theorem 25.38 Groenewold–van Hove

open in the book · parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1342 · p. 888

Rests on

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 definition 6.5 Open cover ¶
    - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
      - proves proof ch:10-hilbert-spaces@proof-11 ¶
    - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      - depends_on definition 5.41 Adjoint ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on theorem 12.46 Riesz representation ¶
      - proves proof ch:10-hilbert-spaces@proof-20 ¶
  - 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 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 ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-21 ¶
    - 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 definition 12.41 The operator classes ¶ ↺
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
    - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-23 ¶
    - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
      - depends_on corollary 12.19 Double complement; the density criterion ¶
      - depends_on definition 12.41 The operator classes ¶ ↺
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - proves proof ch:10-hilbert-spaces@proof-30 ¶
    - 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 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 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.43 Group homomorphism ¶
      - depends_on definition 4.21 Group ¶ ↺
      - depends_on definition 3.43 Map ¶
    - depends_on definition 4.48 Symmetric group ¶
      - depends_on definition 3.47 Bijective map ¶
- depends_on definition 5.147 Invariant subspace ¶
  - depends_on definition 5.7 Vector subspace ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 ¶

Supports

depends_on corollary 25.41 Quantization is not a functor ¶
- depends_on definition 25.42 Ordering rule ¶
  - depends_on definition 25.44 Weyl and normal ordering ¶
    - depends_on definition 25.50 Star product and Moyal bracket ¶
    - depends_on definition 25.52 Wigner function ¶
      - depends_on definition A.605 The analytic continuation of the overlaps ¶
    - depends_on example 25.45 Weyl ordering collapses ¶
    - depends_on proposition 25.46 Normal ordering disagrees already at degree two ¶
  - depends_on proposition 25.43 The size of the ambiguity ¶
    - depends_on proposition 25.48 Where the ambiguity does and does not bite ¶
depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
depends_on remark 24.55 What polarization costs ¶

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`	→	eq:pq-dirac-rule	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1356
`depends_on`	→	Stone–von Neumann	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1356
`depends_on`	→	Schur's second lemma	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1356
`depends_on`	←	Quantization is not a functor	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1555
`depends_on`	←	The half-integer, and the honest status of the construction	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1979
`depends_on`	←	What polarization costs	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1881
`proves`	←	ch:08-poisson-quantum-bridge@proof-16	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1452