definition 14.8 Casimir element

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:331 · p. 545

ground object -- no derivation owed

Rests on

depends_on definition 14.5 Universal enveloping algebra ¶
- depends_on equation 14.5 eq:lie-structconst ¶
  - step_from equation 14.4 eq:lie-gengrlie ¶
    - step_from equation 14.3 eq:lie-generators ¶
    - step_from equation 14.2 eq:lie-varincor ¶

Supports

depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
- depends_on proposition 14.55 The fundamental, the antifundamental and the adjoint ¶
  - depends_on corollary A.377 The adjoint representation is irreducible ¶
  - depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶
  - depends_on theorem 14.57 $\vect{3}\otimes\bar{\vect{3}} = \vect{1}\oplus\vect{8}$ ¶
    - depends_on proposition 102.6 Mesons are $q\bar{q}$, baryons are $qqq$ ¶
      - assumes experiment Deep Inelastic Scattering ¶
      - depends_on phenomenon 116.38 The quark spins do not account for the proton spin ¶
      - depends_on phenomenon 102.9 The $\Delta^{++}$ violates the spin-statistics theorem ¶
      - depends_on phenomenon 102.7 The nucleon magnetic moments come out right ¶
      - depends_on proposition 116.29 The Gross–Llewellyn Smith sum rule in the parton model ¶
  - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
    - depends_on corollary 102.13 The selection rule of the eightfold way ¶
- depends_on theorem A.354 Harish-Chandra, quoted: the labels separate ¶
depends_on lemma A.347 Central is the same as invariant ¶
- depends_on proposition A.348 Symmetrization identifies the invariants ¶
  - depends_on lemma A.349 Lifting generators through the filtration ¶
depends_on lemma 14.87 A rescaled Casimir stays central ¶
depends_on proposition 14.60 The quadratic invariant ¶
- depends_on theorem 14.65 The two Casimir operators of the Poincaré algebra in $3+1$ dimensions ¶
  - depends_on example A.394 $D = 4$ ¶
depends_on theorem A.346 Racah ¶
- depends_on example A.359 The Lorentz algebra ¶
- depends_on example A.357 Rank one: $\mathfrak{su}(2)$ ¶
- depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶ ↺
depends_on theorem A.354 Harish-Chandra, quoted: the labels separate ¶ ↺
depends_on theorem 14.16 Invariant tensors give Casimir operators ¶
- depends_on corollary A.328 The Casimir elements are not accidentally zero ¶
- depends_on corollary 14.17 The quadratic Casimir ¶
  - depends_on lemma 14.41 Ladder algebra ¶
    - depends_on proposition 14.44 Existence: every allowed $j$ is realized ¶
      - depends_on proposition 14.45 Which representations descend to $\SO(3,\R)$ ¶
    - depends_on theorem 14.43 The irreducible representations of $\mathfrak{su}(2)$ ¶
      - depends_on example A.394 $D = 4$ ¶ ↺
      - depends_on example A.357 Rank one: $\mathfrak{su}(2)$ ¶ ↺
      - depends_on proposition 14.44 Existence: every allowed $j$ is realized ¶ ↺
      - depends_on theorem 14.47 Clebsch–Gordan series ¶
  - depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶
    - depends_on example A.358 Rank two: $\mathfrak{su}(3)$ ¶ ↺
- depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
depends_on theorem 14.21 Racah ¶
- depends_on proposition 14.22 The rank of $\mathfrak{so}(p,q)$ ¶
  - depends_on lemma 14.68 The algebra fixing a timelike momentum ¶
    - depends_on definition A.384 The mass shell and the standard momentum ¶
      - depends_on lemma A.387 Standard boost and transitivity ¶
      - depends_on lemma A.386 The little group ¶
      - depends_on lemma A.388 The invariant measure ¶
      - depends_on theorem A.385 Labels of a massive representation ¶
      - depends_on theorem A.392 Wigner–Mackey classification; quoted ¶
    - depends_on lemma A.386 The little group ¶ ↺
    - depends_on theorem A.385 Labels of a massive representation ¶ ↺
- depends_on proposition 14.54 The two Casimir operators of $\mathfrak{su}(3)$ ¶ ↺
- depends_on theorem A.385 Labels of a massive representation ¶ ↺

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`	→	Universal enveloping algebra	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:339
`depends_on`	←	A Casimir acts as a number on an irreducible representation	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:580
`depends_on`	←	Central is the same as invariant	declared	appendices/A-long-proofs.tex:16937
`depends_on`	←	A rescaled Casimir stays central	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3803
`depends_on`	←	The quadratic invariant	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2754
`depends_on`	←	Racah	declared	appendices/A-long-proofs.tex:16923
`depends_on`	←	Harish-Chandra, quoted: the labels separate	declared	appendices/A-long-proofs.tex:17204
`depends_on`	←	Invariant tensors give Casimir operators	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:490
`depends_on`	←	Racah	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:642