Modular localization of massive particles with “any” spin in d=2+1

Abstract

We discuss a concept of particle localization which is motivated from quantum field theory, and has been proposed by Brunetti, Guido and Longo and by Schroer. It endows the single particle Hilbert space with a family of real subspaces indexed by the space-time regions, with certain specific properties reflecting the principles of locality and covariance. We show by construction that such a localization structure exists also in the case of massive anyons in d = 2 + 1, i.e. for particles with positive mass and with arbitrary spin s ∈ R. The construction is completely intrinsic to the corresponding ray representation of the (proper orthochronous) Poincaré group. Our result is of particular interest since there are no free fields for anyons, which would fix a localization structure in a straightforward way. We present explicit formulas for the real subspaces, expected to turn out useful for the construction of a quantum field theory for anyons. In accord with well-known results, only localization in string-like, instead of pointlike or bounded, regions is achieved. We also prove a single-particle PCT theorem, exhibiting a PCT operator which acts geometrically correctly on the family of real subspaces.

Local Anyonic Quantum Fields on the Circle Leading to Cone-Local Anyons in Two Dimensions

“…For the covering of the full Lorentz groupL ↑ + one would get a representation on Fock space of the form U (Λ) ∼ e isQΩ(Λ) U 0 (Λ) where Ω(Λ) is now an operator on the Hilbert space instead of a mere constant ω as for the group U (1) (see e.g. [22] for a possible representation ofL ↑ + on the mass shell). Another problem is that the method of considering multiplication operators on the one-particle Hilbert space as implementable Bogoliubov transformations leads to problems concerning the Hilbert-Schmidt property for theories in more than one dimension (see e.g.…”

“…To account for this fact the fields can be interpreted to be localized in "generalized cones" (or "paths of cones", see e.g. [22]) which are defined in the following way. A usual cone C in two dimensions is determined by a point x ∈ R 2 (its apex) and an interval I on the circle, specifying the asymptotic directions contained in C. Hence one can denote a cone by the pair C = ( x, I), where the width of I determining the opening angle of C should be smaller than π.…”

“…However, to speak of "proper" Anyons in the sense of algebraic quantum field theory one would need field algebras localized in spacelike cones C, extending to infinity only in a connected compact set of spacelike directions. More precisely, the localization regions for Anyons can be labelled by pathsC of such spacelike cones depending also on a kind of winding number which determines the commutation relations of mutually spacelike separated fields [3,9,10,12,22](instead of heaving a mere sign-function in the exchange phase). The spin statistics theorem for Anyons [12,23] then forces these fields to transform under a representation of the (covering of the) Poincaré group P ↑ + with arbitrary real spin s ∈ R. It would be desirable to find an explicit construction for such cone-localized Anyon fields for any spin s in 2 + 1 dimensions.…”

Jorge A. Swieca’s contributions to quantum field theory in the 60s and 70s and their relevance in present research

“…According to the spin-statistics theorem this is not possible in higher dimensions. In d=1+2 QFT the (braid-group) statistics is determined in terms of the (anomalous) spin and this connection is already pre-empted in the setting of Wigner's classification of one-particle states [70]. The statistics in the sense of field commutation relations is also intrinsic in d=1+1 conformal theories.…”

Peculiarities of massive vector mesons and their zero mass limits

“…Formally this corresponds to the possibility of gauge changes in (1); p(λ) changes preserve relative localization and a fortiori do not change the particle content in the presence of interactions. In contrast to the "virtual" strings of anyons/plektons [24], which, similar to cuts in complex function theory, may be displaced as long as crossings are prevented, the strings of string-local fields are "real". Needless to mention that the string-local potentials are the only vector potentials which permit a m → 0 limit.…”