postulate 26.47 Dirac quantization

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1818 · p. 919

ground object -- no derivation owed

Rests on

depends_on postulate 25.27 Dirac's correspondence rule ¶
- depends_on equation 22.41 eq:ham-poisson-bracket ¶
- depends_on theorem 25.3 The bracket is a Lie bracket ¶
  - depends_on equation 22.41 eq:ham-poisson-bracket ¶ ↺
  - depends_on equation 22.56 eq:ham-symplectic-matrix ¶
  - proves proof ch:08-poisson-quantum-bridge@proof-1 ¶
depends_on theorem 26.21 Properties of the Dirac bracket ¶
- 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 5.15 Basis ¶
    - depends_on definition 22.14 Phase-space coordinate transformation ¶
      - depends_on definition 22.2 Hamiltonian ¶
  - 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 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 5.113 Non-degenerate form ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - 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 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 ¶
depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
- depends_on definition 26.8 Total Hamiltonian ¶ ↺
- depends_on proposition 26.10 The four outcomes ¶
  - depends_on definition 26.9 Consistency conditions ¶
    - depends_on equation 26.7 eq:cd-evolution ¶
    - depends_on equation 26.2 eq:cd-primary ¶
  - depends_on remark 26.7 The regularity assumption, stated once ¶
  - proves proof ch:09-constrained-hamiltonian-dynamics@proof-3 ¶
- proves proof ch:09-constrained-hamiltonian-dynamics@proof-5 ¶

Supports

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 proposition A.655 Uniqueness up to a canonical transformation ¶
      - depends_on theorem A.653 Existence of a nilpotent BRST charge ¶
    - 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 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`	→	Dirac's correspondence rule	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1834
`depends_on`	→	Properties of the Dirac bracket	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1834
`depends_on`	→	First-class constraints generate gauge transformations	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1834
`depends_on`	←	The BRST charge	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:2055
`depends_on`	←	Consistency of the state conditions	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1857