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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References

  1. Atiyah M.F., Macdonald I.G., Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass. - London - Don Mills, Ont., 1969.
  2. Butzer P.L., Schmidt M., Stark E.L., Vogt L., Central factorial numbers; their main properties and some applications, Numer. Funct. Anal. Optim. 10 (1989), 419-488.
  3. Curtright T.L., Van Kortryk T.S., On rotations as spin matrix polynomials, arXiv:1408.0767.
  4. Euler L., Problema algebraicum ob affectiones prorsus singulares memorabile, Commentatio 407 Indicis Enestoemiani, Novi Comm. Acad. Sci. Petropolitanae 15 (1770), 75-106.
  5. Lehrer-Ilamed Y., On the direct calculations of the representations of the three-dimensional pure rotation group, Math. Proc. Cambridge Philos. Soc. 60 (1964), 61-66.
  6. Macon N., Spitzbart A., Inverses of Vandermonde matrices, Amer. Math. Monthly 65 (1958), 95-100.
  7. Nelson T.J., Spin-matrix polynomials and the Veneziano formula, Phys. Rev. 184 (1969), 1954-1955.
  8. Nikitin A.G., Laplace-Runge-Lenz vector for arbitrary spin, J. Math. Phys. 54 (2013), 123506, 14 pages, arXiv:1308.4279.
  9. Pinsky M.A., Introduction to Fourier analysis and wavelets, Graduate Studies in Mathematics, Vol. 102, Amer. Math. Soc., Providence, RI, 2009.
  10. Rodrigues O., Des lois géometriques qui regissent les déplacements d' un systéme solide dans l' espace, et de la variation des coordonnées provenant de ces déplacement considerées indé;pendent des causes qui peuvent les produire, J. Math. Pures Appl. 5 (1840), 380-400.
  11. Schwinger J., On angular momentum, report NYO-3071, Harvard University, Nuclear Development Associates, Inc. (US), 1952.
  12. Torruella A.J., Representations of the three-dimensional rotation group by the direct method, J. Math. Phys. 16 (1975), 1637-1642.
  13. van Wageningen R., Explicit polynomial expressions for finite rotation operators, Nuclear Phys. 60 (1964), 250-263.
  14. Weber T.A., Williams S.A., Spin-matrix polynomials and the rotation operator for arbitrary spin, J. Math. Phys. 6 (1965), 1980-1983.
  15. Wigner E.P., Group theory and its application to the quantum mechanics of atomic spectra, Pure and Applied Physics, Vol. 5, Academic Press, New York - London, 1959.
  16. Williams S.A., Draayer J.P., Weber T.A., Spin-matrix polynomial development of the Hamiltonian for a free particle of arbitrary spin and mass, Phys. Rev. 152 (1966), 1207-1212.
  17. Zygmund A., Trigonometric series: Vols. I, II, Cambridge University Press, London - New York, 1968.


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