Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Option A and its constraints

A separate Newton–wave cubic form › Option A and its constraints · lines 1373–1398

V-B@1#eq:cubicclassical

For free scalars with Schwartz Cauchy data, polarize the stress:

T_{μν} (ϕ_{1}, ϕ_{2}) = \partial_{(μ} ϕ_{1} \partial_{ν)} ϕ_{2} - \frac{1}{2} η_{μν} (\partial_{α} ϕ_{1} \partial^{α} ϕ_{2} + m^{2} ϕ_{1} ϕ_{2}) .

It is conserved. Define

C_{f} (ϕ_{1}, ϕ_{2}, ϕ_{3}) = \frac{1}{3} cyc \sum B_{f} (R^{A} T (ϕ_{1}, ϕ_{2}), ϕ_{3}) .

Newton potentials are smooth with polynomially controlled derivatives; multiplying by derivatives of the Schwartz scalar gives an admissible finite-time response source. Hence this classical functional is defined.

Changing the Newton boundary convention or the constraint-compatible initial data changes the inverse. Those changes are not gauge or prescription equivalences proved here.

Displayed formulas in this passage

V-B@1#eq:cubicclassical · line 1384
$C_{f} (ϕ_{1}, ϕ_{2}, ϕ_{3}) = \frac{1}{3} cyc \sum B_{f} (R^{A} T (ϕ_{1}, ϕ_{2}), ϕ_{3}) .$
unlabeled · line 1376
$T_{μν} (ϕ_{1}, ϕ_{2}) = \partial_{(μ} ϕ_{1} \partial_{ν)} ϕ_{2} - \frac{1}{2} η_{μν} (\partial_{α} ϕ_{1} \partial^{α} ϕ_{2} + m^{2} ϕ_{1} ϕ_{2}) .$

Option A and its constraints

Displayed formulas in this passage

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