41 Representations of the Lorentz Group

Having seen the importance in quantum mechanics of understanding the representations of the rotation group $SO(3)$ and its double cover $Spin(3)=SU(2)$ one would like to also understand the representations of the Lorentz group $SO(3,1)$ and its double cover $Spin(3,1)=SL(2,\mathbf{C})$ . One diference from the $SO(3)$ case is that all non-trivial finite dimensional irreducible representations of the Lorentz group are non-unitary (there are infinite dimensional unitary irreducible representations, of no known physical significance, which we will not discuss). While these finite dimensional representations themselves only provide a unitary action of the subgroup $Spin(3)\subset Spin(3,1)$ , they will later be used in the construction of quantum field theories whose state spaces will have a unitary action of the Lorentz group.

中文目录

Chapter contents

• 41.1 Representations of the Lorentz group
• 41.2 Dirac \gamma matrices and Clif(3, 1)
• 41.3 For further reading

Previous chapter: 40 Minkowski Space and the Lorentz Group

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原书 PDF · 印刷页 435、436、437、438、439、440、441、442、443

来源版本：2025-10-20

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