Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 084, 15 pages      arXiv:1402.3541      https://doi.org/10.3842/SIGMA.2014.084

A Compact Formula for Rotations as Spin Matrix Polynomials

Thomas L. Curtright a, David B. Fairlie b and Cosmas K. Zachos c
a) Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA
b) Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK
c) High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439-4815, USA

Received May 07, 2014, in final form August 07, 2014; Published online August 12, 2014

Abstract
Group elements of ${\rm SU}(2)$ are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. The simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.

Key words: spin matrices; matrix exponentials.

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