The interplay of the polar decomposition theorem and the Lorentz group
نویسنده
چکیده
Abstract: 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)↑. 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,C) is preserved by the covering homomorphism of SL(2,C) onto SO(1, 3)↑.
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