Manuscript IX-E@1 · IX-E.tex · section · 8f8d5e7f

E5: involution is a separate equation

E2–E5: analytical obligations and contacts › E5: involution is a separate equation · lines 954–980

IX-E@1#eq:antitime, IX-E@1#eq:time-star

Define anti-time-ordering by

T_{2} (d) = S (d) + S (σ d) - T_{2} (d) .

The required identity is

T_{2} (d)^{*} = T_{2} (γ d) .

Using (eq:A), equations (eq:antitime)– (eq:time-star) are exactly

X (γ d) = - X (d)^{*} .

Symmetry instead requires $X (σ d) = X (d)$ . The two equations are different.

Contacts must solve the overlap, ordered-restriction, admission, symmetry, and involution equations simultaneously. Averaging a candidate is not proof that its old overlap or causal restrictions survive. The algebraic theorem isolates these constraints, but does not solve the sharp distributional contact problem.

Displayed formulas in this passage

IX-E@1#eq:antitime · line 957
$T_{2} (d) = S (d) + S (σ d) - T_{2} (d) .$
IX-E@1#eq:time-star · line 963
$T_{2} (d)^{*} = T_{2} (γ d) .$
unlabeled · line 969
$X (γ d) = - X (d)^{*} .$