Superselection structure and statistics in Three-Dimensional local quantum theory.

Jürg Martin Fröhlich, Fabrizio Gabbiani, P. A. Marchetti

S. S. Horushy, V. N. Sushko

Jürg Martin Fröhlich, Fabrizio Gabbiani

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.

Carsten Binnenhei

The appearance of the Cuntz algebras Od is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. In view of this apparent discrepancy, we develop a comprehensive theory of quasi-free endomorphisms of the CAR and CCR algebras which give rise to representations of the Cuntz algebras Od on Fock space. The number d is the statistics dimension of the endomorphism. It can be any power of 2 (including 1 and infinite) in the CAR case, but takes only the values 1 and infinite in the CCR case. We obtain necessary and sufficient conditions for implementability of quasi- free endomorphisms. By studying extensions of partial Fock states, we find that the semigroup of implementable quasi-free endomorphisms has a simple structure: It can be written as a product of a group of automorphisms which are close to the identity, and the semigroup of endomorphisms which leave the given Fock state invariant. We describe the construction of the implementers of a quasi-free endomorphism in terms of annihilation and creation operators in is a generic feature of local quantum field theory. This fact has been discovered by S. Doplicher and J. E. Roberts within the algebraic theory of superselection sectors. Generators of the Cuntz algebras arise as charged field operators which implement localized endomorphisms of the observable algebra. Whereas the existence of such operators can be derived from first principles, little is known about the actual construction of these fields in concrete models. We compare the structure of the semigroup of (gauge invariant) quasi-free endomorphisms with the generic superselection structure of quantum field theory.

Sergio Doplicher

Howard Barnum, Alexander Wilce

Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category.