B.18 Chapters 40 to 42

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

Problem 1:

If $P_{0}, P_{j}, L_{j}, K_{j}$ are the operators in any Lie algebra representation of the Poincar´e group corresponding to the basis elements $t_{0}, t_{j}, l_{j}, k_{j}$ of the Lie algebra of the group, show that the operator

- P_{0}^{2} + P_{1}^{2} + P_{2}^{2} + P_{3}^{2}

commutes with $P_{0}, P_{j}, L_{j}, K_{j}$ , and thus is a Casimir operator for the Poincar´e Lie algebra.

Problem 2:

Show that the Lie algebra $so (4, C)$ is sl(2, $C$ ) ⊕ sl(2, $C$ ). Within this Lie algebra, identify the sub-Lie algebras of the groups Spin(4), Spin(3, 1) and $S p in (2, 2)$

Problem 3:

Find an explicit realization of the Cliford algebra Clif(4, 0) in terms of 4 by 4 matrices $(γ$ matrices for this case) and use this to realize the group $S p in (4)$ as a group of 4 by 4 matrices (hint: recall that the Lie algebra of the spin group is given by products of two generators). Use these matrices to explicitly construct the representations of $S p in (4)$ on two kinds of half-spinors, on complexified vectors $(C^{4})$ , and the adjoint representation on the Lie algebra.

Problem 4:

The Pauli-Lubanski operator is the four-component operator

W_{0} = - P \cdot L, W = - P_{0} L + P \times K

(same notation as in problem 1) Show that

W^{2} = - W_{0} W_{0} + W_{1} W_{1} + W_{2} W_{2} + W_{3} W_{3}

commutes with the energy-momentum operator $(P_{0}, P)$

Show that $W^{2}$ is a Casimir operator for the Poincar´e Lie algebra.

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

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

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：7f317d1896fa748af4cb9570310ed7801a8bef7ef97dd5d1d2ea35f0099d5952

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书架

B.18 Chapters 40 to 42