Construction of Haag-Kastler nets for factorizing S-matrices with - PowerPoint PPT Presentation

Construction of Haag-Kastler nets for factorizing S-matrices with poles Yoh Tanimoto (partly joint with H. Bostelmann and D. Cadamuro) University of Rome “Tor Vergata” Supported by Rita Levi Montalcini grant of MIUR June 4th 2018, Cortona Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 1 / 17

Towards more 2d QFTs Construct Haag-Kastler nets for integrable models for scalar factorizing S-matrices with poles (bound states). Massive, non-perturbative, interacting quantum field theories in d = 2. Methods and results Take the conjectured S-matrix with poles as an input, construct first observables localized in wedges , then prove the existence of local observables indirectly. Observables in wedge : � φ ( ξ ) = z † ( ξ ) + χ ( ξ ) + z ( ξ ) (c.f. Lechner ‘08, φ ( f ) = z † ( f + ) + z ( f + ) for S-matrix without poles). Observables in double cones by intersection. Duality, solitons, bound states, quantum groups... Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 2 / 17

Overview of the strategy Haag-Kastler net ( {A ( O ) } , U , Ω): local observables A ( O ), spacetime symmetry U and the vacuum Ω. Wedge-algebras first: construct A ( W R ) , U , Ω from wedge-local fields , then take the intersection A ( D a , b ) = U ( a ) A ( W R ) U ( a ) ∗ ∩ U ( b ) A ( W R ) ′ U ( b ) ∗ The intersection is large enough if modular nuclearity or wedge-splitting holds. φ ′ such that [ e i � φ ( ξ ) , e i � Wedge-local observables: � φ, � φ ′ ( η ) ] = 0. Examples : scalar analytic factorizing S-matrix (Lechner ’08), twisting by inner symmetry (T. ’14), diagonal S-matrix (Alazzawi-Lechner ’17)... More example? S-matrices with poles . Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 3 / 17

Standard wedge and double cone t t a W R + a a D 0 , a x x W R Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 4 / 17

Analytic factorizing S-matrix Pointlike fields are hard. Larger regions contain better observables. Wedge : W R / L := { ( t , x ) : x > ±| t |} . Wedge-local fields in integrable models (Schroer, Lechner) S : factorizing S-matrix ( without poles ). z † , z : Zamolodchikov-Faddeev algebra (creation and annihilation operators defined on S -symmetric Fock space ). φ ( f ) = z † ( f + ) + z ( f + ), supp f ⊂ W L , is localized in W L . The full QFT The observables A ( W L ) in W L are generated by φ ( f ). For diamonds D a , b , define A ( D a , b ) = A ( W L + a ) ∩ A ( W R + b ). Examine the boost operator to show the existence of local operators ( modular nuclearity (Buchholz, D’antoni, Longo, Lechner)). Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 5 / 17

Wedge observables for analytic S-matrix Input: analytic function S : R + i (0 , π ) → C , S ( θ ) = S ( θ ) − 1 = S ( − θ ) = S ( θ + π i ) , θ ∈ R . S -symmetric Fock space: H 1 = L 2 ( R , d θ ), H n = P n H ⊗ n 1 , where P n is the projection onto S -symmetric functions: Ψ n ( θ 1 , · · · , θ n ) = S ( θ k +1 − θ k )Ψ n ( θ 1 , · · · , θ k +1 , θ k , · · · , θ n ) . S -symmetrized creation and annihilation operators (ZF-algebra): z † ( ξ ) = Pa † ( ξ ) P , z ( ξ ) = Pa ( ξ ) P , P = � n P n . φ ( f ) = z † ( f + ) + z ( J 1 f − ), Wedge-local field (Lechner ‘03): � f ± ( θ ) = dx e ± ix · p ( θ ) f ( x ) , p ( θ ) = ( m cosh θ, m cosh θ ) , J 1 is the one-particle CPT operator, φ ′ ( g ) = J φ ( g j ) J , g j ( x ) = g ( − x ). If supp f ⊂ W L , supp g ⊂ W R , then [ e i φ ( f ) , e i φ ′ ( g ) ] = 0. Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 6 / 17

S-matrix with poles If S has a pole: [ φ ( f ) , φ ′ ( g )]Ψ 1 ( θ 1 ) = � � f + ( θ ) g − ( θ ) S ( θ 1 − θ ) − f + ( θ + π i ) g − ( θ + π i ) S ( θ 1 − θ + π i ) � − d θ × Ψ 1 ( θ 1 ) obtains the residue of S and does not vanish. Example (the Bullough-Dodd model): poles at θ = π i 3 , 2 π i 3 , residues − R , R � � � � � � θ − (1 − ε ) π θ − (1+ ε ) π i tanh 1 θ + 2 π i tanh 1 tanh 1 2 3 2 3 2 3 � · S ε ( θ ) = � � � � � , θ + (1 − ε ) π i θ + (1+ ε ) π i tanh 1 θ − 2 π i tanh 1 tanh 1 2 3 2 3 2 3 � � � � where 0 < ε < 1 θ + π i θ − π i 2 . S ε ( θ ) = S ε S ε . 3 3 New wedge-local field? Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 7 / 17

