Radiative corrections to the binding energy for a spin 1 / 2 charged - PowerPoint PPT Presentation

Radiative corrections to the binding energy for a spin 1 / 2 charged particle (Toulon 2014) Semjon Wugalter Joint works with Jean-Marie Barbaroux (University of Toulon ) Semjon Wugalter NRQED binding energy

• “Quantitative estimates on the binding energy for Hydrogen in non-relativistic QED. II. The spin case.”, arXiv :1306 :4464 (2013) J.-M. Barbaroux, S. W. • “Contribution of the Spin-Zeeman term to the binding energy for hydrogen in non-relativistic QED”, Annals of the University of Bucharest (2013) J.-M. Barbaroux, S.W. • “On the ground state energy of the translation invariant Pauli-Fierz model. II.”, Documenta Mathematica (2012). J.-M. Barbaroux., S.W. • “Non-analyticity of the ground state energy of the Hamiltonian for Hydrogen atom in non-relativistic QED”, Journal of Physics A : Mathematical and Theoretical (2010) J.-M. Barbaroux., S. W. • “Quantitative estimates on the binding energy for Hydrogen in non-relativistic QED”, Annales Henri Poincaré (2010). J.-M. Barbaroux., Thomas Chen, Vitali Vougalter, S.W. • “On the ground state energy of the translation invariant Pauli-Fierz model”, Proc. Amer. Math. Soc., 136 (3), 1057-1064 (2008). J.-M. Barbaroux., Thomas Chen, Vitali Vougalter, S.W. • “Quantitative estimates on the enhanced binding for the Pauli-Fierz operator”, J. Math. Phys., vol. 46, no12 (2005). J.-M. Barbaroux., Helmut Linde, S. W. • “Binding conditions for atomic N-electron systems in non-relativistic QED”, Ann. Henri Poincaré. 4 (6), 1101 - 1136 (2003). J.-M. Barbaroux., Thomas Chen, S. W. Semjon Wugalter NRQED binding energy

Pauli-Fierz Operator 1 Ground state - Binding energy 2 Unitary transform - Self-Energy - Change of units 3 Preliminary results 4 Increase of the binding energy - spin case 5 Semjon Wugalter NRQED binding energy

Description We study the Hamiltonian in NRQED for an atom. ◮ System with 1 electron, described as quantum, non relativistic , pointwise particle with charge − e and spin 1 2 ◮ The electron interacts with the quantized magnetic field ◮ One static pointwise nucleus, with positive charge - The electron interacts with the field of the nucleus via the Coulomb potential. ◮ One also study the free case (“self-energy” operator). Semjon Wugalter NRQED binding energy

Introduction - Schrödinger Hamiltonian Hamiltonian One electron interacting with a pointwise nucleus of charge e , ( Z = 1). H p = − ∆ + V Coulomb Potential : V ( x ) = − e 2 | x | Electron mass m = 1 / 2 ; Planck constant � = 1 ; velocity of light c = 1. Fine structure constant : α = e 2 ≈ 1 / 137 Binding energy Energy necessary to remove the electron to spatial infinity : Σ( 0 ) − Σ( V ) = inf σ ( − ∆) − inf σ ( − ∆ − α/ | x | ) . Semjon Wugalter NRQED binding energy

Introduction - Schrödinger Hamiltonian � R 3 1 | x | | ψ ( x ) | 2 d x ≤ �∇ ψ � � ψ � ⊲ Coulomb uncertainty principle : ⊲ � ψ, ( − ∆ + V ) ψ � ≥ �∇ ψ � 2 − α �∇ ψ � � ψ � � 2 � �∇ ψ � 2 − e 2 − α 2 4 � ψ � 2 = 2 � ψ � � �� � ≥ 0 ⊲ Σ( V ) = inf σ ( − ∆ + V ) = − α 2 4 . Binding energy Σ( 0 ) − Σ( V ) = inf σ ( − ∆) − inf σ ( − ∆ + V ) = α 2 4 . Semjon Wugalter NRQED binding energy

