18 Semi-direct Products
The theory of a free particle is largely determined by its group of symmetries, the group of symmetries of three dimensional space, a group which includes a subgroup of spatial translations, and a subgroup SO(3) of rotations. The second subgroup acts non-trivially on the first, since the direction of a translation is rotated by an element of . In later chapters dealing with special relativity, these groups get enlarged to include a fourth dimension, time, and the theory of a free particle will again be determined by the action of these groups, now on space-time, not just space. In chapters 15 and 16 we studied two groups acting on phase space: the Heisenberg group and the symplectic group . In this situation also, the second group acts non-trivially on the first by automorphisms (see 16.19).
This situation of two groups, with one acting on the other by automorphisms, allows one to construct a new sort of product of the two groups, called the semidirect product, and this will be the topic for this chapter. The general theory of such a construction will be given, but our interest will be in certain specific examples: the semi-direct product of and , the semi-direct product of and , and the Poincar´e group (which will be discussed later, in chapter 42). This chapter will just be concerned with the groups and their Lie algebras, with their representations the topics of later chapters (19, 20 and 42).
Chapter contents
- 18.1 An example: the Euclidean group
- 18.2 Semi-direct product groups
- 18.3 Semi-direct product Lie algebras
- 18.4 For further reading
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 204、205、206、207、208、209
来源版本：2025-10-20
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