O(4) Symmetry and Angular Momentum Theory in Four Dimensions

Abstract

The theory of angular momentum in four-dimensional space is developed by studying in detail the properties of O(4), the group of rotations in four dimensions. Properties of O(4) Clebsch-Gordan coefficients are reviewed. Making use of the isomorphism O(4) ≈ O(3) ⊗ O(3), general expressions are derived for O(4) recoupling coefficients in terms of the corresponding ones in O(3) (6j-, 9j-,… symbols). The Wigner-Eckart theorem is discussed for O(4) and explicit formulas are given for the associated tensor calculus. The Pauli principle is generalized to four-dimensional space by establishing a connection between O(4) tensor character and statistics (Bose or Fermi). The equations for computing coefficients of fractional parentage in O(4) are derived. The applicability of this formalism to atomic and molecular physics problems is discussed.