31 Fermionic Quantization and Spinors

In this chapter we’ll begin by investigating the fermionic analog of the notion of quantization, which takes functions of anticommuting variables on a phase space with symmetric bilinear form $(\cdot ,\cdot )$ and gives an algebra of operators with generators satisfying the relations of the corresponding Cliford algebra. We will then consider analogs of the constructions used in the bosonic case which there gave us the Schr¨odinger and Bargmann-Fock representations of the Weyl algebra on a space of states.

We know that for a fermionic oscillator with $d$ degrees of freedom, the algebra of operators will be $\mathrm{Cliff}(2d,\mathbf{C})$ , the algebra generated by annihilation and creation operators $a_{Fj},a_{F_j}†$ . These operators will act on ${\mathcal{H}}_F={\mathcal{F}}_d^{+}$ , a complex vector space of dimension $2^d$ , and this will provide a fermionic analog of the bosonic ${\Gamma }_{BF}^{\prime }$ acting on ${\mathcal{F}}_d.$ . Since the spin group consists of invertible elements of the Cliford algebra, it has a representation on ${\mathcal{F}}_d^{+}$ . This is known as the “spinor representation”, and it can be constructed by analogy with the construction of the metaplectic representation in the bosonic case. We’ll also consider the analog in the fermionic case of the Schr¨odinger representation, which turns out to have a problem with unitarity, but finds a use in physics as “ghost” degrees of freedom.

中文目录

Chapter contents

• 31.1 Quantization of pseudo-classical systems
  • 31.1.1 Quantization of the pseudo-classical spin
• 31.2 The Schr¨odinger representation for fermions: ghosts
• 31.3 Spinors and the Bargmann-Fock construction
• 31.4 Complex structures, U(d) SO(2 d) and the spinor representation
• 31.5 An example: spinors for SO(4)
• 31.6 For further reading

Previous chapter: 30 Anticommuting Variables and Pseudo-classical Mechanics

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原书 PDF · 印刷页 335、336、337、338、339、340、341、342、343、344、345、346、347、348

来源版本：2025-10-20

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