ABSTRACT - DRUM - University of Maryland

one-dimensional representation is considered, there is no reason to exclude higherdimensional
representations. The whole subject of this thesis is about the physical
realization of a two-dimensional irreducible representation of the fundamental group
π 1 (C N ) for d = 2.
The physical universe has d = 3 (as far as condensed matter system is concerned)
and π 1 (C N ) = S N [16]. It is known mathematically that S N has two onedimensional
representations: the trivial one, corresponding to Bose-Einstein statistics
and the “alternating” one corresponding to Fermi-Dirac statistics. Higher dimensional
representations are possible but they are just disguised versions of bosonic
and fermionic statistics with internal degrees of freedom [17].
If d = 2, π 1 (C N ) is no longer isomorphic to S N . Since the wordlines of particles
are just curves in (2+1)-dimensional spacetime and exchanging particles “braids” the
wordlines, π 1 (C N ) is called the braid group, denoted by B N and exchange statistics
is often referred as braiding statistics. To represent the braid group, we need N −
1 generators σ i which are physically nothing but counterclockwise braiding two
neighboring particles, subject to the following relations:
σ i σ j = σ j σ i , |i − j| > 1
σ i σ i+1 σ i = σ i+1 σ i σ i+1 .
(1.2)
Recall in the beginning of this section we gave the textbook argument why
there are only bosonic and fermionic statistics when d = 3. The argument translates
to the mathematical statement that if we supplement the definition of the braid
group (1.2) with σi 2 = 1, the braid group reduces to the permutation group.
The study of unitary representations of the braid group is a rich subject of
6