The interplay of the polar decomposition theorem and the Lorentz group

Details The interplay of the polar decomposition theorem and the Lorentz group The interplay of the polar decomposition theorem and the Lorentz group

2002-11-20

arxiv.org/abs/math-ph/0211047v1

Physics

Mathematical Physics

7 pages, no figures, LaTeX 2e

Scientific paper

It is shown that the polar decomposition theorem of operators in (real) Hilbert spaces gives rise to the known decomposition in boost and spatial rotation part of any matrix of the orthochronous proper Lorentz group $SO(1,3)\uparrow$. This result is not trivial because the polar decomposition theorem is referred to a positive defined scalar product while the Lorentz-group decomposition theorem deals with the indefinite Lorentz metric. A generalization to infinite dimensional spaces can be given. It is finally shown that the polar decomposition of $SL(2,\bC)$ is preserved by the covering homomorphism of $SL(2,\bC)$ onto $SO(1,3)\spa\uparrow$

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