Jens Mund

String-localized quantum fields and modular localization

We study free, covariant, quantum (Bose) fields that are associated with irreducible representations of the Poincaré group and localized in semi-infinite strings extending to spacelike infinity. Among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point-like localized fields. For the massive representations and the massless representations of finite helicity, all string-localized free fields can be written as an integral, along the string, of point-localized tensor or spinor fields. As a special case we discuss the string-localized vector fields associated with the point-like electromagnetic field and their relation to the axial gauge condition in the usual setting.

The Spin-Statistics Theorem for Anyons and Plektons in d = 2+1

We prove the spin-statistics theorem for massive particles obeying braid group statistics in three-dimensional Minkowski space. We start from first principles of local relativistic quantum theory. The only assumption is a gap in the mass spectrum of the corresponding charged sector, and a restriction on the degeneracy of the corresponding mass.

An Algebraic Jost-Schroer Theorem for Massive Theories

We consider a purely massive local relativistic quantum theory specified by a family of von Neumann algebras indexed by the space-time regions. We assume that, affiliated with the algebras associated to wedge regions, there are operators which create only single particle states from the vacuum (so-called polarization-free generators) and are well-behaved under the space-time translations. Strengthening a result of Borchers, Buchholz and Schroer, we show that then the theory is unitarily equivalent to that of a free field for the corresponding particle type. We admit particles with any spin and localization of the charge in space-like cones, thereby covering the case of string-localized covariant quantum fields.

String-Localized Quantum Fields, Modular Localization, and Gauge Theories

The concept of modular localization introduced by Brunetti, Guido and Longo, and Schroer, can be used to construct quantum fields. It combines Wigner’s particle concept with the Tomita-Takesaki modular theory of operator algebras. I report on the construction of free fields which are localized in semi-infinite strings extending to space-like infinity (mainly in collaboration with B. Schroer and J. Yngvason). Particular applications are: The first local (in the above sense) construction of fields for Wigner’s massless “infinite spin” particles; String-localized vector/tensor potentials for Photons and Gravitons, respectively; Massive vector bosons. Some speculative ideas are be presented concerning the perturbative construction of gauge theories (and quantum gravity) completely within a Hilbert space, trading gauge dependence with dependence on the direction of the localization string.

Braid Group Statistics Implies Scattering in Three-Dimensional Local Quantum Physics

AbstractIt is shown that quantum fields for massive particles with braid group statistics (Plektons) in three-dimensional space-time cannot be free, in a quite elementary sense: They must exhibit elastic two-particle scattering into every solid angle, and at every energy. This also implies that for such particles there cannot be any operators localized in wedge regions which create only single particle states from the vacuum and which are well-behaved under the space-time translations (so-called temperate polarization- free generators). These results considerably strengthen an earlier “NoGo-theorem for ’free’ relativistic Anyons”.As a by-product we extend a fact which is well-known in quantum field theory to the case of topological charges (i.e., charges localized in space-like cones) in d ≥ 4, namely: If there is no elastic two-particle scattering into some arbitrarily small open solid angle element, then the 2-particle S-matrix is trivial.

The Chirality Theorem

AbstractWe show how chirality of the weak interactions stems from string independence in the string-local formalism of quantum field theory.

Relations between positivity, localization and degrees of freedom: The Weinberg–Witten theorem and the van Dam–Veltman–Zakharov discontinuity

Abstract The problem of accounting for the quantum degrees of freedom in passing from massive higher-spin potentials to massless ones, and the inverse problem of “fattening” massless tensor potentials of helicity ±h to their massive s = | h | counterparts, are solved – in a perfectly ghost-free approach – using “string-localized fields”. This approach allows to overcome the Weinberg–Witten impediment against the existence of massless | h | ≥ 2 energy–momentum tensors, and to qualitatively and quantitatively resolve the van Dam–Veltman–Zakharov discontinuity concerning, e.g., very light gravitons, in the limit m → 0 .

String Chopping and Time-ordered Products of Linear String-localized Quantum Fields

For a renormalizability proof of perturbative models in the Epstein–Glaser scheme with string-localized quantum fields, one needs to know what freedom one has in the definition of time-ordered products of the interaction Lagrangian. This paper provides a first step in that direction. The basic issue is the presence of an open set of n-tuples of strings which cannot be chronologically ordered. We resolve it by showing that almost all such string configurations can be dissected into finitely many pieces which can indeed be chronologically ordered. This fixes the time-ordered products of linear field factors outside a nullset of string configurations. (The extension across the nullset, as well as the definition of time-ordered products of Wick monomials, will be discussed elsewhere).

Canonical interacting quantum fields on two-dimensional de Sitter space

Abstract We present the P ( φ ) 2 model on de Sitter space in the canonical formulation. We discuss the role of the Noether theorem and we provide explicit expressions for the energy-stress tensor of the interacting model.