Loop braid group
Encyclopedia
The loop braid group is a mathematical group structure
that is used in some models of theoretical physics
to model the exchange of particle
s with loop-like topologies within three dimensions of space and time.
The basic operations which generate a loop braid group for n loops are exchanges of two adjacent loops, and passing one adjacent loop through another. The topology forces these generators to satisfy some relations, which determine the group.
To be precise, the loop braid group on n loops is defined as the motion group of n disjoint circles embedded in a compact three dimensional "box" diffeomorphic to the three dimensional disk. A motion is a loop in the configuration space, which consists of all possible ways of embedding n circles into the 3-disk. This becomes a group in the same way as loops in any space can be made into a group; first, we define equivalence classes of loops by letting paths g and h be equivalent iff they are related by a (smooth) homotopy, and then we define a group operation on the equivalence classes by concatenation of paths. Dahm was able to show that there is an injective homomorphism from this group into the automorphism group of the free group on n generators, so it is natural to identify the group with this subgroup of the automorphism group (described in a paper by Goldsmith). One may also show that the loop braid group is isomorphic to the welded braid group, as is done for example in a paper by Baez, Crans, and Wise, which also gives some presentations of the loop braid group using the work of Lin.
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
that is used in some models of theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
to model the exchange of particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s with loop-like topologies within three dimensions of space and time.
The basic operations which generate a loop braid group for n loops are exchanges of two adjacent loops, and passing one adjacent loop through another. The topology forces these generators to satisfy some relations, which determine the group.
To be precise, the loop braid group on n loops is defined as the motion group of n disjoint circles embedded in a compact three dimensional "box" diffeomorphic to the three dimensional disk. A motion is a loop in the configuration space, which consists of all possible ways of embedding n circles into the 3-disk. This becomes a group in the same way as loops in any space can be made into a group; first, we define equivalence classes of loops by letting paths g and h be equivalent iff they are related by a (smooth) homotopy, and then we define a group operation on the equivalence classes by concatenation of paths. Dahm was able to show that there is an injective homomorphism from this group into the automorphism group of the free group on n generators, so it is natural to identify the group with this subgroup of the automorphism group (described in a paper by Goldsmith). One may also show that the loop braid group is isomorphic to the welded braid group, as is done for example in a paper by Baez, Crans, and Wise, which also gives some presentations of the loop braid group using the work of Lin.