Abstract
In integrable models of quantum field theory, local fields are normally constructed by means of the bootstrap-formfactor program. However, the convergence of their n-point functions is unclear in this setting. An alternative approach uses fully convergent expressions for fields with weaker localization properties in spacelike wedges, and deduces existence of observables in bounded regions from there, but yields little information about their explicit form. We propose a new, hybrid construction: We aim to describe pointlike local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with a net of local von Neumann algebras that is known from the wedge-local approach. This is shown to work at least in the Ising model.
Highlights
Quantum field theory (QFT), describing the behaviour of subatomic particles at relativistic speeds, is usually formulated in terms of the eponymous quantum fields: A quantum field Φ(x)— and we will restrict to scalar Bose fields in all what follows—is a quantum observable localized at the spacetime point x, where localization manifests itself in commutativity at spacelike distances:[Φ(x), Φ(y)] = 0 if x is spacelike separated from y. (1)localization at a single point x is an unphysical over-idealization
In integrable models of quantum field theory, local fields are normally constructed by means of the bootstrap-formfactor program
Products Φ(x)Φ(y) of fields do not exist in general, and there is no notion of spectral projections oreigenvalues of Φ(x), all of which would be fundamental to a physical interpretation
Summary
Quantum field theory (QFT), describing the behaviour of subatomic particles at relativistic speeds, is usually formulated in terms of the eponymous quantum fields: A quantum field Φ(x)— and we will restrict to scalar Bose fields in all what follows—is a quantum observable localized at the spacetime point x, where localization manifests itself in commutativity at spacelike distances:. We show that the Φ(g) exist as (unbounded) operators that are affiliated with the algebras (Sec. 3), bypassing the treatment of products of fields or of their n-point functions In this way, we ameliorate (though not eliminate) the convergence problem of the series, breaking it down to a problem that is tractable at least in the simplest example, the massive Ising model. 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†(θ1) · · · z†(θk)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. Details of the construction can be found in [3]
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