Direct construction of pointlike observables in the Ising model
aa r X i v : . [ m a t h - ph ] J un Direct construction of pointlike observables in the Ising model
Daniela Cadamuro
Relativistic quantum field theories are described by their set of local observ-ables. These are linear bounded or unbounded operators associated with regionsof Minkowski space. They form ∗ -algebras that are expected to satisfy, e.g., theHaag-Kastler axioms, which are relevant to their interpretation as physical “mea-surements”.The problem of constructing models of quantum field theory, i.e., exhibitingalgebras of local observables with such properties, is a notoriously hard task dueto the complicated structure of local observables in the presence of interaction.Quantum integrable models in 1+1-dimensional Minkowski space are simplifiedmodels of interaction, rendering the mathematical structure of quantum field the-ory more accessible. In these models, the scattering of n particles is the product oftwo particle scattering processes, namely the S -matrix is said to be “factorizing”,a property connected to integrability. Examples include the Ising model, the O ( N )nonlinear sigma models and the Sine-Gordon model.We are interested in studying the content of local observables in these theories.This can be investigated in various mathematical frameworks: as Wightman fields[1], as algebras of bounded operators [2], or as closed operators affiliated with thosealgebras. For example, the task of constructing the Wightman n -point functionsin integrable models from a given S -matrix has been widely studied, see, e.g., [3],but convergence of the associated series expansions has not been established sofar, despite some progress [4].An alternative approach considers fields localized in unbounded wedge-shapedregions as an intermediate step to the construction of sharply localized objects,which is handled indirectly [5, 6, 7, 8], thus avoiding explicit computation of point-like fields. The existence proof of local observables is reduced to an abstract condi-tion on the underlying wedge algebras. While the generators of the wedge algebrasare explicitly known, the passage to the von Neumann algebras includes the weaklimit points of this set. These limit points include the elements of local algebras,but of these much less is known. Our task is to gain more information on the struc-ture of these local observables, in the form of smeared pointlike quantum fields.For this, we use a novel approach: instead of considering the n -point functions ofpointlike fields and verifying the Wightman axioms, we establish them as closedoperators affiliated with the local algebras above.In joint work with H. Bostelmann [9] we characterize the local observables interms of a family of coefficient functions f [ A ] m,n in the following series expansion:(1) A = ∞ X m,n =0 Z d m θθθd n ηηηm ! n ! f [ A ] m,n ( θθθ, ηηη ) z † ( θ ) · · · z † ( θ m ) z ( η ) · · · z ( η n ) , where z † , z are “interacting” creators and annihilators fulfilling a deformed versionof the CCR relations which involves the scattering function. ue to the form of this expansion, local observables are defined as quadraticforms in a suitable class. We denote H ω,f the dense space of finite particle numberstates Ψ fulfilling the condition k e ω ( H/µ ) Ψ k < ∞ , where H is the Hamiltonian, µ > ω : [0 , ∞ ) → [0 , ∞ ) is a function with the properties of [10,Definition 2.1]. In (1), A is a quadratic form on H ω,f × H ω,f such that k Q k Ae − ω ( H/µ ) Q k k + k Q k e − ω ( H/µ ) AQ k k < ∞ for any k ∈ N , where Q k is the projector onto the space of k or fewer particles.We denote this class of quadratic forms by Q ω .In order to characterize the coefficients f [ A ] m,n in terms of the localization of A in spacetime, we need a notion of locality which is adapted to quadratic forms inthe class Q ω : We say that A ∈ Q ω is ω -local in the double cone O x,y := W x ∩ W ′ y (where W x denotes the right wedge with edge at x and W ′ y the left wedge withedge at y , with x to the left of y ) if and only if [ A, φ ( f )] = [ A, φ ′ ( g )] = 0 forall f ∈ D ω ( W ′ y ) and all g ∈ D ω ( W x ), as a relation in Q ω . Here φ, φ ′ are the leftand right wedge-local fields, respectively, D ω ( W x ) is the space of smooth functionscompactly supported in W x with the property that θ e ω (cosh θ ) f ± ( θ ) is boundedand square integrable ( f ± is positive and negative frequency part of the Fouriertransform, respectively.)The notion of ω -locality is weaker than the usual notion of locality in the net of C ∗ -algebras A ( O x,y ). It does not imply that A commutes with unitary operators e iφ ( f ) − , or with an element B ∈ A ( W x ): if A is just a quadratic form, it would notbe possible to write down these commutators in a meaningful way. We thereforeclarify how ω -locality is related to the usual locality [11]: Proposition 1. (i)
