Manuscript VI@1 · VI.tex · section · b5b60afb

Bare functional coefficients are not solution-jet coefficients

Bare dressing, solution coefficients, and external response › Bare functional coefficients are not solution-jet coefficients · lines 1205–1246

VI@1#eq:bare-O1, VI@1#eq:bare-O2

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For perturbative quantization one usually expands a functional of independent fields $(h, φ)$ , with $γ = η + κh$ , before substituting interacting solutions. Let

v_{0} [h] = v [h, 0] .

The bare geometric coefficients are

O_{f}^{(0)} [h, φ] O_{f}^{(1)} [h, φ] O_{f}^{(2)} [h, φ] = \int f φ d^{4} x, = - \int f w^{μ} [h] \partial_{μ} φ d^{4} x, = \int f [(w^{ν} \partial_{ν} w^{μ} - v_{0}^{μ}) \partial_{μ} φ + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} φ] d^{4} x .

Round parentheses label bare functional coefficients. Square brackets in Theorem thm:descent label coefficients after substitution of a solution jet.

The relation between the two expansions is explicit:

v [h, j] = w [j] + v_{0} [h] .

Substituting

h ⟼ h + κj, φ ⟼ ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2}

into the bare expansion gives exactly (eq:O0-jet)–(eq:psi2). In particular, one must not insert a sourced $j$ into a bare second-order term and then count the same dynamical correction again through an interacting-field map.

Displayed formulas in this passage

VI@1#eq:bare-O1 · line 1215
$O_{f}^{(0)} [h, φ] O_{f}^{(1)} [h, φ] O_{f}^{(2)} [h, φ] = \int f φ d^{4} x, = - \int f w^{μ} [h] \partial_{μ} φ d^{4} x, = \int f [(w^{ν} \partial_{ν} w^{μ} - v_{0}^{μ}) \partial_{μ} φ + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} φ] d^{4} x .$
VI@1#eq:bare-O2 · line 1215
$O_{f}^{(0)} [h, φ] O_{f}^{(1)} [h, φ] O_{f}^{(2)} [h, φ] = \int f φ d^{4} x, = - \int f w^{μ} [h] \partial_{μ} φ d^{4} x, = \int f [(w^{ν} \partial_{ν} w^{μ} - v_{0}^{μ}) \partial_{μ} φ + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} φ] d^{4} x .$
unlabeled · line 1211
$v_{0} [h] = v [h, 0] .$
unlabeled · line 1234
$v [h, j] = w [j] + v_{0} [h] .$
unlabeled · line 1238
$h ⟼ h + κj, φ ⟼ ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2}$