Manuscript Psy02-A@1 · Psy02-A.tex · theorem · d92ce402

Theorem: Nonzero prescription-independent response

A separate normalization-independent response benchmark › An exactly evaluated invariant · lines 1779–1810 · proof included or linked

Psy02-A@1#thm:response-invariant, Psy02-A@1#eq:response-value

For every permitted response,

⟨ R, F_{*} ⟩ = - \frac{ℏ t _{0}}{2 π ^{2}} \neq = 0.

Proof. The test vanishes near the origin, so its response is independent of $c$ . On $r = t$ ,

F_{*} (t, t ω) = 0, \partial_{r} F_{*} (t, r ω) ∣_{r = t} = - 2 t b (t)^{2} .

Equation (eq:finite-part) gives

⟨ R, F_{*} ⟩ = \frac{γ}{4} (4 π) \int_{0}^{\infty} - 2 t b (t)^{2} d t = - 2 π γ t_{0} .

Substitute $γ = ℏ/ (4 π^{3})$ .

Displayed formulas in this passage

Psy02-A@1#eq:response-value · line 1782
$⟨ R, F_{*} ⟩ = - \frac{ℏ t _{0}}{2 π ^{2}} \neq = 0.$
unlabeled · line 1794
$F_{*} (t, t ω) = 0, \partial_{r} F_{*} (t, r ω) ∣_{r = t} = - 2 t b (t)^{2} .$
unlabeled · line 1801
$⟨ R, F_{*} ⟩ = \frac{γ}{4} (4 π) \int_{0}^{\infty} - 2 t b (t)^{2} d t = - 2 π γ t_{0} .$