30 Anticommuting Variables and Pseudo-classical Mechanics

The analogy between the algebras of operators in the bosonic (Weyl algebra) and fermionic (Cliford algebra) cases can be extended by introducing a fermionic analog of phase space and the Poisson bracket. This gives a fermionic analog of classical mechanics, sometimes called “pseudo-classical mechanics”, the quantization of which gives the Cliford algebra as operators, and spinors as state spaces. In this chapter we’ll introduce “anticommuting variables” ${\xi }_j$ that will be the fermionic analogs of the variables $q_j,p_j$ . These objects will become generators of the Cliford algebra under quantization, and will later be used in the construction of fermionic state spaces, by analogy with the Schr¨odinger and Bargmann-Fock constructions in the bosonic case.

中文目录

Chapter contents

• 30.1 The Grassmann algebra of polynomials on anticommuting generators
• 30.2 Pseudo-classical mechanics and the fermionic Poisson bracket
• 30.3 Examples of pseudo-classical mechanics
  • 30.3.1 The pseudo-classical spin degree of freedom
  • 30.3.2 The pseudo-classical fermionic oscillator
• 30.4 For further reading

Previous chapter: 29 Clifford Algebras and Geometry

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来源版本：2025-10-20

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