29 Clifford Algebras and Geometry

Cliford Algebras and Geometry

The definitions given in chapter 28 of Weyl and Cliford algebras were purely algebraic, based on a choice of generators and relations. These definitions do though have a more geometrical formulation, with the definition in terms of generators corresponding to a specific choice of coordinates. For the Weyl algebra, the geometry involved is symplectic geometry, based on a non-degenerate antisymmetric bilinear form. We have already seen that in the bosonic case quantization of a phase space ${\mathbf{R}}^{2d}$ depends on the choice of a non-degenerate antisymmetric bilinear form Ω which determines the Poisson brackets and thus the Heisenberg commutation relations. Such a Ω also determines a group $Sp(2d,\mathbf{R})$ ， which is the group of linear transformations of ${\mathbf{R}}^{2d}$ preserving Ω.

The Cliford algebra also has a coordinate invariant definition, based on a more well known structure on a vector space ${\mathbf{R}}^n$ , that of a non-degenerate symmetric bilinear form, $i$.e., an inner product. In this case the group that preserves the inner product is an orthogonal group. In the symplectic case antisymmetric forms require an even number of dimensions, but this is not true for symmetric forms, which also exist in odd dimensions.

中文目录

Chapter contents

• 29.1 Non-degenerate bilinear forms
• 29.2 Cliford algebras and geometry
  • 29.2.1 Rotations as iterated orthogonal reflections
  • 29.2.2 The Lie algebra of the rotation group and quadratic elements of the Cliford algebra
• 29.3 For further reading

Previous chapter: 28 Weyl and Clifford Algebras

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原书 PDF · 印刷页 318、319、320、321、322、323、324、325

来源版本：2025-10-20

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