Manuscript XIV-E@1 · XIV-E.tex · section · e89fd2a5

The candidate sum of line and attachment phases

The elementary product phase is not a common symbol calculus › The candidate sum of line and attachment phases · lines 883–916

XIV-E@1#eq:phase

Back to manuscript

Consider the smooth homogeneous candidate

Φ (a, b, z, y; t_{R}, t_{1}, t_{2}, κ) = - t_{R} q_{R} + t_{1} q_{1} + t_{2} q_{2} + κ_{μ} (a^{μ} - z^{μ}),

where

(t_{R}, t_{1}, t_{2}) \in (0, \infty)^{3}, κ \in R^{4} .

Its signs encode the copied oriented conormals. A simultaneous reversal of the Fourier convention reverses the whole expression and changes none of the rank conclusions.

Formula (eq:phase) is the elementary candidate obtained by adding the selected three line phases and the metric attachment phase. It is not asserted to represent the full six-branch sum. Its validity must be tested before amplitudes can be compared using it.

For a nondegenerate phase, the differentials of its auxiliary critical equations are independent on the critical set. This is the standard local criterion for the principal-symbol construction for Lagrangian distributions Wunsch (Section 8.2).

For the present calculation, call a candidate clean if its auxiliary critical set is smooth and its tangent space equals the kernel of the differential of those critical equations. This is the tangent-space requirement needed for clean critical-set reduction. It is weaker than full independence but still fails below.

Displayed formulas in this passage

XIV-E@1#eq:phase · line 886
$Φ (a, b, z, y; t_{R}, t_{1}, t_{2}, κ) = - t_{R} q_{R} + t_{1} q_{1} + t_{2} q_{2} + κ_{μ} (a^{μ} - z^{μ}),$
unlabeled · line 892
$(t_{R}, t_{1}, t_{2}) \in (0, \infty)^{3}, κ \in R^{4} .$