proposition 101.63 Parameter counting, and why three generations permit $CP$ violation

Rests on

• depends_on equation 101.67 eq:weak-ckm-current ¶
• depends_on theorem 101.62 The mixing matrix is unitary ¶
  • depends_on proposition 5.155 Averaging trick ¶
    • depends_on definition 5.18 Inner product ¶
      • depends_on definition 4.33 Vector space ¶
        • depends_on definition 4.8 Commutativity; abelian structure ¶
        • depends_on definition 4.32 Field ¶
        • depends_on definition 4.31 Module ¶
    • depends_on definition 5.145 Equivalent representations ¶
      • depends_on definition 5.141 Representation of a group ¶
        • depends_on definition 4.21 Group ¶
        • depends_on definition 4.43 Group homomorphism ¶
        • depends_on definition 4.48 Symmetric group ¶
    • depends_on definition 5.144 Unitary representation ¶
      • depends_on definition 5.18 Inner product ¶ ↺
      • depends_on definition 5.141 Representation of a group ¶ ↺
      • depends_on equation 5.171 eq:rep-unitary-group ¶
    • depends_on equation 5.183 eq:rep-unitary-invariance ¶
    • depends_on proposition 5.28 Gram–Schmidt ¶
      • depends_on definition 5.14 Linear independence ¶
        • depends_on definition 5.5 Linear combination ¶
      • depends_on definition 5.18 Inner product ¶ ↺
      • depends_on definition 5.27 Orthonormal basis ¶
        • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.25 Unit vector ¶
      • depends_on equation 5.44 eq:lin-norm-assoc ¶
      • proves proof ch:03-linear-algebra-representations@proof-5 ¶
    • proves proof ch:03-linear-algebra-representations@proof-66 ¶
  • depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
    • depends_on corollary 5.29 Orthogonal decomposition ¶
      • depends_on definition 5.18 Inner product ¶ ↺
      • depends_on definition 5.7 Vector subspace ¶
        • depends_on definition 5.5 Linear combination ¶ ↺
      • depends_on proposition 5.28 Gram–Schmidt ¶ ↺
      • proves proof ch:03-linear-algebra-representations@proof-6 ¶
    • depends_on definition 5.148 Irreducible representation ¶
      • 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 5.149 Totally reducible representation ¶
      • depends_on definition 5.147 Invariant subspace ¶ ↺
      • depends_on equation 5.52 eq:lin-orthogonal-complement ¶
    • depends_on lemma 5.150 Invariance of the orthogonal complement ¶
      • depends_on definition 5.147 Invariant subspace ¶ ↺
      • depends_on definition 5.144 Unitary representation ¶ ↺
      • depends_on equation 4.25 eq:alg-hom-inverse ¶
      • depends_on equation 5.52 eq:lin-orthogonal-complement ¶ ↺
      • depends_on equation 5.183 eq:rep-unitary-invariance ¶ ↺
      • proves proof ch:03-linear-algebra-representations@proof-63 ¶
    • proves proof ch:03-linear-algebra-representations@proof-65 ¶
  • proves proof ch:03-weak-interactions@proof-43 ¶
• depends_on theorem 101.59 Glashow–Iliopoulos–Maiani ¶
  • depends_on equation 101.62 eq:weak-cabibbo-rotation ¶
  • depends_on proposition 101.58 A single rotated doublet produces a flavour-changing neutral current ¶
    • depends_on equation 101.62 eq:weak-cabibbo-rotation ¶ ↺
    • proves proof ch:03-weak-interactions@proof-41 ¶
  • proves proof ch:03-weak-interactions@proof-42 ¶
• proves proof ch:03-weak-interactions@proof-44 ¶

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