definition 26.50 The BRST charge

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:2029 · p. 922

ground object -- no derivation owed

Rests on

depends_on equation 26.16 eq:cd-first-class-algebra ¶
depends_on postulate 26.47 Dirac quantization ¶
- 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 22.14 Phase-space coordinate transformation ¶
    - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on definition 26.12 First and second class ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
      - 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 ¶
      - 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 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 proposition A.655 Uniqueness up to a canonical transformation ¶ ↺
  - 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 lemma A.652 Nilpotency, order by order ¶ ↺
depends_on theorem 26.51 Existence and uniqueness of the BRST charge ¶

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`	→	eq:cd-first-class-algebra	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:2055
`depends_on`	→	Dirac quantization	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:2055
`depends_on`	←	Gradings and brackets	declared	appendices/A-long-proofs.tex:31555
`depends_on`	←	Existence and uniqueness of the BRST charge	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:2074