theorem 26.21 Properties of the Dirac bracket

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:611 · p. 906

Rests on

depends_on definition 26.20 Dirac bracket ¶
- 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 7.9 Real function ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
  - 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 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 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 ¶
depends_on definition 26.8 Total Hamiltonian ¶
- depends_on definition 22.29 Poisson bracket ¶ ↺
- depends_on proposition 26.6 The canonical Hamiltonian is defined on the constraint surface ¶
  - depends_on equation 26.4 eq:cd-canonical-hamiltonian ¶
  - depends_on proposition 26.5 Primary constraints ¶
    - depends_on definition 26.3 Singular Lagrangian ¶
      - depends_on equation 22.2 eq:ham-hessian ¶
      - depends_on proposition 22.3 Invertibility of the Legendre map ¶
    - depends_on remark 26.4 The constant-rank theorem is owed by Part II ¶
    - proves proof ch:09-constrained-hamiltonian-dynamics@proof-1 ¶
  - proves proof ch:09-constrained-hamiltonian-dynamics@proof-2 ¶
depends_on proposition 26.19 Second-class constraints come in pairs ¶ ↺
proves proof ch:09-constrained-hamiltonian-dynamics@proof-8 ¶
proves proof ch:09-constrained-hamiltonian-dynamics@prooflink-1 ¶

Supports

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.651 Acyclicity of $\delta$ ¶
      - depends_on lemma A.648 $\delta$ is the ghost bracket with $\Omega_{(0)}$, and $\delta^{2}=0$ ¶
    - depends_on lemma A.652 Nilpotency, order by order ¶
      - depends_on proposition A.654 The Yang–Mills case ¶
      - depends_on proposition A.655 Uniqueness up to a canonical transformation ¶
      - depends_on theorem A.653 Existence of a nilpotent BRST charge ¶
  - depends_on theorem 26.51 Existence and uniqueness of the BRST charge ¶
- depends_on proposition 26.48 Consistency of the state conditions ¶

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Dirac bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:628
`depends_on`	→	Total Hamiltonian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:628
`depends_on`	→	Second-class constraints come in pairs	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:628
`depends_on`	←	Dirac quantization	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1834
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-8	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:631
`proves`	←	ch:09-constrained-hamiltonian-dynamics@prooflink-1	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:680