Manuscript IX-B@2 · IX-B-v2.tex · section · 8ce915d6

Inherited parent involution and descendant exchange

Involution and the three continuity statements › Inherited parent involution and descendant exchange · lines 1286–1318

IX-B@2#eq:adjoint

Let $τ (x, y, x^{'}) = (x^{'}, y, x)$ , and define

⟨ U^{†}, Φ ⟩ = ⟨ U, Φ \circ τ ⟩ .

VII-D proves the parent identities

(T_{ε}^{ρ})^{†} = T_{ε}^{ρ}, r^{†} = r, (u_{μ}^{ρ})^{†} = u_{μ}^{ρ} .

We use these as inherited symmetry statements VIID (Section “Adjoint symmetry').

For a differential–multiplier operation $O$ , define its exchanged-conjugate operation by

O^{†} U = (O (U^{†}))^{†} .

Then, using (eq:adjoint),

(O u_{μ}^{ρ})^{†} = O^{†} u_{μ}^{ρ} .

This is the descendant exchange rule. A first-observation derivative is exchanged with the corresponding third-observation derivative, with conjugation of smooth coefficients; it is not generally an individually self-adjoint operation.

Displayed formulas in this passage

IX-B@2#eq:adjoint · line 1294
$(T_{ε}^{ρ})^{†} = T_{ε}^{ρ}, r^{†} = r, (u_{μ}^{ρ})^{†} = u_{μ}^{ρ} .$
unlabeled · line 1289
$⟨ U^{†}, Φ ⟩ = ⟨ U, Φ \circ τ ⟩ .$
unlabeled · line 1305
$O^{†} U = (O (U^{†}))^{†} .$
unlabeled · line 1310
$(O u_{μ}^{ρ})^{†} = O^{†} u_{μ}^{ρ} .$