theorem 25.53 Properties of the Wigner function

open in the book · parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1879 · p. 894

Rests on

depends_on equation 25.53 eq:pq-wigner ¶
depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
- depends_on definition 12.2 Hilbert space ¶
  - depends_on definition 5.18 Inner product ¶
    - 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 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 ¶
  - depends_on equation 5.44 eq:lin-norm-assoc ¶
- depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on definition 5.18 Inner product ¶ ↺
  - proves proof ch:03-linear-algebra-representations@proof-4 ¶
- proves proof ch:10-hilbert-spaces@proof-1 ¶
depends_on theorem 17.32 Fourier inversion ¶
- depends_on definition 17.29 Fourier transform; the treatise convention ¶
- depends_on equation 17.42 eq:ft-gaussian ¶
- proves proof ch:15-fourier-integral-transforms@proof-20 ¶
proves proof ch:08-poisson-quantum-bridge@proof-20 ¶

Every logical edge within two steps of this node.

type	direction	node	provenance	where
`depends_on`	→	eq:pq-wigner	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1902
`depends_on`	→	Cauchy–Schwarz and continuity of the inner product	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1902
`depends_on`	→	Fourier inversion	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1902
`depends_on`	←	Overlaps and the resolution of unity	declared	appendices/A-long-proofs.tex:28914
`proves`	←	ch:08-poisson-quantum-bridge@proof-20	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1906