remark 24.57 The half-integer, and the honest status of the construction

Rests on

• depends_on proposition 23.38 Bohr–Sommerfeld and Einstein–Brillouin–Keller quantization ¶
  • depends_on definition 23.28 Action variables ¶
    • depends_on definition 16.99 Abbreviated action ¶
      • depends_on definition 16.96 Action; Hamilton's principle ¶
        • depends_on definition 21.31 Lagrangian ¶
        • depends_on equation 16.11 eq:calcvar-basic-functional ¶
      • depends_on equation 16.86 eq:calcvar-hamilton-action ¶
    • depends_on definition 23.13 Additive separability ¶
      • depends_on definition 23.5 Complete integral ¶
        • depends_on equation 23.2 eq:hj-equation ¶
      • depends_on equation 23.15 eq:hj-characteristic-equation ¶
  • depends_on remark 9.79 Where the one half comes from ¶
  • depends_on theorem 9.78 Semiclassical quantisation of a well ¶
    • depends_on corollary 9.62 The semiclassical wave ¶
      • depends_on equation 9.62 eq:slt-wkb-ansatz ¶
      • depends_on equation 9.61 eq:slt-wkb-kappa ¶
      • depends_on proposition 9.61 Eikonal and transport equations ¶
        • depends_on equation 9.64 eq:slt-wkb-series ¶
        • depends_on proposition 9.60 Exact Riccati form ¶
        • proves proof ch:07-odes-sturm-liouville@proof-31 ¶
      • proves proof ch:07-odes-sturm-liouville@proof-32 ¶
    • depends_on equation 9.59 eq:slt-wkb-schrodinger ¶
    • depends_on theorem 9.72 Connection formulae ¶
      • depends_on corollary 9.62 The semiclassical wave ¶ ↺
      • depends_on equation 9.79 eq:slt-wkb-airy-ai ¶
      • depends_on equation 9.80 eq:slt-wkb-airy-bi ¶
      • depends_on equation 9.76 eq:slt-wkb-airy-equation ¶
      • depends_on equation 9.74 eq:slt-wkb-airy-scale ¶
      • proves proof ch:07-odes-sturm-liouville@proof-37 ¶
    • proves proof ch:07-odes-sturm-liouville@proof-40 ¶
  • proves proof ch:06-hamilton-jacobi@proof-10 ¶
• depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
  • depends_on definition 23.28 Action variables ¶ ↺
  • depends_on definition 24.49 Prequantum datum ¶
    • depends_on definition 14.103 Connection $1$-form ¶
      • depends_on definition 14.97 Principal bundle ¶
        • depends_on definition 14.95 Fibre bundle ¶
      • depends_on notation 14.101 Algebra-valued forms and their bracket ¶
    • depends_on definition 14.105 Curvature $2$-form ¶
      • depends_on definition 14.103 Connection $1$-form ¶ ↺
      • depends_on notation 14.101 Algebra-valued forms and their bracket ¶ ↺
    • depends_on definition 24.8 Symplectic manifold ¶
      • depends_on definition 13.105 Closed form ¶
        • depends_on definition 13.103 Exterior derivative ¶
        • depends_on definition 13.98 $k$-form ¶
      • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on definition 24.2 Symplectic vector space ¶
        • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  • depends_on theorem 23.31 Liouville–Arnold ¶
    • depends_on equation 22.41 eq:ham-poisson-bracket ¶
    • depends_on theorem 23.30 Motion in action–angle variables ¶
      • depends_on definition 23.28 Action variables ¶ ↺
      • depends_on proposition 22.18 The mixed generating functions ¶
        • depends_on definition 22.15 Canonical transformation ¶
        • depends_on equation 22.18 eq:ham-generating ¶
        • proves proof ch:05-hamiltonian-mechanics@proof-6 ¶
      • depends_on theorem 22.6 Hamilton's equations ¶
        • depends_on definition 22.2 Hamiltonian ¶
        • depends_on equation 21.39 eq:lag-el-conservative ¶
        • depends_on equation 21.43 eq:lag-generalized-momentum ¶
        • proves proof ch:05-hamiltonian-mechanics@proof-2 ¶
      • proves proof ch:06-hamilton-jacobi@proof-8 ¶
    • depends_on theorem 24.21 Liouville ¶
      • depends_on definition 24.20 Liouville volume ¶
        • depends_on definition 13.102 Wedge product ¶
        • depends_on definition 24.8 Symplectic manifold ¶ ↺
        • depends_on theorem 24.12 Darboux ¶
      • depends_on proposition 7.105 Clairaut–Schwarz ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-64 ¶
      • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
        • depends_on definition 24.14 Hamiltonian vector field ¶
        • depends_on definition 24.8 Symplectic manifold ¶ ↺
        • depends_on proposition 13.128 Cartan's magic formula ¶
        • proves proof ch:07-symplectic-geometry@proof-5 ¶
      • proves proof ch:07-symplectic-geometry@proof-8 ¶
    • proves proof ch:06-hamilton-jacobi@prooflink-1 ¶
  • proves proof ch:07-symplectic-geometry@proof-18 ¶
• depends_on theorem 25.38 Groenewold–van Hove ¶
  • depends_on equation 25.31 eq:pq-dirac-rule ¶
  • depends_on theorem 25.34 Stone–von Neumann ¶
    • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
      • depends_on definition 12.41 The operator classes ¶
        • depends_on definition 6.9 Compact set ¶
        • depends_on proposition 12.21 Characterization of orthogonal projections ¶
        • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      • depends_on definition 6.6 Continuous map ¶
        • depends_on definition 3.53 Preimage ¶
        • depends_on definition 6.2 Open set ¶
        • depends_on definition 6.1 Topological space ¶
    • depends_on equation 25.33 eq:pq-ccr ¶
    • depends_on theorem 12.66 Stone ¶
      • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      • depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
        • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
        • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
        • depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
        • proves proof ch:10-hilbert-spaces@proof-33 ¶
      • depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
        • depends_on definition 12.58 Projection-valued measure ¶
        • depends_on proposition 12.43 Norm of a self-adjoint operator ¶
        • depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
        • proves proof ch:10-hilbert-spaces@prooflink-2 ¶
      • proves proof ch:10-hilbert-spaces@prooflink-3 ¶
    • proves proof ch:08-poisson-quantum-bridge@prooflink-1 ¶
  • depends_on theorem 5.160 Schur's second lemma ¶
    • depends_on definition 5.53 Isomorphism ¶
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶
    • 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 5.147 Invariant subspace ¶
      • depends_on definition 5.7 Vector subspace ¶
        • depends_on definition 5.5 Linear combination ¶
      • 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-68 ¶
  • proves proof ch:08-poisson-quantum-bridge@proof-16 ¶

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