We demonstrate that the anyon statistics and three-loop statistics of various
2d and
3d topological phases can be derived using semiclassical nonlinear sigma model field theories with a topological $\ensuremath{\Theta}$ term. In our formalism, the braiding statistics has a natural geometric meaning: The braiding process of anyons or loops leads to a nontrivial field configuration in the space-time, which will contribute a braiding phase factor due to the $\ensuremath{\Theta}$ term.
The statistics of fields in low dimensions is studied from the point of view of the braid group Bn of n strings. Explicit representations MR for the N-component model, N=2 to 5, are derived by solving the Yang-Baxter-like braid group relations for the statistical matrix R, which describes the transformation of the bilinear product of two N-component fields under the transposition of coordinates. When R2≠1 the statistics is neither Bose-Einstein nor Fermi-Dirac; it is strange. It is shown that for each N, the N+1 parameter family of solutions obtained is the most general one under a given set of constraints including “charge” conservation. Extended Nth order (N>2) Alexander-Conway relations for link polynomials are derived. They depend nonhomogeneously only on one of the N+1 parameters. The N=3 and 4 ones agree with those previously derived by Akutsu et al. Flat connections ω defining integrable systems of the N-component model are derived from the representations. The monodromy of the solution of such a system also carries a representation Mω of Pn⊂Bn. For N=2, Mω=MR, but the equality may not hold in general. The connections also lead directly to solutions of the classical Yang-Baxter equation. A generalization of Kohno’s monodromy representation of Bn associated with the algebra sl(N,C) is given. Applications of the braid group representations to statistical models, conformal field theory and many-body systems of extended objects are briefly discussed.
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.
General kinematical restrictions on quantum-theories of N identical anyons have been established by using braid groups. For an orientable compact surface (without boundary) of genus g, the statistical parameter θ is a rational multiple of π, θ=πp/v (p and v mutually prime integers), and the number of components of the wavefunction is (a multiple of) vg. The particle-number is N=rv+1−g (r integer) for spinless (S=0) and N=rv for spinning (S=θ/2π) anyons. The restrictions for spinning anyons are consistent with results for fractional quantum Hall systems, nonlinear O(3) field theory, and Chern-Simons theory. The multi-component structure, which appears for g≥1, reflects an internal collective degree of freedom. A comparison with the multi-component wavefunctions needed to describe systems with fractionally charged quasi-particles yields consistency relations between charge and statistics. Non-orientable surfaces do not allow fractional statistics.
The aim of this paper is to show a connection between an extended theory of statistical experiments on the one hand and the foundation of quantum theory on the other hand. The main aspects of this extension are: One assumes a hyperparameter space
Φ common to several potential experiments, and a basic symmetry group G associated with this space. The parameter θ_{a} of a single experiment, looked upon as a parametric function
θa(⋅) on
Φ, is said to be permissible if G induces in a natural way a new group on the image space of the function. If this is not the case, it is arranged for by changing from G to a subgroup
Ga. The Haar measure of this subgroup (confined to the spectrum; see below) is the prefered prior when the parameter is unknown. It is assumed that the hyperparameter itself can never be estimated, only a set of parametric functions. Model reduction is made by restricting the space of complex `wave' functions, also regarded as functions on
Φ, to an irreducible invariant subspace
M under G. The spectrum of a parametric function is defined from natural group-theoretical and statistical considerations. We establish that a unique operator can be associated with every parametric functions
θa(⋅), and essentially all of the ordinary quantum theory formalism can be retrieved from these and a few related assumptions. In particular the concept of spectrum is consistent. The scope of the theory is illustrated on the one hand by the example of a spin 1/2 particle and a related EPR discussion, on the other hand by a simple macroscopic example.