theorem 14.65 The two Casimir operators of the Poincaré algebra in $3+1$ dimensions

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2895 · p. 574

Rests on

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 definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on proposition 13.93 Algebra of densities ¶
      - depends_on definition 13.92 Tensor density ¶ ↺
      - proves proof ch:11-manifolds-tensors-curvature@proof-17 ¶
  - 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 ¶
      - step_from equation 14.4 eq:lie-gengrlie ¶
- 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 ¶

Supports

depends_on example A.394 $D = 4$ ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Pauli–Lubanski vector	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2903
`depends_on`	→	Properties of $W$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2903
`depends_on`	→	The quadratic invariant	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2903
`depends_on`	←	$D = 4$	declared	appendices/A-long-proofs.tex:19324
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-31	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2906