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

Lemma: Punctured Wick-square kernel

A separate normalization-independent response benchmark › Model and punctured response · lines 1591–1634 · proof included or linked

Psy02-A@1#lem:response-kernel, Psy02-A@1#eq:cone-response

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The declared first-order response satisfies

R^{\circ} (z) = γ Θ (t) δ^{'} (s (z)), γ = \frac{ℏ}{4 π ^{3}}, z \neq = 0.

It is real, retarded, and homogeneous of degree $- 4$ on the punctured space.

Proof. Vacuum Wick contraction has exactly two connecting pairings, so

ω_{0} (Q (z) Q (0)) = 2 ℏ^{2} W (z)^{2} .

These positive-frequency products are defined in the free Wick framework. On the nonzero future null cone, $t > 0$ and $d s \neq = 0$ . The boundary-value identity

(u - i 0)^{- 2} - (u + i 0)^{- 2} = - 2 i π δ^{'} (u)

therefore pulls back to $s$ . It gives

W (z)^{2} - W (- z)^{2} = - \frac{i}{8 π ^{3}} δ^{'} (s (z)) (t > 0) .

Substitute in (eq:response-definition). Away from the origin, the commutator is supported on the nonzero null cones; the time cut creates no extra singular product there. Under positive dilation, $s$ has degree two and $δ^{'} (s)$ degree $- 4$ .

Displayed formulas in this passage

Psy02-A@1#eq:cone-response · line 1594
$R^{\circ} (z) = γ Θ (t) δ^{'} (s (z)), γ = \frac{ℏ}{4 π ^{3}}, z \neq = 0.$
unlabeled · line 1610
$ω_{0} (Q (z) Q (0)) = 2 ℏ^{2} W (z)^{2} .$
unlabeled · line 1617
$(u - i 0)^{- 2} - (u + i 0)^{- 2} = - 2 i π δ^{'} (u)$
unlabeled · line 1623
$W (z)^{2} - W (- z)^{2} = - \frac{i}{8 π ^{3}} δ^{'} (s (z)) (t > 0) .$

Nearby in this section

section: Model and punctured response