Rotation matrices corresponding to complex angular momenta

Aleksei G. Sitenko

An expression is given for the integral over the domain of Euler angles of a product of any number of rotation matrices. As a result a scheme has been set up in which any number of angular momenta may be coupled on an equal footing. Explicit algebraic exp

Daniel Grucker

For a two-dimensional complex vector space, the spin matrices can be calculated directly from the angular momentum commutator definition. The 3 Pauli matrices are retrieved and 23 other triplet solutions are found. In the three-dimensional space, we show that no matrix fulfills the spin equations and preserves the norm of the vectors. By using a Clifford geometric algebra it is possible in the four-dimensional spacetime (STA) to retrieve the 24 different spins 1/2. In this framework, spins 1/2 are rotations characterized by multivectors composed of 3 vectors and 3 bivectors. Spins 1 can be defined as rotations characterized by 4 vectors, 6 bivectors and 4 trivectors which result in unit multivectors which preserve the norm. Let us note that this simple derivation retrieves the main spin properties of particle physics.

J. O. Eeg, J. Wroldsen

We give manifest rotation symmetric expressions for eigenfunctions for spin s , orbital angular momentum l and total angular momentum j = l + s ,...,| l - s | in terms of (2 j + 1) × (2 s + 1) multipole transition matrices (MTM). These matrices, which are irreducible tensor matrices, have an algebra together with ordinary spin matrices for spin s and spin j . Explicit expressions for MTM's and their algebra are given for angular momenta ⩽ 3. By means of some examples we show that within this formalism angular integrations in central field problems will be simplified considerably. Thus the formalism turns out to be very useful for instance for calculations within the MIT-bag and also within spin-spin interactions in atomic physics.