Coupling to the quantized radiation field : Pauli-Fierz Hamiltonian Hamiltonian (Coulomb gauge) N = 1 electron Coulomb potential case : Pauli-Fierz operator − i ∇ x ⊗ I f + √ α A ( x ) α Z � � 2 H PF = − | x | ⊗ I f � �� � � �� � kinetic energy Coulomb electrostatic potential + ( q − 1 ) √ ασ · B ( x ) + I el ⊗ H f � �� � � �� � Zeeman term radiation field energy operator Free case : “self-energy” operator T self.en. = ( − i ∇ x + √ α A ( x )) 2 +( q − 1 ) √ ασ · B ( x ) + H f � �� � � �� � ���� kinetic energy Zeeman term Field energy System of units : Electron mass m = 1 / 2 ; Planck const. � = 1 ; speed of light c = 1 ; fine structure constant : α = e 2 ≈ 1 / 137 Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian Hilbert space H = H part ⊗ F s ◮ H part = L 2 ( R 3 , C q ) : Hilbert space for N = 1 electron. R 3 is configuration space, C q for spin q = 1 : “spinless” particule ; q = 2 : electron (with spin) ◮ F s : Bosonic Fock space   ∞ � � n L 2 ( R 3 , C 2 ) F s = Ω f C ⊕   s � �� � n = 1 one photon space (momentum variable × 2 polarizations transv.) � �� � F ( n ) n -photon space s ◮ Vacuum : Ω f . Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian ◮ creation/annihilation operators : a ∗ λ ( k ) , a λ ( k ) . Fulfils C.C.R : λ ′ ( k )] = δ λ,λ ′ δ ( k − k ′ ) , [ a ♯ λ ( k ) , a ♯ [ a λ ( k ) , a ∗ λ ′ ( k )] = 0, a λ ( k )Ω f = 0 ◮ Field energy : � � ω ( k ) a ∗ H f = λ ( k ) a λ ( k ) d k , ω ( k ) = | k | λ = 1 , 2 H f = � n H f ( n ) , H f Ω f = 0, ( H ( n ) Φ ( n ) )( k 1 , k 2 , · · · , k n ) = � n j = 1 | k j | Φ ( n ) ( k 1 , k 2 , · · · , k n ) f ◮ Photon number operator : � � a ∗ N f = λ ( k ) a λ ( k ) d k λ = 1 , 2 i.e., ( N f Φ) ( n ) ( k 1 , · · · , k n ) = n Φ ( n ) ( k 1 , · · · , k n ) . Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian ◮ Magnetic vector potential : (Coulomb gauge) A ( x ) = A − ( x ) + A + ( x ) � χ Λ ( | k | ) � ǫ λ ( k ) e ik . x a λ ( k ) d k + h . c . = 1 2 π | k | 2 λ = 1 , 2 ◮ Polarization vectors : ǫ λ ( k ) , ǫ 1 ( k ) · ǫ 2 ( k ) = 0, k · ǫ λ ( k ) = 0. ◮ UV (Ultraviolet) cutoff : χ Λ ( | k | ) ◮ Coupling between electron and quantized magnetic field √ ασ · B , with B = Curl A , � χ Λ ( | k | ) � 2 π | k | 1 / 2 k × i ε λ ( k ) e ikx a λ ( k ) d k + h . c . , B ( x ) = R 3 λ = 1 , 2 and σ = ( σ 1 , σ 2 , σ 3 ) , σ i are 2 × 2 Pauli matrices. Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian H PF =( P −√ α A ( x )) 2 + V +( q − 1 ) √ ασ · B ( x )+ H f , V = − α | x | , P = i ∇ x And also H PF = H p + H f + H I ( α ) where H p = ( − ∆ + V ) ⊗ I f H f = field energy operator H I ( α )= interaction = − 2Re √ α P . A ( x )+ α A ( x ) 2 + √ ασ · B ( x ) Semjon Wugalter NRQED binding energy

