Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

The four terms of a possible successor construction

The boundary of the quantum claim › The four terms of a possible successor construction · lines 1874–1912

VII-A@1#eq:quantumfour

If a quantum retarded-product construction is subsequently established, and if the interaction and composite insertion have expansions

V = κ V_{1} + κ^{2} V_{2} + \dots, O = O^{(0)} + κ O^{(1)} + κ^{2} O^{(2)} + \dots,

then the coefficient of order $κ^{2}$ has the formal bookkeeping pattern

O^{(2)} + R_{1} (V_{1}; O^{(1)}) + R_{1} (V_{2}; O^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; O^{(0)}) .

Here (eq:quantumfour) records obligations; it is not a definition of products whose existence has been proved.

A connected two-point response would additionally contain the product of the two first-order observable coefficients. A renormalized construction would have to name the state, the gravitational gauge-fixing and ghost sector, and the convention relating powers of $ℏ$ to graph loops.

The Action Ward Identity framework treats dependence on the action rather than on a representative Lagrangian density DF. Applying that framework here would still require an extension to the declared harmonic-coordinate kernels. Integration by parts in the classical formulas is not a proof of that extension.

Displayed formulas in this passage

VII-A@1#eq:quantumfour · line 1887
$O^{(2)} + R_{1} (V_{1}; O^{(1)}) + R_{1} (V_{2}; O^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; O^{(0)}) .$
unlabeled · line 1879
$V = κ V_{1} + κ^{2} V_{2} + \dots, O = O^{(0)} + κ O^{(1)} + κ^{2} O^{(2)} + \dots,$