definition 26.20 Dirac bracket

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:596 · p. 906

ground object -- no derivation owed

Rests on

depends_on definition 22.29 Poisson bracket ¶
- depends_on definition 7.97 Partial derivative; gradient ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
- depends_on definition 22.14 Phase-space coordinate transformation ¶
  - depends_on definition 22.2 Hamiltonian ¶
    - depends_on definition 21.34 Generalized momentum ¶
    - depends_on definition 21.31 Lagrangian ¶
depends_on proposition 26.19 Second-class constraints come in pairs ¶
- depends_on definition 26.12 First and second class ¶
  - depends_on definition 26.11 The final constraint set ¶
    - depends_on proposition 26.10 The four outcomes ¶
      - depends_on definition 26.9 Consistency conditions ¶
      - depends_on remark 26.7 The regularity assumption, stated once ¶
      - proves proof ch:09-constrained-hamiltonian-dynamics@proof-3 ¶
    - depends_on remark 26.7 The regularity assumption, stated once ¶ ↺
  - depends_on definition 22.29 Poisson bracket ¶ ↺
- 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 4.33 Vector space ¶
  - 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 ¶
- proves proof ch:09-constrained-hamiltonian-dynamics@proof-7 ¶

Supports

depends_on definition 26.23 Gauge conditions ¶
- depends_on proposition 26.24 The Faddeev–Popov determinant ¶
  - depends_on remark 26.26 What the factor is doing ¶
depends_on example 26.22 Holonomic constraints ¶
depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
depends_on theorem 26.21 Properties of the Dirac bracket ¶
- depends_on postulate 26.47 Dirac quantization ¶
  - depends_on definition 26.50 The BRST charge ¶
    - depends_on notation A.646 Gradings and brackets ¶
      - depends_on definition A.647 Koszul–Tate differential ¶
      - depends_on lemma A.652 Nilpotency, order by order ¶
    - depends_on theorem 26.51 Existence and uniqueness of the BRST charge ¶
  - depends_on proposition 26.48 Consistency of the state conditions ¶

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`	→	Poisson bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:608
`depends_on`	→	Second-class constraints come in pairs	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:608
`depends_on`	←	Gauge conditions	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:792
`depends_on`	←	Holonomic constraints	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:705
`depends_on`	←	Jacobi identity for the Dirac bracket	declared	appendices/A-long-proofs.tex:29694
`depends_on`	←	Properties of the Dirac bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:628