Abstract
The Lie group structure of the Lorentz group is explored. Its generators and its Lie algebra are exhibited, via the study of infinitesimal Lorentz transformations. The exponential map is introduced and it is shown that the study of the Lorentz group can be reduced to that of its Lie algebra. Finally, the link between the restricted Lorentz group and the special linear group \(\mathrm{SL}(2, \mathbb{C})\) is established via the spinor map. The Lie algebras of these two groups are shown to be identical (up to some isomorphism).
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- 1.
One can show that they are then necessarily differentiable.
- 2.
Hausdorff means separated: any two distinct points admit disjoint open neighbourhoods.
- 3.
A topological space is second-countable iff there exists a countable family \((\mathcal{U}_{k})_{k\in \mathbb{N}}\) of open sets such that any open set of can be written as the union (possibly infinite) of some members of this family. This property excludes “unreasonably large” manifolds. In particular, it allows for a differentiable manifold of dimension n to be embedded smoothly in the Euclidean space \({\mathbb{R}}^{2n}\) (Whitney theorem).
- 4.
A homeomorphism between two topological spaces is a bijective continuous map, whose inverse is continuous as well.
- 5.
Let us recall that the index i ranges from 1 to 3.
- 6.
Let us recall that GL(E) stands for the general linear group of E, formed by all the invertible endomorphisms (automorphisms) of E (cf. p. 170).
- 7.
Sophus Lie (1842–1899): Norwegian mathematician, essentially known for the foundation of the theory of Lie groups. As an adolescent, he contemplated some military career, but his strong myopia forced him to choose an academic career instead! During a stay in Paris in 1870, at the contact of Camille Jordan, he started the study of continuous groups of transformations. After the declaration of war of France to Prussia in July 1870, he was arrested near Paris, being suspected to be a German spy: his mathematical notes had been mistaken for coded messages! He was released thanks to the intervention of the mathematician Gaston Darboux.
- 8.
\(\mathrm{Mat}(2, \mathbb{C})\) stands for the set of 2 ×2 matrices with complex coefficients.
- 9.
But not over \(\mathbb{C}\), since multiplying H 11 by \(\lambda = \mathrm{i}\) would lead to a violation of the first condition in (7.48).
- 10.
\(\mathbb{I}_{2}:=\mathrm{ diag}(1, 1)\) is the 2 ×2 identity matrix.
- 11.
- 12.
As for that of the Lorentz group, the Lie algebra of \(\mathrm{SL}(2, \mathbb{C})\) is denoted by the same letters than the group but in lower case.
- 13.
Considering \(\mathrm{SL}(2, \mathbb{C})\) and SOo(3, 1) as differentiable manifolds (of dimension 6) over \(\mathbb{R}\), this mapping is actually the differential of the spinor map taken at the point \(\mathbb{I}_{2}\); accordingly, it goes from the vector space tangent to \(\mathrm{SL}(2, \mathbb{C})\) at \(\mathbb{I}_{2}\), i.e. \(\mathrm{sl}(2, \mathbb{C})\), to the vector space tangent to SOo(3, 1) at \(\boldsymbol{\mathcal{S}}(\mathbb{I}_{2}) =\mathrm{ Id}\), i.e. so(3, 1) (cf. Remark 7.4 p. 224).
- 14.
Felix Klein (1849–1925): German mathematician, who authored numerous works in group theory and non-Euclidean geometry; in 1872, he proposed the famous
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Gourgoulhon, É. (2013). Lorentz Group as a Lie Group. In: Special Relativity in General Frames. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37276-6_7
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