋 ( 𝑔 𝑔 𝑔 ) 𝑔 P 𝑤 𝑤 P P ◮ For 𝜏 > 0 , we have 𝜌 𝑄 𝑄,𝜏 ≃ 𝜌 vac , but 𝜌 𝑄 𝑄 ̸≃ 𝜌 vac 𝑄 𝑄 A possible solution: regularize the map 𝛽 𝑤 P ,σ ⇒ Infravacuum state 𝑤 𝑤 P P 6/13

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 𝛽 𝑤 P P P ◮ 𝛿 have poor localization property in front of the vacuum: ⃒ ⃒ ⃒ ⃒ 𝜌 vac ∘ 𝛿 A ( 𝒫 ′ ) ̸≃ 𝜌 vac A ( 𝒫 ′ ) , 𝜌 vac ∘ 𝛿 A ( 𝒟 ′ ) ̸≃ 𝜌 vac ⃒ ⃒ ⃒ ⃒ 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: 𝜌 𝑈 ∘ 𝛿 A ( 𝒟 ′ ) ≃ 𝜌 𝑈 ⃒ ⃒ A ( 𝒟 ′ ) ◮ automorphism of the algebra 𝛽 𝑈 : 𝛽 𝑈 ( 𝑋 ( 𝑔 )) = 𝑋 ( 𝑈𝑔 ) 7/13

The symplectic map T ◮ Recall ℒ := ⋃︁ 𝜗> 0 𝑀 2 tr ,𝜗 ( R 3 ; C 3 ) 𝑈 = 𝑈 1 1+ 𝐾 + 𝑈 2 1 − 𝐾 ◮ 𝑈 : ℒ → ℒ , 2 2 𝑜 𝑜 (︁ 1 ∑︂ ∑︂ )︁ 1 1 𝑈 1 := 1 1+ s- lim ( 𝑐 𝑗 − 1) 𝑅 𝑅 𝑗 , 𝑅 𝑈 2 := 1 1+ s- lim − 1 𝑅 𝑅 𝑅 𝑗 𝑐 𝑗 𝑜 →∞ 𝑜 →∞ 𝑗 =1 𝑗 =1 tr ( R 3 ; C 3 ) , ∑︁ ◮ 𝑅 𝑅 𝑅 i orthogonal projectors on 𝑀 2 𝑅 𝑅 i = 1 i 𝑅 ◮ 𝑗 large means “low energy” ◮ 𝑐 i decay with 𝑗 large ◮ 𝑈 modify the low energy behaviour of wave functions in ℒ , and in particular of 𝑤 𝑤 𝑤 𝑄 𝑄,𝜏 , in such a way that 𝑄 𝑄,𝜏 ∈ 𝑀 2 tr ( R 3 ; C 3 ) . lim 𝜏 → 0 𝑈𝑤 𝑤 𝑤 𝑄 𝑄 8/13

Infravacuum state ◮ Idea: Instead of 𝜕 𝑄 𝑄 , consider a modified state 𝜕 𝑄 𝑄,𝑈 defined by 𝑄 𝑄 (︁ )︁ 𝑄,𝑈 ( 𝐵 ) := lim 𝜏 → 0 ⟨ Φ 𝑄 𝛽 𝑈 ( 𝛽 𝑤 P ,σ ( 𝐵 )) Φ 𝑄 𝑄,𝜏 ⟩ 𝜕 𝑄 𝑄,𝜏 , 𝜌 vac 𝑄 𝑄 𝑤 𝑤 P 𝑄 P (︁ )︁ 𝑄,𝜏 ) * ) = lim 𝜏 → 0 ⟨ Φ 𝑄 𝑄,𝜏 , 𝜌 vac 𝛽 𝑈 ( 𝑋 ( 𝑤 𝑤 𝑤 𝑄 𝑄,𝜏 ) 𝐵𝑋 ( 𝑤 𝑤 𝑤 𝑄 Φ 𝑄 𝑄,𝜏 ⟩ 𝑄 𝑄 𝑄 𝑄 (︁ 𝑄,𝜏 ) * )︁ = lim 𝜏 → 0 ⟨ Φ 𝑄 𝑋 ( 𝑈𝑤 𝑤 𝑄,𝜏 ) 𝛽 𝑈 ( 𝐵 ) 𝑋 ( 𝑈𝑤 𝑤 Φ 𝑄 𝑄,𝜏 ⟩ 𝑄,𝜏 , 𝜌 vac 𝑤 𝑄 𝑤 𝑄 𝑄 𝑄 𝑄 𝑄 ◮ Fact: lim 𝜏 → 0 𝑈𝑤 𝑄 ∈ 𝑀 2 tr ( R 3 ; C 3 ) 𝑤 𝑄,𝜏 := 𝑈𝑤 𝑤 𝑤 𝑄 𝑤 𝑄 𝑄 𝑄 𝑄 ′ . ◮ Result: 𝜌 𝑄 𝑄,𝑈 ≃ 𝜌 𝑄 𝑄 ′ ,𝑈 for any 𝑄 𝑄 𝑄 ̸ = 𝑄 𝑄 𝑄 𝑄 9/13

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 are well-defined. 𝑄 ◮ choose A to be the algebra of observables of the free electromagnetic field → local and relativistic A ( 𝒫 ) := 𝑔 b )) | supp 𝑔 𝐹 𝑔 𝐶 𝑔 𝐷 * { 𝑓 𝑗 ( 𝐹 𝐹 ( 𝑔 𝑔 e )+ 𝐶 𝐶 ( 𝑔 𝑔 𝑓,𝑐 ∈ 𝒠 ( R 4 , R 3 ) } 𝑔 𝑔 𝑓 , supp 𝑔 𝑔 𝑔 𝑐 ⊂ 𝒫 ,𝑔 𝑔 ◮ Result: if A := ⋃︁ 𝒫⊂ R 4 A ( 𝒫 ) (quasi-local algebra), 𝑄 ′ , but with A ( 𝑊 + ) := ⋃︁ then 𝜌 𝑄 𝑄 ̸≃ 𝜌 𝑄 𝒫⊂ 𝑊 + A ( 𝒫 ) , 𝑄 𝑄 ⃒ ⃒ 𝑄 ′ ∈ 𝑇 𝜌 𝑄 A ( 𝑊 + ) ≃ 𝜌 𝑄 A ( 𝑊 + ) for any 𝑄 𝑄,𝑄 𝑄 𝑄 ⃒ ⃒ 𝑄 𝑄 𝑄 ′ 𝑄 ( 𝑊 + : forward light cone) 10/13

Restriction to the light cone Recall that (︁ 𝑄,𝜏 ) * )︁ 𝑄 ( 𝐵 ) = lim 𝜏 → 0 ⟨ Φ 𝑄 𝑋 ( 𝑤 𝑤 𝑄,𝜏 ) 𝐵𝑋 ( 𝑤 𝑤 Φ 𝑄 𝑄,𝜏 ⟩ 𝜕 𝑄 𝑄,𝜏 , 𝜌 vac 𝑤 𝑄 𝑤 𝑄 𝑄 𝑄 𝑄 𝑄 𝑄 Idea of proof: ◮ Use Huygens principle: A ( 𝑊 − ) ⊂ A ( 𝑊 + ) ′ ◮ Approximate 𝑤 𝑤 𝑄,𝜏 with functions in the symplectic space of the 𝑤 𝑄 𝑄 backward light cone 𝑊 − . ◮ Then 𝑋 ( 𝑤 𝑤 𝑄,𝜏 ) and 𝐵 ∈ A ( 𝑊 + ) approximately commute, hence 𝑤 𝑄 𝑄 𝜕 𝑄 𝑄 lives in the vacuum representation. 𝑄 ⃒ ⃒ 𝑄 ′ ∈ 𝑇 . ◮ Hence 𝜌 𝑄 A ( 𝑊 + ) ≃ 𝜌 vac ≃ 𝜌 𝑄 A ( 𝑊 + ) for any 𝑄 𝑄 𝑄,𝑄 𝑄 ⃒ ⃒ 𝑄 𝑄 𝑄 𝑄 ′ 11/13