Admissible presymplectic reduction and Hamiltonian generators

Let $\mathcal{S}$ be a specified finite-dimensional smooth solution manifold with specified admissible variations and boundary conditions, and let $\mathrm{\Omega }$ be a smooth closed two-form on $\mathcal{S}$, obtained by restricting an integrated form ${\mathrm{\Omega }}_{\mathrm{\Sigma }}$ from the chosen covariant variation data. The pair $(\mathcal{S},\mathrm{\Omega })$ is presymplectic when $\mathrm{\Omega }$ is allowed to be degenerate. At $p\in \mathcal{S}$, its pointwise null space is $\ker {\mathrm{\Omega }}_p:=\{v\in T_p\mathcal{S}:{\mathrm{\Omega }}_p(v,w)=0\text{ for every }w\in T_p\mathcal{S}\}\mathrm{.}$ A presymplectic reduction is defined here only when there is a specified smooth Hausdorff manifold $\mathcal{R}$ and a surjective submersion $q:\mathcal{S}\to \mathcal{R}$ with connected fibers such that $\ker (d{q}_p)=\ker {\mathrm{\Omega }}_p(p\in \mathcal{S}),$ where $\ker (d{q}_p)$ consists of tangent vectors mapped to zero by the differential. This condition makes the quotient fibers exactly the connected directions annihilated by $\mathrm{\Omega }$. The reduction further requires a smooth closed nondegenerate two-form $\bar{\mathrm{\Omega }}$ on $\mathcal{R}$ satisfying $q^{\ast }\bar{\mathrm{\Omega }}=\mathrm{\Omega }$. For a smooth vector field $X$ and smooth function $H$ on a specified phase space, the Hamiltonian convention is ${\iota }_X\mathrm{\Omega }=dH$.

Symplectic form and symplectic manifold

Symplectic reduction of a coisotropic vector subspace

Hamiltonian vector field and Hamiltonian function

Covariant presymplectic current

Closedness of the integrated presymplectic form on smooth field families

Symplectic form and symplectic manifold

Symplectic reduction of a coisotropic vector subspace

Hamiltonian vector field and Hamiltonian function

Poisson bracket on a symplectic manifold

Immersions, submersions, and constant-rank maps

The finite-dimensional proposition for a coisotropic vector subspace applies to tangent linear models only. This definition does not establish a smooth quotient of an infinite-dimensional gravitational solution space. No falloff, boundary condition, constant-rank property, or existence of such a quotient is asserted here. These classical identities construct no measure, Hilbert space, or anomaly-free quantization. Thermal first-law and entropy interpretations are assigned to thermodynamics and general relativity; topological quantum field theory quantization is assigned to quantum field theory.