theorem 12.75 The canonical commutation relation admits no bounded solution

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2047 · p. 435

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- 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 ¶
- depends_on definition 12.35 Bounded operator; operator 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 ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - proves proof ch:10-hilbert-spaces@proof-4 ¶
- proves proof ch:10-hilbert-spaces@proof-19 ¶
proves proof ch:10-hilbert-spaces@proof-37 ¶

Supports

depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
depends_on corollary 25.30 The relation cannot be realized by matrices ¶
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 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.589 Cyclic subspaces and the rank of the average ¶
      - 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 lemma A.590 The Gram matrix is universal ¶ ↺
- 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 ¶

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`	→	Bounded operator; operator norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2055
`depends_on`	→	$\mathcal{B}(\mathcal{H})$ is a Banach algebra	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2055
`depends_on`	←	Position and momentum are unbounded, and cannot be everywhere defined	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2091
`depends_on`	←	The relation cannot be realized by matrices	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1117
`depends_on`	←	Weyl system	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3009
`proves`	←	ch:10-hilbert-spaces@proof-37	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2058