Book contents
- Frontmatter
- Contents
- Preface
- List of symbols
- 1 Linear transformations
- 2 The theory of matrix transformations
- 3 Elements of abstract group theory
- 4 Unitary and orthogonal groups
- 5 The point groups of finite order
- 6 Theory of group representations
- 7 Construction of symmetry-adapted linear combinations based on the correspondence theorem
- 8 Subduced and induced representations
- 9 Elements of continuous groups
- 10 The representations of the rotation group
- 11 Single- and double-valued representations of point groups
- 12 Projective representations
- 13 The 230 space groups
- 14 Representations of the space groups
- 15 Applications of unirreps of space groups to energy bands and vibrational modes of crystals
- 16 Time reversal, anti-unitary point groups and their co-representations
- 17 Anti-unitary space groups and their co-representations
- Appendix Character tables of the crystal point groups
- References
- Index
10 - The representations of the rotation group
Published online by Cambridge University Press: 12 November 2009
- Frontmatter
- Contents
- Preface
- List of symbols
- 1 Linear transformations
- 2 The theory of matrix transformations
- 3 Elements of abstract group theory
- 4 Unitary and orthogonal groups
- 5 The point groups of finite order
- 6 Theory of group representations
- 7 Construction of symmetry-adapted linear combinations based on the correspondence theorem
- 8 Subduced and induced representations
- 9 Elements of continuous groups
- 10 The representations of the rotation group
- 11 Single- and double-valued representations of point groups
- 12 Projective representations
- 13 The 230 space groups
- 14 Representations of the space groups
- 15 Applications of unirreps of space groups to energy bands and vibrational modes of crystals
- 16 Time reversal, anti-unitary point groups and their co-representations
- 17 Anti-unitary space groups and their co-representations
- Appendix Character tables of the crystal point groups
- References
- Index
Summary
As has been shown in Section 9.4, the parameter space of the proper rotation group SO(3, r) is doubly connected so that there exist single-valued as well as double-valued representations for SO(3, r). In this chapter, it will be shown that the special unitary group SU(2) is simply connected and homomorphic to SO(3, r) with two-to-one correspondence. Accordingly, the representations of SU(2) are single-valued, but these provide all the single-and double-valued representations of SO(3, r). In particular, SU(2) itself provides a 2 × 2 matrix representation of SO(3, r) that is double-valued. As was mentioned at the end of Section 4.1, an element of SU(2) is called a spinor transformation, because of the role it plays in the theory of the spinning electron. Moreover, the double-valued unirreps of SO(3, r) given by the unirreps of SU(2) are called the spinor representations of SO(3, r). We shall begin with a discussion on the special unitary group SU(2).
The structure of SU(2)
The generators of SU(2)
Previously, in Section 4.1, we have shown that an element of the special unitary group in n dimensions SU(n) is expressed by a matrix of the form U0 = exp K0, where K0 is an anti-Hermitian traceless matrix.
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- Publisher: Cambridge University PressPrint publication year: 1999