proposition 12.111 The Weyl relation is a covariance statement

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:3041 · p. 446

Rests on

depends_on definition 12.109 Weyl system ¶
- 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 theorem 12.75 The canonical commutation relation admits no bounded solution ¶
  - depends_on definition 12.35 Bounded operator; operator norm ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-37 ¶
- 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 ¶
depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
- depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
- depends_on theorem 12.66 Stone ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-34 ¶
depends_on theorem 12.66 Stone ¶ ↺
proves proof ch:10-hilbert-spaces@proof-50 ¶

Supports

depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Weyl system	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3051
`depends_on`	→	The generator is symmetric, and generates the motion	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3051
`depends_on`	→	Stone	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3051
`depends_on`	←	Commutator of the momentum with a function of the position	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3111
`proves`	←	ch:10-hilbert-spaces@proof-50	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3055