theorem 26.14 First-class constraints generate gauge transformations

open in the book · parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:399 · p. 903

Rests on

depends_on definition 26.8 Total Hamiltonian ¶
- 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 26.6 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 26.10 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 26.16 Extended Hamiltonian ¶
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 ¶
depends_on theorem 26.17 Counting the physical degrees of freedom ¶
- depends_on corollary A.642 The constraints are first class ¶
- 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 proposition 26.32 Constraint structure of the free Maxwell field ¶
  - depends_on proposition 26.34 Proca constraints are second class ¶
    - depends_on proposition 113.3 Degrees of freedom before and after ¶
  - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
- depends_on proposition 26.34 Proca constraints are second class ¶ ↺
- depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
- depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
  - depends_on notation A.636 Densities and smearing ¶
    - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
      - depends_on lemma A.638 Naturality ¶
  - depends_on proposition 26.43 The hypersurface-deformation algebra ¶

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`	→	Total Hamiltonian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:411
`depends_on`	→	The four outcomes	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:411
`depends_on`	←	Extended Hamiltonian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:476
`depends_on`	←	Dirac quantization	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:1834
`depends_on`	←	Counting the physical degrees of freedom	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:490
`proves`	←	ch:09-constrained-hamiltonian-dynamics@proof-5	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:414