Self-adjointness Hamiltonian • The Hamiltonian H PF is self-adjoint, with domain D ( H part. + H f ) • Stability of the first kind : inf σ ( H PF ) > −∞ • Stability of the second kind : N electrons and M nuclei with charge Z k ( k = 1 , ..., M ) inf σ ( H PF ) ≥ − C (Λ , max { Z k } ) ( M + N ) Ground state inf σ ( H PF ) is an eigenvalue of multiplicity q . Semjon Wugalter NRQED binding energy

Binding energy Hamiltonian H = L 2 ( R 3 ) ⊗ F H PF = T + V ⊗ I f sur T = ( − i ∇ x ⊗ I f + √ α A ( x )) 2 + I el ⊗ H f − c n . o . α Binding energy : Σ α ( 0 ) − Σ α ( V ) = inf σ ( T ) − inf σ ( T + V ) Remark : H PF = T + V = − ∆ x + V + H f + ( − 2Re √ α A ( x ) · i ∇ x + α A ( x ) 2 − c n . o . α ) � �� � := H I ( α ) Semjon Wugalter NRQED binding energy

What is expected ? H PF = − ∆ x + V + H f + H I ( α ) The free particle binds a larger quantity of (low-energetic) 1 photons than the confined particle. The binding energy should increase : 2 Σ α ( 0 ) − Σ α ( V ) > Σ( 0 ) − Σ( V ) = α 2 4 Semjon Wugalter NRQED binding energy

Unitary transform U = e iP f · x The e − momentum variable is shifted by P f = � � k a ∗ λ ( k ) a λ ( k ) d k . λ The photon “position” is shifted by x . • U ( i ∇ x ) U ∗ = i ∇ x − P f ( i ∇ x acquires the meaning of the total momentum, i.e., momentum of particle + field). • U A ( x ) U ∗ = A ( 0 ) U B ( x ) U ∗ = B ( 0 ) . and • U T U ∗ = ( P − P f − √ α A ( 0 )) 2 + ( q − 1 ) σ. B ( 0 ) + H f − c n . o . α , P := i ∇ x . • U ( T + V ) U ∗ = U H PF U ∗ = P − P f − √ α A ( 0 ) � 2 + ( q − 1 ) σ. B ( 0 ) + H f � − α | x | − c n . o . α � �� � T Semjon Wugalter NRQED binding energy

Unitary transform Equivalent Hamiltonian H := U ( T + V ) U ∗ P − P f − √ α A ( 0 ) − α � 2 + ( q − 1 ) σ. B ( 0 ) + H f � = | x | − c n . o α � �� � T + ( P f + √ α A ( 0 )) 2 + ( q − 1 ) σ. B ( 0 ) + H f − c n . o α = ( P 2 − α | x | ) � �� � � �� � Self-Energy with total momentum 0, T ( 0 ) Schrödinger operator − 2Re P . ( P f + √ α A ( 0 )) Semjon Wugalter NRQED binding energy

Operator Self-Energy Self-energy at fixed total momentum : The operator T = − ∆ + H f + H I ( α ) commutes with the total momentum P tot , � � k a ∗ P tot = ( i ∇ x ⊗ I f ) + ( I el ⊗ P f ) , P f = λ ( k ) a λ ( k ) d k λ � ⊕ T ( p ) acting on H 0 ≃ C q ⊗ F T ≃ R 3 T ( p ) d p Theorem ( [F’74], [CF’07], [C’08] ) inf σ ( T ( 0 )) = inf σ ( T ) = Σ α ( 0 ) Σ α ( 0 ) is an eigenvalue of T ( 0 ) : T ( 0 )Ψ GS 0 = Σ α ( 0 )Ψ GS 0 . Semjon Wugalter NRQED binding energy