Formulation of the Racah-Wigner calculus using category theory.
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
The Racah-Wigner calculus is formulated as a ring category. This leads to a deeper understanding and a frame for the structure of the calculus. This formulation is natural and results in some key concepts of the calculus defined differently. Every construct corresponds to a categorical principle giving more insight into its role within the calculus. This in turn results in a calculus that is easier to perform calculations in. The definition of a ring category is given and coherence proved. The central concept of a categorical product is extended introducing the concepts of inside and outside projection, associated inclusion and component summation. This is used to define the notion of coupling. The natural isomorphisms of the category define recoupling. The rules of diagram projection are derived which allows one to obtain recoupling coefficient equations from the underlying recoupling structure. The process of group chain factorisation, Racah factorisation and the Wigner-Eckart theorem are given a categorical formulation. The Wigner symbols are defined and their properties with respect to the action on isotypical irreps of permutation, dual conjugation and change of coupling choice are given. Hence all the properties are derived within a categorical structure. In particular the underlying recoupling of 3j, 6j and 9j symbols are found to be transpositions. The recoupling diagrams corresponding to the Biedenharn-Elliott and Racah backcoupling equations are derived. Finally some time is spent investigating calculation of recoupling coefficients resulting in a recursion algorithm for partitionable groups, a new concept defined here.