The Generic Superintegrable System on the 3-Sphere and the 9j Symbols of su(1, 1)

Abstract

The 9j symbols of su(1,1) are studied within the framework of the generic superintegrable system on the 3-sphere. The canonical bases corresponding to the binary coupling schemes of four su(1,1) representations are constructed explicitly in terms of Jacobi polynomials and are seen to correspond to the separation of variables in different cylindrical coordinate systems. A triple integral expression for the 9j coefficients exhibiting their symmetries is derived. A double integral formula is obtained by extending the model to the complex three-sphere and taking the complex radius to zero. The explicit expression for the vacuum coefficients is given. Raising and lowering operators are constructed and are used to recover the relations between contiguous coefficients. It is seen that the 9j symbols can be expressed as the product of the vacuum coefficients and a rational function. The recurrence relations and the difference equations satisfied by the 9j coefficients are derived.