remark 24.55 What polarization costs

Rests on

• depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶
  • depends_on definition 24.53 Real polarization and polarized sections ¶
    • 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 definition 24.49 Prequantum datum ¶
      • depends_on definition 14.103 Connection $1$-form ¶
        • depends_on definition 14.97 Principal 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 24.50 Prequantum operator ¶
    • depends_on definition 24.14 Hamiltonian vector field ¶
      • depends_on definition 24.8 Symplectic manifold ¶ ↺
      • depends_on equation 22.4 eq:ham-pdot ¶
      • depends_on equation 22.3 eq:ham-qdot ¶
    • depends_on definition 24.49 Prequantum datum ¶ ↺
  • depends_on equation 24.35 eq:sym-prequantum-local ¶
  • proves proof ch:07-symplectic-geometry@proof-17 ¶
• 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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