On the Restricted Lorentz Group and Groups Homomorphically Related to It
A study is made of the real restricted Lorentz group, L, and of its relationship to the group, SL(2C), of complex unimodular two-dimensional matrices, and to the group, O3, of orthogonal transformations in a complex space of three dimensions. The discussion of SL(2C) gives important formulas in concise forms and their proofs in an elegant and economical manner, it deals with the nontrivial matter of proving the internal consistency of the formalism. To illustrate the practical utility of the theory, the product of two nonparallel pure Lorentz transformations is studied. In the discussion of O3, explicit formulas realizing the isomorphism of O3 and L are obtained. These formulas are applied, for illustrative purposes, to the derivation of the transformation properties under L of the electromagnetic field vectors, regarded as a complex three-vector (E + iH). A result analogous to the fuctorization of the general element of L into a spatial rotation and a pure Lorentz transformation, and to the polar decomposition of the general element of SL(2C), is derived for O3. Insight into the relationship of O3 to L is provided by considering the unimodular matrix description of the complex Lorentz group, and the contrasting specializations of it, which lead to the unimodular matrix descriptions of its subgroups, O3 and L.
- Research Organization:
- Univ. of Rochester, NY (United States)
- Sponsoring Organization:
- USDOE
- NSA Number:
- NSA-17-008999
- OSTI ID:
- 4730250
- Report Number(s):
- NYO-10229; JMAPAQ; 0022-2488
- Journal Information:
- Journal of Mathematical Physics, Vol. 3, Issue 6; Other Information: NYO-10229. Orig. Receipt Date: 31-DEC-63; ISSN 0022-2488
- Publisher:
- American Institute of Physics (AIP)
- Country of Publication:
- Country unknown/Code not available
- Language:
- English
Similar Records
From the rotation group to the Poincare group
Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles