Manuscript VII-H@1 · VII-H.tex · lemma · d20d2a9f

Lemma: Connected two-insertion coefficient

The complete first-order contraction and its domain · lines 665–690 · proof included or linked

VII-H@1#lem:wick, VII-H@1#eq:wick-square, VII-H@1#eq:punctured-response

For the calibrated Wick square,

ω_{0} (Q (x) Q (y)) = 2 ℏ^{2} W (x - y)^{2} .

Consequently the punctured retarded response is

R^{\circ} (z) = 2 i ℏΘ (t) (W (z)^{2} - W (- z)^{2}), z \neq = 0.

Proof. Vacuum normal ordering removes self-contractions at each insertion. There are exactly two pairings connecting the two fields at $x$ to the two fields at $y$ . Each contributes $ℏ^{2} W (x - y)^{2}$ . Terms with an uncontracted normal-ordered field have zero vacuum expectation. This proves (eq:wick-square); subtraction in the commutator gives (eq:punctured-response).

Displayed formulas in this passage

VII-H@1#eq:wick-square · line 668
$ω_{0} (Q (x) Q (y)) = 2 ℏ^{2} W (x - y)^{2} .$
VII-H@1#eq:punctured-response · line 674
$R^{\circ} (z) = 2 i ℏΘ (t) (W (z)^{2} - W (- z)^{2}), z \neq = 0.$

Lemma: Connected two-insertion coefficient

Displayed formulas in this passage

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