B.21 Chapter 47

B.21 Chapter 47

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation

Problem 1:

Show that complex-valued solutions of the Dirac equation 47.2 correspond in the non-relativistic limit (energies small compared to the mass) to solutions of the Pauli-Schr¨odinger equation 34.3. Hint: write solutions in the form

$$\psi (t,\mathbf{x})=e^{-imt}\varphi (t,\mathbf{x})$$

Problem 2:

For two copies of the Majorana fermion theory, with the same mass m and field operators ${\hat{\Psi }}_1,{\hat{\Psi }}_2$ , show that the theory has an $SO(2)$ symmetry, and find the Lie algebra representation operator $\hat{Q}$ for this symmetry. Compute the commutators

$$[\hat{Q},{\hat{\Psi }}_1],[\hat{Q},{\hat{\Psi }}_2]$$

Problem 3:

Show that, for $m=0$ , the Majorana fermion theory has an $SO(2)$ symmetry given by the action

$$\Psi (\mathbf{x})\to \cos \theta \Psi (\mathbf{x})+\sin \theta {\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3\Psi (\mathbf{x})$$

What is the corresponding Lie algebra representation operator (this is called the “axial charge”)?

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Previous section: B.20 Chapters 45 and 46

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原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

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