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Hamilton’s Turns for the Lorentz Group

Simon, R and Chaturvedi, S and Srinivasan, V and Mukunda, N (2006) Hamilton’s Turns for the Lorentz Group. In: International Journal of Theoretical Physics, 45 (11). pp. 2075-2094.

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Abstract

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by "turns," which are equivalence classes of directed great circle arcs on the unit sphere $S^2$, in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram lawfor the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group SU(1, 1) = Sp(2,R) = SL(2,R), the double cover of SO(2, 1). The present work develops a theory of turns for SL(2,C), the double and universal cover of SO(3, 1) and SO(3,C), rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Springer.
Keywords: Hamilton’s turns;Lorentz group
Department/Centre: Division of Physical & Mathematical Sciences > Centre for High Energy Physics
Date Deposited: 22 Aug 2008
Last Modified: 27 Aug 2008 12:52
URI: http://eprints.iisc.ernet.in/id/eprint/11349

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