The Lorentz Group and Some of Its Representations
As we have seen in the last chapter, all laws of nature that can be written as the vanishing of some 4-tensor (field) manifestly satisfy the principle of (Einsteinian) Relativity. The essential point here is that the 4-tensor spaces are linear spaces on w
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The Lorentz Group and Some of Its Representations
As we have seen in the last chapter, all laws of nature that can be written as the vanishing of some 4-tensor (field) manifestly satisfy the principle of (Einsteinian) Relativity. The essential point here is that the 4-tensor spaces are linear spaces on which the Lorentz group acts as a group of linear transformations. This will be characterized formally in sect. 6.4 where we introduce the concept of representation of a group. From a more systematic point of view we may then ask whether tensors are the only type of quantities that allow such a linear action. In chap. 8, we will answer this question in the positive-but in this investigation a new type of quantities will emerge that on the one hand turns out, in chap. 9, to be essential if the question is asked from a quantum mechanical point of view, and on the other hand also proves very helpful even in the classical, tensorial regime. These are the spinors and spinor fields. For a systematic investigation of all possible representations it is, however, necessary to study the Lorentz group itself more closely, since the structure of the group has implications on the structure of the set of all its representations. The study of the group itself as well as the introduction of the basic notions of representation theory is the purpose of the present chapter. At this point it becomes ad visible to gradually refresh the basic concepts of abstract group theory (a short account of which is found in Appendix A).
6.1
The Lorentz Group as a Lie Group
In chap. 3 we found all transformations leaving invariant the 4-dimensional line element ds 2 . Apart from space-time translations (which will be taken up again only in chap. 9) these are homogeneous transformations or
i; =
Lx
(6.1.1)
satisfying the (pseudo )orthogonality relations L
i
m
k
L n Tlik = Tlmn
or
(6.1.2)
Because of Tlmn = Tlnm these are 10 relations restricting the 16 matrix elements of L; and these relations are independent from each other, so that only 6 matrix elements can be chosen independently. This follows, e.g., from the fact that we were able, in sect. 1.5, to associate to any L satisfying eq. (6.1.2) the 6 components v, a which uniquely characterize L and are allowed to vary arbitrarily over the admissible domain Ivi < 1, lal :=:; Jr. (Note that the latter restrictions are inequalities, whereas the former restrictions by orthogonality are equalities!) R. U. Sexl et al., Relativity, Groups, Particles © Springer-Verlag Wien 2001
6.1 Lorentz Group as a Lie Group
135
A slightly more direct argument which at the same time is characteristic of Lie groups (to be defined below) would be as follows. Let L be a solution of eq. (6.1.2); then for every infinitesimal change L -+ L + 6L it follows from eq. (6.1.2) (6.1.3) or, because of TJ T = TJ:
LT TJ 6L ::::::: TJ L -1 6L is an antisymmetric matrix. As such, it has 6 independent elements; and conversely, every infinitesimal antisymmetric matrix 6A defines, via (6.1.4) increments for L that will satis
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