definition 24.49 Prequantum datum

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1680 · p. 868

ground object -- no derivation owed

Rests on

depends_on definition 14.103 Connection $1$-form ¶
- depends_on definition 14.97 Principal bundle ¶
  - depends_on definition 14.95 Fibre bundle ¶
- depends_on notation 14.101 Algebra-valued forms and their bracket ¶
depends_on definition 14.105 Curvature $2$-form ¶
- depends_on definition 14.103 Connection $1$-form ¶ ↺
- depends_on notation 14.101 Algebra-valued forms and their bracket ¶ ↺
depends_on definition 24.8 Symplectic manifold ¶
- depends_on definition 13.105 Closed form ¶
  - depends_on definition 13.103 Exterior derivative ¶
    - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on equation 13.239 eq:mfd-dual-basis ¶
  - depends_on definition 13.98 $k$-form ¶ ↺
- depends_on definition 13.98 $k$-form ¶ ↺
- depends_on definition 24.2 Symplectic vector space ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-54 ¶

Supports

depends_on definition 24.53 Real polarization and polarized sections ¶
- depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶
  - depends_on remark 24.55 What polarization costs ¶
depends_on definition 24.50 Prequantum operator ¶
- depends_on proposition 24.51 Prequantization is a Lie-algebra homomorphism ¶
  - depends_on example 24.52 The prequantum line and the Schrödinger representation ¶
- depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶ ↺
depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
- depends_on remark 24.57 The half-integer, and the honest status of the construction ¶

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`	→	Connection $1$-form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1701
`depends_on`	→	Curvature $2$-form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1701
`depends_on`	→	Symplectic manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1701
`depends_on`	←	Real polarization and polarized sections	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1815
`depends_on`	←	Prequantum operator	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1713
`depends_on`	←	The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1900