example A.394 $D = 4$

open in the book · appendices/A-long-proofs.tex:19284 · p. 2984

example -- no derivation owed by its kind

Rests on

depends_on theorem A.385 Labels of a massive representation ¶
- depends_on definition A.384 The mass shell and the standard momentum ¶
  - depends_on lemma 14.68 The algebra fixing a timelike momentum ¶
    - depends_on equation 14.23 eq:lie-sopq-defining ¶
    - depends_on proposition 14.22 The rank of $\mathfrak{so}(p,q)$ ¶
      - depends_on definition 14.20 Cartan subalgebra and rank ¶
      - depends_on equation 14.90 eq:lie-algebrasopq ¶
      - depends_on theorem 14.21 Racah ¶
      - proves proof ch:12-lie-groups-fibre-bundles@proof-7 ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-32 ¶
  - depends_on notation 14.1 not:lie-indices ¶
- depends_on lemma 14.68 The algebra fixing a timelike momentum ¶ ↺
- depends_on theorem 14.21 Racah ¶ ↺
- proves proof app:A-long-proofs@proof-238 ¶
depends_on theorem 14.65 The two Casimir operators of the Poincaré algebra in $3+1$ dimensions ¶
- depends_on definition 14.63 Pauli–Lubanski vector ¶
  - depends_on equation 14.86 eq:lie-sopq-gen ¶
  - depends_on equation 14.82 eq:lie-transl-gen ¶
- depends_on lemma 14.64 Properties of $W$ ¶
  - depends_on definition 14.63 Pauli–Lubanski vector ¶ ↺
  - depends_on equation 14.90 eq:lie-algebrasopq ¶ ↺
  - depends_on equation 14.97 eq:lie-jp-comm ¶
  - depends_on equation 14.84 eq:lie-transl-comm ¶
  - depends_on lemma 14.62 The Levi-Civita symbol is invariant ¶
    - depends_on equation 13.223 eq:mfd-epsilon-density ¶
    - depends_on example 13.94 The two densities this chapter uses ¶
      - depends_on definition 13.92 Tensor density ¶
      - depends_on proposition 13.93 Algebra of densities ¶
    - depends_on notation 14.1 not:lie-indices ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@proof-29 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-30 ¶
- depends_on proposition 14.60 The quadratic invariant ¶
  - depends_on definition 14.8 Casimir element ¶
    - depends_on definition 14.5 Universal enveloping algebra ¶
      - depends_on equation 14.5 eq:lie-structconst ¶
  - depends_on equation 14.97 eq:lie-jp-comm ¶ ↺
  - depends_on equation 14.84 eq:lie-transl-comm ¶ ↺
  - proves proof ch:12-lie-groups-fibre-bundles@proof-28 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-31 ¶
depends_on theorem 14.43 The irreducible representations of $\mathfrak{su}(2)$ ¶
- depends_on equation 14.44 eq:lie-su2-algebra ¶
- depends_on lemma 14.41 Ladder algebra ¶
  - depends_on corollary 14.17 The quadratic Casimir ¶
    - depends_on corollary 14.12 Total antisymmetry of the structure constants ¶
      - depends_on definition 14.10 Killing form ¶
      - depends_on lemma 14.11 Invariance of the Killing form ¶
      - proves proof ch:12-lie-groups-fibre-bundles@proof-3 ¶
    - depends_on theorem 14.13 Cartan's criterion ¶
      - depends_on definition 14.10 Killing form ¶ ↺
      - proves proof ch:12-lie-groups-fibre-bundles@prooflink-2 ¶
    - depends_on theorem 14.16 Invariant tensors give Casimir operators ¶
      - depends_on definition 14.8 Casimir element ¶ ↺
      - depends_on definition 14.14 Invariant symmetric tensor ¶
      - depends_on definition 14.5 Universal enveloping algebra ¶ ↺
      - proves proof ch:12-lie-groups-fibre-bundles@proof-4 ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-5 ¶
  - depends_on definition 14.40 Ladder operators and the Casimir ¶
    - depends_on equation 14.44 eq:lie-su2-algebra ¶ ↺
  - depends_on equation 14.15 eq:lie-killing-components ¶
  - depends_on equation 14.44 eq:lie-su2-algebra ¶ ↺
  - proves proof ch:12-lie-groups-fibre-bundles@proof-16 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-17 ¶

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type	direction	node	provenance	where
`depends_on`	→	Labels of a massive representation	declared	appendices/A-long-proofs.tex:19324
`depends_on`	→	The two Casimir operators of the Poincaré algebra in $3+1$ dimensions	declared	appendices/A-long-proofs.tex:19324
`depends_on`	→	The irreducible representations of $\mathfrak{su}(2)$	declared	appendices/A-long-proofs.tex:19324