Manuscript X-L-ST@1 · X-L-ST.tex · section · 20a32a22

Domain and formal background split

The nonlinear classical field differential › Domain and formal background split · lines 656–695

X-L-ST@1#eq:split, X-L-ST@1#eq:sr, X-L-ST@1#eq:snonlinear

Let $P (U)$ be the graded algebra of smooth-coefficient polynomial jet expressions on unrestricted smooth germs in $U$ . Let $F_{ml} (U)$ be the algebra generated by compactly smeared local polynomial functionals under finite pointwise products.

The latter is multilocal: a product of two single-integral functionals is generally not a single-integral local functional. Both algebras are completed coefficientwise in the even formal parameter $κ$ .

Fix smooth classical backgrounds $h, k$ and write

q ˉ = η + κh + κ^{2} k, q = q ˉ + κ r .

The formal inverse-metric coefficients are permitted coefficientwise. Neither $q$ nor $q ˉ$ is the host metric $G$ .

Define

sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, s c^{μ} = κ c^{ν} \partial_{ν} c^{μ}, = b, s b = 0, s h = s k = 0.

The differential commutes with spacetime derivatives and uses the left graded Leibniz rule. In particular,

s q = κ L_{c} q, s = s_{0} + κ s_{1} + κ^{2} s_{2} .

Displayed formulas in this passage

X-L-ST@1#eq:split · line 670
$q ˉ = η + κh + κ^{2} k, q = q ˉ + κ r .$
X-L-ST@1#eq:sr · line 680
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, s c^{μ} = κ c^{ν} \partial_{ν} c^{μ}, = b, s b = 0, s h = s k = 0.$
X-L-ST@1#eq:snonlinear · line 680
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, s c^{μ} = κ c^{ν} \partial_{ν} c^{μ}, = b, s b = 0, s h = s k = 0.$
unlabeled · line 691
$s q = κ L_{c} q, s = s_{0} + κ s_{1} + κ^{2} s_{2} .$

Domain and formal background split

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