theorem 102.12 Which quark combinations can be colour singlets

open in the book · parts/11-qft-standard-model/04-qcd.tex:502 · p. 1995

Rests on

depends_on definition 102.10 Colour ¶
depends_on proposition 14.55 The fundamental, the antifundamental and the adjoint ¶
- depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
  - 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 14.6 eq:lie-expgen ¶
  - depends_on remark 14.6 What $U(\mathfrak{g})$ is for ¶
  - depends_on theorem 5.158 Schur's first lemma ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.147 Invariant subspace ¶
      - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.141 Representation of a group ¶
    - depends_on definition 5.148 Irreducible representation ¶
      - depends_on definition 5.147 Invariant subspace ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-67 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-6 ¶
- depends_on definition 14.51 Gell-Mann basis ¶
  - depends_on definition 14.49 The special unitary group in three dimensions ¶
- depends_on proposition 14.53 Killing form of $\mathfrak{su}(3)$ ¶
  - depends_on definition 14.51 Gell-Mann basis ¶ ↺
  - depends_on equation 14.15 eq:lie-killing-components ¶
  - depends_on proposition 14.52 Structure constants and the symmetric tensor ¶
    - depends_on definition 14.51 Gell-Mann basis ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@proof-22 ¶
    - proves proof ch:12-lie-groups-fibre-bundles@prooflink-5 ¶
  - depends_on theorem 14.13 Cartan's criterion ¶
    - depends_on definition 14.10 Killing form ¶
      - depends_on equation 14.5 eq:lie-structconst ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@prooflink-2 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-23 ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-25 ¶
depends_on theorem 5.158 Schur's first lemma ¶ ↺
proves proof ch:04-qcd@proof-5 ¶

Supports

depends_on corollary 102.13 The selection rule of the eightfold way ¶

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`	→	Colour	declared	parts/11-qft-standard-model/04-qcd.tex:508
`depends_on`	→	The fundamental, the antifundamental and the adjoint	declared	parts/11-qft-standard-model/04-qcd.tex:508
`depends_on`	→	Schur's first lemma	declared	parts/11-qft-standard-model/04-qcd.tex:508
`depends_on`	←	The selection rule of the eightfold way	declared	parts/11-qft-standard-model/04-qcd.tex:551
`proves`	←	ch:04-qcd@proof-5	declared	parts/11-qft-standard-model/04-qcd.tex:511