theorem A.653 Existence of a nilpotent BRST charge

open in the book · appendices/A-long-proofs.tex:31859 · p. 3110

Rests on

depends_on equation 26.16 eq:cd-first-class-algebra ¶
depends_on lemma A.651 Acyclicity of $\delta$ ¶
- depends_on definition A.647 Koszul–Tate differential ¶
  - depends_on equation 26.16 eq:cd-first-class-algebra ¶ ↺
  - depends_on notation A.646 Gradings and brackets ¶
    - depends_on definition 26.50 The BRST charge ¶
      - depends_on equation 26.16 eq:cd-first-class-algebra ¶ ↺
      - depends_on postulate 26.47 Dirac quantization ¶
    - depends_on equation 26.97 eq:cd-brst-charge ¶
    - depends_on equation 26.16 eq:cd-first-class-algebra ¶ ↺
- depends_on lemma A.650 Flattening the constraints ¶
  - depends_on remark 26.4 The constant-rank theorem is owed by Part II ¶
  - depends_on remark 26.7 The regularity assumption, stated once ¶
  - depends_on theorem A.286 Implicit function theorem ¶
    - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on equation 6.12 eq:top-euclidean-metric ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-62 ¶
    - proves proof app:A-long-proofs@proof-185 ¶
  - proves proof app:A-long-proofs@proof-390 ¶
- depends_on remark A.649 The regularity hypothesis, in the form the proof needs ¶
- proves proof app:A-long-proofs@proof-391 ¶
depends_on lemma A.652 Nilpotency, order by order ¶
- depends_on equation 26.98 eq:cd-brst-nilpotent ¶
- depends_on lemma A.648 $\delta$ is the ghost bracket with $\Omega_{(0)}$, and $\delta^{2}=0$ ¶
  - depends_on definition A.647 Koszul–Tate differential ¶ ↺
  - depends_on equation A.1034 eq:app-brst-existence-first-two ¶
  - depends_on equation A.1030 eq:app-brst-existence-graded ¶
  - proves proof app:A-long-proofs@proof-389 ¶
- depends_on notation A.646 Gradings and brackets ¶ ↺
- proves proof app:A-long-proofs@proof-392 ¶
proves proof app:A-long-proofs@proof-393 ¶

Supports

depends_on proposition A.655 Uniqueness up to a canonical transformation ¶

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	appendices/A-long-proofs.tex:31867
`depends_on`	→	Acyclicity of $\delta$	declared	appendices/A-long-proofs.tex:31867
`depends_on`	→	Nilpotency, order by order	declared	appendices/A-long-proofs.tex:31867
`depends_on`	←	Uniqueness up to a canonical transformation	declared	appendices/A-long-proofs.tex:31981
`proves`	←	app:A-long-proofs@proof-393	declared	appendices/A-long-proofs.tex:31871