definition A.585 The Gaussian average of the Weyl operators

open in the book · appendices/A-long-proofs.tex:28017 · p. 3072

ground object -- no derivation owed

Rests on

depends_on definition A.580 Weyl operator ¶
- 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 definition 12.109 Weyl system ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
  - 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 5.19 Norm ¶
    - 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 ¶
      - 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 ¶
- depends_on equation 12.71 eq:hilbert-weyl-relation ¶
depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
- depends_on lemma A.254 Riemann integral of a continuous curve ¶
  - depends_on definition 12.2 Hilbert space ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
  - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
    - depends_on definition 12.2 Hilbert space ¶ ↺
    - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-4 ¶
    - proves proof ch:10-hilbert-spaces@proof-1 ¶
  - proves proof app:A-long-proofs@proof-157 ¶
- depends_on lemma A.582 Joint strong continuity ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
  - depends_on lemma A.581 Composition law ¶
    - depends_on definition A.580 Weyl operator ¶ ↺
    - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    - depends_on equation 12.71 eq:hilbert-weyl-relation ¶ ↺
    - proves proof app:A-long-proofs@proof-347 ¶
  - proves proof app:A-long-proofs@proof-348 ¶
- depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
- proves proof app:A-long-proofs@proof-349 ¶

Supports

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 ¶

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`	→	Weyl operator	declared	appendices/A-long-proofs.tex:28034
`depends_on`	→	Absolutely convergent operator-valued integrals	declared	appendices/A-long-proofs.tex:28034
`depends_on`	←	The Gaussian average is a non-zero projector that absorbs the Weyl operators	declared	appendices/A-long-proofs.tex:28082