A Lorentz boost as a product of two space-time reflections and some additional results about Clifford algebra
Résumé
This is a technical clarifying note consisting of two parts. In the first part we derive the expression for a boost in two representations of the homogeneous Lorentz group, viz. the two-dimensional representation SL(2,C) and the four-dimensional Dirac representation in its Cartan-Weyl form. The derivation is purely algebraic. It uses the development of a Clifford algebra for a group of isometries of a vector space, whereby the group is generated by reflections. We prove that a boost can be obtained as a product of two space-time reflections, in perfect analogy with the way a rotation in R 3 can be obtained as a product of two reflections. The derivation does therefore not rely on physical considerations as in Einstein's approach. It is purely based on symmetry arguments. The second part deals with the justification of the definition of a Clifford algebra given in certain mathematical textbooks, which immediately introduce a basis of multi-vectors 1, ej, ej 1 ∧ ej 2 , ej 1 ∧ ej 2 ∧ ej 3 , • • • for this algebra. Rather than as a bemusing "postulate" that descends from heaven, we will present the introduction of this basis as an obvious result of a logical construction of the group representation theory. This will provide the reader with a much better understanding of what is going on behind the scenes of the formalism. We prove that this basis of multi-vectors 1, ej, ej 1 ∧ ej 2 , ej 1 ∧ ej 2 ∧ ej 3 , • • • is orthogonal in terms of a scalar product whose use is very natural in vector spaces of matrices.
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