The bound state operator � � � � S : two-particle S-matrix, poles θ = π i 3 , 2 π i θ + π i θ − π i 3 , S ( θ ) = S S 3 3 P n : S -symmetrization, H = � P n H ⊗ n 1 , H 1 = L 2 ( R ), Dom ( χ 1 ( ξ )) : to be defined � � � � � θ + π i θ − π i ( χ 1 ( ξ ))Ψ 1 ( θ ) := 2 π | R | ξ Ψ 1 , R = Res ζ = 2 π i 3 S ( ζ ) 3 3 New observables : � χ ( ξ ) := χ n ( ξ ) , χ n ( ξ ) = nP n ( χ 1 ( ξ ) ⊗ ✶ ⊗ · · · ⊗ ✶ ) P n , � (= z † ( ξ ) + χ ( ξ ) + z ( ξ )) , φ ( ξ ) := φ ( ξ ) + χ ( ξ ) φ ′ ( η ) := J � � χ ′ ( η ) = J χ ( J 1 η ) J . φ ( J 1 η ) J , Theorem (Cadamuro-T. arXiv:1502.01313) ξ : L 2 bounded analytic in R + i (0 , π ) “real”, η : L 2 bounded analytic in R + i ( − π, 0) “real”, then � � φ ( ξ )Φ , � φ ′ ( η )Ψ � = � � φ ′ ( η )Φ , � φ ( ξ )Ψ � on a dense domain. Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 8 / 17

The one-particle bound state operator ξ ( ζ ): analytic in R + i (0 , π ), ξ ( θ + π i ) = ξ ( θ ) (“real”). H 1 = L 2 ( R ) D 0 = H 2 ( − π 3 , π 3 ): L 2 -analytic functions in R + i ( − π 3 , π 3 ) � 2 π | R | ξ ( θ + π i 3 )Ψ 1 ( θ − π i ( χ 1 ( ξ ))Ψ 1 ( θ ) := 3 ) What are self-adjoint extensions of χ 1 ( ξ )? Many extensions : n ± ( χ 1 ( ξ )) = “half of the zeros” of ξ Choose ξ = ξ 2 0 , no zeros, no singular part (Beurling decomposition). �� d θ P ( θ + 2 π i � Set ξ + ( θ + π i 3 ) log | ξ ( θ + π i 3 ) | 3 ) = exp , where P ( θ ) is the Poisson kernel for { ζ : π 3 < Re ζ < 2 π 3 } . 1 χ 1 ( ξ ) := M ∗ ξ + ∆ 1 M ξ + is self-adjoint and a natural extension of the 6 1 1 Ψ 1 )( θ ) = Ψ 1 ( θ − π i above, M ξ + is unitary, (∆ 6 3 ). Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 9 / 17

Towards proof of strong commutativity 1 Note: χ 1 ( ξ ) = M ∗ ξ + ∆ 1 M ξ + have different domains for different ξ . 6 � χ ( ξ ) := χ n ( ξ ) , χ n ( ξ ) = nP n ( χ 1 ( ξ ) ⊗ ✶ ⊗ · · · ⊗ ✶ ) P n � � 1 = nM ∗⊗ n P n M ⊗ n ∆ 1 ⊗ ✶ ⊗ · · · ⊗ ✶ 6 ξ + P n ξ + . If χ ( ξ ) + χ ′ ( η ) is self-adjoint , then... χ ( ξ ) + χ ′ ( η ) + cN is self-adjoint. T ( ξ, η ) := � φ ( ξ ) + � φ ′ ( η ) + cN is self-adjoint by Kato-Rellich. (= χ ( ξ ) + χ ′ ( η ) + cN + φ ( ξ ) + φ ′ ( η )) [ T ( ξ, η ) , � φ ( ξ )] = [ cN , � φ ( ξ )] = [ cN , φ ( ξ )] is small, � � φ ( ξ )Ψ � ≤ � T ( ξ, η )Ψ � . use Driessler-Fr¨ ohlich theorem (weak ⇒ strong commutativity : [ e i � φ ( ξ ) , e i � φ ′ ( η ) ] = 0) with T ( ξ, η ) as the reference operator. Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 10 / 17

Self-adjointness of χ n ( ξ ) + χ ′ n ( η ) We exhibit the proof for 1 1 1 χ 2 ( ξ ) ∼ 1 ) M ∗ = P 2 (∆ 1 ⊗ ✶ ) P 2 6 ⊂ ∆ 1 ⊗ ✶ + M S ( ✶ ⊗ ∆ 6 6 on H 1 ⊗ H 1 . S Dom = L 2 -functions Ψ( θ 1 , θ 2 ) analytic in θ 1 in R + i ( − π i 3 , 0) and s.t. S ( θ 1 − θ 2 )Ψ( θ 1 , θ 2 ) analytic in θ 2 in R + i ( − π i 3 , 0). Lemma (Kato-Rellich+) If A , B , A + B are self-adjoint, and assume that there is δ > 0 such that Re � A Ψ , B Ψ � > ( δ − 1) � A Ψ �� B Ψ � for Ψ ∈ Dom ( A + B ) . If T is a symmetric operator such that Dom ( A ) ⊂ Dom ( T ) and � T Ψ � 2 < δ � A Ψ � 2 , then A + B + T is self-adjoint. 1 1 1 is self-adjoint. Dom = L 2 -functions Ψ( θ 1 , θ 2 ) both ∆ 1 ⊗ ✶ + ✶ ⊗ ∆ 6 6 analytic in θ 1 and in θ 2 . Y. Tanimoto (Tor Vergata University) Integrable QFT with bound states 04/06/2018, Cortona 11 / 17