Characterization of Local Observables in Integrable Quantum Field Theories

Abstract

Integrable quantum field theories in 1+1 dimensions have recently become amenable to a rigorous construction, but many questions about the structure of their local observables remain open. Our goal is to characterize these local observables in terms of their expansion coefficients in a series expansion by interacting annihilators and creators, similar to form factors. We establish a rigorous one-to-one characterization, where locality of an observable is reflected in analyticity properties of its expansion coefficients; this includes detailed information about the high-energy behaviour of the observable and the growth properties of the analytic functions. Our results hold for generic observables, not only smeared pointlike fields, and the characterizing conditions depend only on the localization region -we consider wedges and double cones -and on the permissible high energy behaviour.

On the status of pointlike fields in integrable QFTs

“…when h L is supported spacelike to the left and h R to the right of g. Since φ changes the particle number only by one, and hence preserves particle-number cutoffs, the commutators in (19) are well-defined in matrix elements for suitable h L , h R . That the commutators do actually vanish is then quite direct to verify; it is a consequence of the form factor equations, see [11,Secs. 4.3 and 5.3].…”

“…This approach solves all mathematical convergence problems, but it has a quite different shortcoming: the explicit form of the local operators A ∈ A(O) remains unclear. Their matrix elements z † (θ 1 ) · · · z † (θ k )Ω, AΩ do fulfill the form factor equations [11]; but in the end, these operators are "constructed" using the axiom of choice, and no further information about their relation to pointlike fields or other generators is available.…”

Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories

“…14) is monotonically increasing for s → 0, it follows that the uniform bound (4.15) holds also forK −λ =K (0) −λ , with 0 ≤ λ ≤ π. This finishes the proof.…”

Negative energy densities in integrable quantum field theories at one-particle level

“…One then constructs an "S-symmetric" Fock space H over H 1 , on which "interacting" annihilation and creation operators z(θ) and z † (θ) act; instead of the CCR, they fulfil the Zamolodchikov-Faddeev relations [Lec08], depending on S. The quantum field theory is constructed on this Fock space. If A is an operator of the theory (of a certain regularity class, including smeared Wightman fields), localized in a bounded spacetime region, then it can be written in a series expansion [BC13,BC15] …”

“…The first set of conditions follows from the general properties of the expansion coefficients F k [A] of a local operator A in integrable models, as derived in [BC15].…”