theorem 14.57 $\vect{3}\otimes\bar{\vect{3}} = \vect{1}\oplus\vect{8}$

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2283 · p. 567

Rests on

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 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-26 ¶
proves proof ch:12-lie-groups-fibre-bundles@prooflink-6 ¶

Supports

depends_on proposition 102.6 Mesons are $q\bar{q}$, baryons are $qqq$ ¶
- assumes experiment Deep Inelastic Scattering ¶
  - cites (data) derived bib 1 Combined measurement and QCD analysis of the inclusive e $\pm$ p scattering cross sections at HERA ¶
  - cites (data) derived bib 1 Combined measurement and QCD analysis of the inclusive e $\pm$ p scattering cross sections at HERA ¶ ↺
  - cites (evidence) derived bib 1 Combined measurement and QCD analysis of the inclusive e $\pm$ p scattering cross sections at HERA ¶ ↺
  - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶
  - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
  - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
  - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
  - cites (data) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
  - cites (evidence) derived bib 12 Combination of measurements of inclusive deep inelastic $e^{\pm}p$ scattering cross sections and QCD analysis of HERA data ¶ ↺
  - cites (context) derived bib 9 Electron-Ion Collider: The next QCD frontier ¶
  - cites (context) derived bib 9 Electron-Ion Collider: The next QCD frontier ¶ ↺
  - cites (context) derived bib 2 Asymptotic freedom in parton language ¶
  - … 102 more
- 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 ¶

Neighborhood

Every logical edge within two steps of this node.

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

Edges

type	direction	node	provenance	where
`depends_on`	→	The fundamental, the antifundamental and the adjoint	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2289
`depends_on`	←	Mesons are $q\bar{q}$, baryons are $qqq$	declared	parts/11-qft-standard-model/04-qcd.tex:320
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-26	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2292
`proves`	←	ch:12-lie-groups-fibre-bundles@prooflink-6	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2170