definition 12.64 Strongly continuous one-parameter unitary group

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1739 · p. 432

ground object -- no derivation owed

Rests on

depends_on definition 12.41 The operator classes ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- depends_on proposition 12.21 Characterization of orthogonal projections ¶
  - depends_on definition 12.20 Orthogonal projection operator ¶
    - depends_on theorem 12.18 Projection theorem ¶
      - depends_on proposition 12.17 The complement is always a closed subspace ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
  - 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 definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.19 Norm ¶
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on definition 5.19 Norm ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - proves proof ch:10-hilbert-spaces@proof-4 ¶
    - proves proof ch:10-hilbert-spaces@proof-19 ¶
  - depends_on theorem 12.46 Riesz representation ¶
    - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.46 Functional ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-24 ¶
  - 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 ¶ ↺

Supports

depends_on definition A.580 Weyl operator ¶
- depends_on definition A.585 The Gaussian average of the Weyl operators ¶
  - depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
    - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶ ↺
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
    - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
- depends_on lemma A.581 Composition law ¶
  - depends_on lemma A.582 Joint strong continuity ¶
    - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
      - depends_on definition A.585 The Gaussian average of the Weyl operators ¶ ↺
  - depends_on lemma A.590 The Gram matrix is universal ¶ ↺
depends_on definition 12.109 Weyl system ¶
- depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
- depends_on definition A.580 Weyl operator ¶ ↺
- depends_on example 12.110 The Schrödinger system ¶
  - depends_on proposition A.592 The vacuum of the Schrödinger system ¶ ↺
  - depends_on theorem A.579 Stone–von Neumann ¶
- depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
  - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
- depends_on theorem A.579 Stone–von Neumann ¶ ↺
- depends_on theorem 12.114 Stone–von Neumann ¶
depends_on lemma A.255 Smoothed vectors lie in the domain ¶
- depends_on proposition A.256 Density ¶
  - depends_on proposition A.259 The generator is self-adjoint ¶
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - depends_on proposition A.263 The two constructions are inverse ¶
depends_on lemma A.581 Composition law ¶ ↺
depends_on lemma A.582 Joint strong continuity ¶ ↺
depends_on proposition A.256 Density ¶ ↺
depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
- depends_on theorem 12.66 Stone ¶
  - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
  - depends_on definition 12.109 Weyl system ¶ ↺
  - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
    - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on proposition A.258 The generator is closed ¶
    - depends_on proposition A.263 The two constructions are inverse ¶ ↺
    - depends_on proposition A.259 The generator is self-adjoint ¶ ↺
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
    - depends_on theorem A.253 Stone ¶
      - depends_on proposition A.263 The two constructions are inverse ¶ ↺
  - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
  - depends_on theorem 25.34 Stone–von Neumann ¶
    - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on example 25.45 Weyl ordering collapses ¶
    - depends_on theorem 25.38 Groenewold–van Hove ¶
      - depends_on corollary 25.41 Quantization is not a functor ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
      - depends_on remark 24.55 What polarization costs ¶
depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
depends_on theorem A.253 Stone ¶ ↺
depends_on theorem 12.66 Stone ¶ ↺
depends_on theorem 25.34 Stone–von Neumann ¶ ↺

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`	→	The operator classes	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1749
`depends_on`	→	Continuous map	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1749
`depends_on`	←	Weyl operator	declared	appendices/A-long-proofs.tex:27740
`depends_on`	←	Weyl system	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3009
`depends_on`	←	Smoothed vectors lie in the domain	declared	appendices/A-long-proofs.tex:12704
`depends_on`	←	Composition law	declared	appendices/A-long-proofs.tex:27761
`depends_on`	←	Joint strong continuity	declared	appendices/A-long-proofs.tex:27852
`depends_on`	←	Density	declared	appendices/A-long-proofs.tex:12748
`depends_on`	←	Exponential of a bounded self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1775
`depends_on`	←	The generator is symmetric, and generates the motion	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1849
`depends_on`	←	Stone	declared	appendices/A-long-proofs.tex:12645
`depends_on`	←	Stone	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1825
`depends_on`	←	Stone–von Neumann	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1272