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Closedness of the integrated presymplectic form on smooth field families

Statement

For a smooth finite-dimensional family of field configurations, let $Θ$ be the parameter-space one-form with values in spacetime three-forms from the chosen variational decomposition. For every finite oriented smooth singular three-chain $Σ$ , the integrated two-form $Ω_{Σ} = \int_{Σ} (- d_{P} Θ)$ is closed on the parameter space: $d_{P} Ω_{Σ} = 0$ . This is a field-space identity and requires no equations of motion. If a smooth solution family is specified, its restricted form is also closed.

Statement references

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §2.2

Covariant presymplectic current

Facts & Assumptions

Given: A smooth finite-dimensional field family and a finite oriented smooth singular three-chain $Σ$ .

[F1]

The current is $ω = - d_{P} Θ$ , and $Ω_{Σ} (δ_{1}, δ_{2}) = \int_{Σ} ω (δ_{1}, δ_{2})$ . Covariant presymplectic current

[F2]

The exterior derivative satisfies $d_{P}^{2} = 0$ . The exterior derivative squares to zero

[F3]

For a fixed finite chain, differentiation of a smooth parameter family of integrated three-forms passes through the integral. Differentiation of a smooth family of finite-chain integrals

Proof

Given: As in [F1]–[F3].

1.1F1F3

In commuting parameter coordinates $p^{a}, p^{b}$ , [F1] and [F3] give $(Ω_{Σ})_{a b} = \partial_{b} (Θ_{Σ})_{a} - \partial_{a} (Θ_{Σ})_{b} = - (d_{P} Θ_{Σ})_{a b}$ , where $Θ_{Σ} = \int_{Σ} Θ$ . The same differentiation formula applies to each coefficient of the parameter-space form $Θ$ , so $Ω_{Σ} = - d_{P} Θ_{Σ}$ .

2.1F2step 1.1∎

Apply [F2] to step 1.1: $d_{P} Ω_{Σ} = - d_{P}^{2} Θ_{Σ} = 0$ . The same computation applies to any smooth subfamily of solutions, proving closedness of its restricted two-form without asserting that a global solution manifold exists.

Closedness of the integrated presymplectic form on smooth field families

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources