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 ofrepresentation 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 thespinors and spinor fields.
KeywordsInvariant Subspace Lorentz Transformation Lorentz Group Reducible Representation Invariant Tensor
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