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Generalized Particle Statistics in Two-Dimensions: Examples from the Theory of Free Massive Dirac Field

Dario Salvitti

Published Jan 1, 2005• 1 citation
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Abstract

Dedicated to the memory of Sabrina Picucci In the framework of algebraic quantum field theory we analyze the anomalous statistics exhibited by a class of automorphisms of the observable algebra of the two-dimensional free massive Dirac field, constructed by fermionic gauge group methods. The violation of Haag duality, the topological peculiarity of a two-dimensional space-time and the fact that unitary implementers do not lie in the global field algebra account for strange behaviour of statistics, which is no longer an intrinsic property of sectors. Since automorphisms are not inner, we exploit asymptotic abelianness of intertwiners in order to construct a braiding for a suitable C ∗-tensor subcategory of End(A). We define two inequivalent classes of path connected bi-asymptopias, selecting only those sets of nets which yield a true generalized statistics operator. 1

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Generalized Particle Statistics in Two-Dimensions: Examples from the Theory of Free Massive Dirac Field

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On the example of Luttinger model and Schwinger model we consider the observable algebra of interacting fermi systems in two--dimensional space--time and construct field algebra related to it as a crossed product with some automorphism group. Fermi statistics results for conveniently chosen automorphisms. The extension of time evolution to the field algebra and its asymptotic behaviour are treated. For the Luttinger model time evolution is asymptotically anticommutative, while for the Schwinger model we find a reformulation of confinement. ?) Work supported in part by "Fonds zur Forderung der wissenschaftlichen Forschung in Osterreich" under grant M083--PHY--1994/95 ) Lise--Meitner Fellow ]) permanent address: Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Sofia 1 1 Introduction The Bose--Fermi duality in one space dimension has been successfully used for solving various problems in (1+1)--dimensional field theories and in 1--dimensional mode...

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Quantum Field Theory of Particles with Braid Group Statistics in 2+1 Dimenions

Jens Mund

The present thesis is concerned with the local quantum physics of relativistic particles and fields in three space-time dimensions, whose statistics is to be described by a representation of the braid group -- so-called plektons or, if the representation is Abelian, anyons. In particular the issue of the existence of free anyonic fields is addressed. In our context, these are operators affiliated with the `local´ field algebras and creating only single particle vectors from the vacuum. (Localizability here refers to regions extending to infinity in some spacelike direction.) Under a mild regularity condition for these fields, we can derive commutation relations which are not compatible with braid group statistics. Further, model-independent results concerning the PCT operator and the connection of spin and statistics are obtained. Assuming the observable algebra to satisfy the Bisognano-Wichmann theorem, a PCT theorem for plektons is derived. For anyons it is shown that the Bisognano-Wichmann property of the observable algebra lifts, in a `twisted´ version, to the field algebra. In particular, the PCT-operator coincides, up to a `twist operator´, with the modular conjugation of the field algebra associated to a standard wedge region. From this fact, the spin-statistics theorem for anyons is derived. The structure of the Hilbert space of scattering states of plektons and anyons is further clarified. It is known to be fixed by the relevant representations of the braid group and by the rules which determine the charges contained in the scattering states (the `fusion´ rules). We present Møller operators W+ and W- from the spaces of outgoing and incoming scattering states into a reference Hilbert space, which translate the physical ray representation of the Poincaré group on the spaces of scattering states into a canonical ray representation on the reference Hilbert space. A new result concerns the `incoming free PCT operator´, i.e. the product of the PCT operator with the S-matrix S = ( W+)* W- : It is exhibited how this model-independent operator acts in the reference Hilbert space. In the case of anyons, the fusion rules are trivial and the relevant representations of the braid group are classified by a finite set of complex numbers, namely the statistics parameters of the particles under consideration. Hence in the anyonic case the above results yield explicit formulae for the ray representation of the Poincaré group and for the incoming free PCT-operator. In view of the Bisognano-Wichmann property, we have thereby constructed the product of the Tomita operator associated to any given wedge region with the S-matrix, i.e. the `incoming free´ Tomita-operator of the anyonic field algebra.

•Jan 1, 1999•0 citations
Dedicated to the memory of Sabrina Picucci In the framework of algebraic quantum field theory we analyze the anomalous statistics exhibited by a class of automorphisms of the observable algebra of the two-dimensional free massive Dirac field, constructed by fermionic gauge group methods. The violation of Haag duality, the topological peculiarity of a two-dimensional space-time and the fact that unitary implementers do not lie in the global field algebra account for strange behaviour of statistics, which is no longer an intrinsic property of sectors. Since automorphisms are not inner, we exploit asymptotic abelianness of intertwiners in order to construct a braiding for a suitable C ∗-tensor subcategory of End(A). We define two inequivalent classes of path connected bi-asymptopias, selecting only those sets of nets which yield a true generalized statistics operator. 1
We present details of a mathematical theory of superselection sectors and their statistics in local quantum theory over (two- and) three-dimensional space-time. The framework for our analysis is algebraic quantum field theory. Statistics of superselection sectors in three-dimensional local quantum theory with charges not localizable in bounded space-time regions and in two-dimensional chiral theories is described in terms of unitary representations of the braid groups generated by certain Yang-Baxter matrices. We describe the beginnings of a systematic classification of those representations. Our analysis makes contact with the classification theory of subfactors initiated by Jones. We prove a general theorem on the connection between spin and statistics in theories with braid statistics. We also show that every theory with braid statistics gives rise to a “Verlinde algebra”. It determines a projective representation of SL(2, ℤ) and, presumably, of the mapping class group of any Riemann surface, even if the theory does not display conformal symmetry.
On the example of Luttinger model and Schwinger model we consider the observable algebra of interacting fermi systems in two--dimensional space--time and construct field algebra related to it as a crossed product with some automorphism group. Fermi statistics results for conveniently chosen automorphisms. The extension of time evolution to the field algebra and its asymptotic behaviour are treated. For the Luttinger model time evolution is asymptotically anticommutative, while for the Schwinger model we find a reformulation of confinement. ?) Work supported in part by "Fonds zur Forderung der wissenschaftlichen Forschung in Osterreich" under grant M083--PHY--1994/95 ) Lise--Meitner Fellow ]) permanent address: Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Sofia 1 1 Introduction The Bose--Fermi duality in one space dimension has been successfully used for solving various problems in (1+1)--dimensional field theories and in 1--dimensional mode...
This chapter gives a brief introduction to rotations in two-dimensional space and finds that rotations in two-dimensional space have some special properties. In particular, spin and statistics have peculiarities not possible in three- or higher-dimensional space—spin is not quantized in integer or half-integer units of h and hence the particles are not necessarily bosons or fermions but may obey any statistics and were thus called anyons by Wilczek. In fact, the concept of fractional statistics had been already put forward by Leinaas and Myrheim in the late 1970s. They found that the origin of the fractional statistics lies in the peculiar topological properties of the configuration space of many identical particles. This space is doubly connected in three or more dimensions but is multiply connected in two dimensions. Later Goldin, Menikoff and Sharp reached a similar conclusion using a completely different method based on the rigorous study of the unitary representations of current algebra and diffeomorphism groups.
The present thesis is concerned with the local quantum physics of relativistic particles and fields in three space-time dimensions, whose statistics is to be described by a representation of the braid group -- so-called plektons or, if the representation is Abelian, anyons. In particular the issue of the existence of free anyonic fields is addressed. In our context, these are operators affiliated with the `local´ field algebras and creating only single particle vectors from the vacuum. (Localizability here refers to regions extending to infinity in some spacelike direction.) Under a mild regularity condition for these fields, we can derive commutation relations which are not compatible with braid group statistics. Further, model-independent results concerning the PCT operator and the connection of spin and statistics are obtained. Assuming the observable algebra to satisfy the Bisognano-Wichmann theorem, a PCT theorem for plektons is derived. For anyons it is shown that the Bisognano-Wichmann property of the observable algebra lifts, in a `twisted´ version, to the field algebra. In particular, the PCT-operator coincides, up to a `twist operator´, with the modular conjugation of the field algebra associated to a standard wedge region. From this fact, the spin-statistics theorem for anyons is derived. The structure of the Hilbert space of scattering states of plektons and anyons is further clarified. It is known to be fixed by the relevant representations of the braid group and by the rules which determine the charges contained in the scattering states (the `fusion´ rules). We present Møller operators W+ and W- from the spaces of outgoing and incoming scattering states into a reference Hilbert space, which translate the physical ray representation of the Poincaré group on the spaces of scattering states into a canonical ray representation on the reference Hilbert space. A new result concerns the `incoming free PCT operator´, i.e. the product of the PCT operator with the S-matrix S = ( W+)* W- : It is exhibited how this model-independent operator acts in the reference Hilbert space. In the case of anyons, the fusion rules are trivial and the relevant representations of the braid group are classified by a finite set of complex numbers, namely the statistics parameters of the particles under consideration. Hence in the anyonic case the above results yield explicit formulae for the ray representation of the Poincaré group and for the incoming free PCT-operator. In view of the Bisognano-Wichmann property, we have thereby constructed the product of the Tomita operator associated to any given wedge region with the S-matrix, i.e. the `incoming free´ Tomita-operator of the anyonic field algebra.