34 The Pauli Equation and the Dirac Operator

In chapter 33 we considered supersymmetric quantum mechanical systems where both the bosonic and fermionic variables that get quantized take values in an even dimensional space ${\mathbf{R}}^{2d}={\mathbf{C}}^d$ . There are then two diferent operators $Q_1$ and $Q_2$ that are square roots of the Hamiltonian operator. It turns out that there are much more interesting quantum mechanics systems that can be defined by quantizing bosonic variables in phase space ${\mathbf{R}}^{2d}$ , and fermionic variables in ${\mathbf{R}}^d$ The operators appearing in such a theory will be given by the tensor product of the Weyl algebra in 2d variables and the Cliford algebra in $d$ variables, and there will be a distinguished operator that provides a square root of the Hamiltonian.

This is equivalent to the fact that introduction of fermionic variables and the Cliford algebra provides the Casimir operator $-|\mathbf{P}{|}^2$ for the Euclidean group $E(3)$ with a square root: the Dirac operator $\partial /.$ This leads to a new way to construct irreducible representations of the group of spatial symmetries, using a new sort of quantum free particle, one carrying an internal “spin” degree of freedom due to the use of the Cliford algebra. Remarkably, fundamental matter particles are well-described in exactly this way, both in the non-relativistic theory we study in this chapter as well as in the relativistic theory to be studied later.

中文目录

Chapter contents

• 34.1 The Pauli-Schr¨odinger equation and free spin 1/2 particles in d=3
• 34.2 Solutions of the Pauli equation and representations of E(3)
• 34.3 The E(3) -invariant inner product
• 34.4 The Dirac operator
• 34.5 For further reading

Previous chapter: 33 Supersymmetry, Some Simple Examples

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 358、359、360、361、362、363、364、365、366、367

来源版本：2025-10-20

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