Manuscript II@2 · II.tex · lemma · 50d64def

Lemma: The A,B,B coefficient

The three labels and the reference value › Restricted Einstein cubic · lines 1218–1242 · proof included or linked

II@2#lem:ABB, II@2#eq:ABB

For incoming momenta $p + q + r = 0$ ,

v_{3} (A_{p}, B_{q}, B_{r}) = - 2 (p^{0} q^{0} + p^{0} r^{0} + q^{0} r^{0}) = (p^{0})^{2} + (q^{0})^{2} + (r^{0})^{2} .

Proof. Polarize (eq:diagonal-cubic). For each choice of undifferentiated labeled wave, the sum of its spatial positions and the two assignments of differentiated waves has coefficient $- 2$ . For instance, with $A$ undifferentiated,

A_{11} (B_{22} B_{33} + B_{33} B_{22}) = - 2.

A time derivative contributes $i k_{0} = - i k^{0}$ . The two derivative factors and the density sign yield the first formula. The second follows from $p^{0} + q^{0} + r^{0} = 0$ .

Displayed formulas in this passage

II@2#eq:ABB · line 1221
$v_{3} (A_{p}, B_{q}, B_{r}) = - 2 (p^{0} q^{0} + p^{0} r^{0} + q^{0} r^{0}) = (p^{0})^{2} + (q^{0})^{2} + (r^{0})^{2} .$
unlabeled · line 1235
$A_{11} (B_{22} B_{33} + B_{33} B_{22}) = - 2.$

Lemma: The A,B,B coefficient

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