Manuscript VII-B@1 · VII-B.tex · section · be632835

No double counting

Classical input and geometric bookkeeping › No double counting · lines 430–470

VII-B@1#eq:classical-bookkeeping

For clarity, consider arbitrary smooth formal curves

H_{int} = h + κk + O (κ^{2}), ϕ_{int} = ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2} + O (κ^{3}) .

Then direct substitution gives

B_{f} [η + κ H_{int}, ϕ_{int}] = \int f ϕ_{0} d^{4} x + κ \int f (ϕ_{1} - w [h]^{μ} \partial_{μ} ϕ_{0}) d^{4} x + κ^{2} \int f [ϕ_{2} - w [h]^{μ} \partial_{μ} ϕ_{1} - w [k]^{μ} \partial_{μ} ϕ_{0} + (w [h]^{ν} \partial_{ν} w [h]^{μ} - v [h, h]^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w [h]^{μ} w [h]^{ν} \partial_{μ} \partial_{ν} ϕ_{0}] d^{4} x + O (κ^{3}) .

Thus the second metric coefficient $k$ enters through the interacting expansion of one graviton field. It is not a second independent quantum field.

Equation (eq:classical-bookkeeping) is an algebraic identity on smooth configurations. It is not a theorem asserting the existence of quantum retarded products. In particular, $ϕ_{1}$ and $ϕ_{2}$ are not included in the bare coefficients (eq:B1)–(eq:B2).

Displayed formulas in this passage

VII-B@1#eq:classical-bookkeeping · line 439
$B_{f} [η + κ H_{int}, ϕ_{int}] = \int f ϕ_{0} d^{4} x + κ \int f (ϕ_{1} - w [h]^{μ} \partial_{μ} ϕ_{0}) d^{4} x + κ^{2} \int f [ϕ_{2} - w [h]^{μ} \partial_{μ} ϕ_{1} - w [k]^{μ} \partial_{μ} ϕ_{0} + (w [h]^{ν} \partial_{ν} w [h]^{μ} - v [h, h]^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w [h]^{μ} w [h]^{ν} \partial_{μ} \partial_{ν} ϕ_{0}] d^{4} x + O (κ^{3}) .$
unlabeled · line 433
$H_{int} = h + κk + O (κ^{2}), ϕ_{int} = ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2} + O (κ^{3}) .$