Let A be a bounded operator; then A is ω -local in O x,y for some x, y ∈ R if and only if A ∈ A ( O x,y ) . (ii) Let A be a closed operator with core H ω,f , and H ω,f ⊂ dom A ∗ . Supposethat ∀ g ∈ D ω R ( R ) : exp( iφ ( g ) − ) H ω,f ⊂ dom A. ( ∗ ) Then A is ω -local in O x,y if and only if it is affiliated with A ( O x,y ) . (iii) In the case S = − , statement (ii) is true even without the condition ( ∗ ) . This proposition gives criteria for affiliation of closed operators to local alge-bras, but in examples, closability of a quadratic form A is difficult to characterizein terms of the coefficients in the expansion (1). Moreover, not much is knownabout the domain of the closed operator. We therefore look for sufficient (but notnecessary) conditions that allow to apply Proposition 1. We will understand (1)as an absolutely convergent sum on a certain domain, using summability condi-tions on the norms of the coefficients f [ A ] m,n . The following proposition provides asufficient criterion for closability of A as an operator: roposition 2. Let A ∈ Q ω . Suppose that for each fixed n , ∞ X m =0 m/ √ m ! (cid:0) k f [ A ] m,n k ωm × n + k f [ A ] n,m k ωn × m (cid:1) < ∞ . Then, A extends to a closed operator A − with core H ω,f , and H ω,f ⊂ dom( A − ) ∗ . To apply Proposition 1 we therefore need to fulfill the condition in Proposition 2and to show ω -locality of A . Hence, we formulate the ω -locality condition in termsof properties of the functions f [ A ] m,n . This is the content of [10, Theorem 5.4],which we summarize briefly: A is localized in the standard double cone O r ofradius r if and only if the coefficients f [ A ] m,n are boundary values of meromorphicfunctions ( F k ) ∞ k =0 on C k (with k = m + n ) with a certain pole structure, whichare S -symmetric, S -periodic, and fulfill certain bounds in the real and imaginarydirections, depending on ω and r , and which fulfill the recursion relations res ζ n − ζ m = iπ F k ( ζ ) = − πi (cid:16) n Y j = m S ( ζ j − ζ m ) (cid:17)(cid:16) − k Y p =1 S ( ζ m − ζ p ) (cid:17) F k − ( ˆ ζ ) . The problem is now to find examples of functions ( F k ) ∞ k =0 fulfilling the aboveconditions of ω -locality and closability via Proposition 2. In the case S = − k ≥
0, let g ∈ D ( O r ) with some r >
0, and let P be a symmetricLaurent polynomial of 2 k variables. We define the analytic functions(2) F [2 k,P,g ]2 k ( ζζζ ) := ˜ g ( p ( ζζζ )) P ( e ζζζ ) X σ ∈ S k sign σ k Y j =1 sinh ζ σ (2 j − − ζ σ (2 j ) , and F [2 k,P,g ] j = 0 for j = 2 k . For these the properties above hold with respect tothis r and for example with ω ( p ) := ℓ log(1 + p ) for some ℓ > F j for odd j , is the non-terminating sequence(3) F [1 ,P,g ]2 j +1 ( ζζζ ) := 1(2 πi ) k ˜ g ( p ( ζζζ )) P j +1 ( e ζζζ ) Y ≤ ℓ PCT, Spin and Statistics, and All That , Benjamin(1964).[2] R. Haag, Local Quantum Physics – Fields, Particles, Algebras , Springer (1996).[3] F. A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory ,World Scientific (1992).[4] H. M. Babujian and M. Karowski, Towards the construction of Wightman functions ofintegrable quantum field theories , Int. J. Mod. Phys. A (2004), 34–49.[5] B. Schroer, Modular localization and the bootstrap-formfactor program , Nucl. Phys. B499 (1997), 547–568.[6] G. Lechner, Construction of Quantum Field Theories with Factorizing S-Matrices , Com-mun. Math. Phys. , 821–860 (2008).[7] S. Alazzawi and G. Lechner, Inverse Scattering and Local Observable Algebras in IntegrableQuantum Field Theories , Commun. Math. Phys. , 913–956 (2017).[8] D. Cadamuro and Y. Tanimoto, Wedge-Local Fields in Integrable Models with Bound States ,Commun. Math. Phys. , 661–697 (2015).[9] H. Bostelmann and D. Cadamuro, An operator expansion for integrable quantum field the-ories , Journal of Physics A: Mathematical and Theoretical , 095401 (2013).[10] H. Bostelmann and D. Cadamuro, Characterization of Local Observables in Integrable Quan-tum Field Theories , Commun. Math. Phys. , 1199–1240 (2015).[11] H. Bostelmann and D. Cadamuro, Explicit construction of local observables in integrablequantum field theories , arXiv:1806.00269 (2018)., arXiv:1806.00269 (2018).