Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

Adjoint symmetry

Complete conversion and the real counterterm class › Adjoint symmetry · lines 2006–2031

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

⟨ U^{†}, Φ ⟩ = ⟨ U, Φ \circ τ ⟩ .

The real multiplier in (eq:Deps) implies

D_{ε} (z) = D_{ε} (- z) .

Together with (eq:Texact), this gives

(T_{ε}^{ρ})^{†} = T_{ε}^{ρ} .

The residue is real and invariant under $τ$ , by interchanging $t, t^{'}$ in (eq:rjoint). Strong passage to the limit gives

r^{†} = r, (u_{μ}^{ρ})^{†} = u_{μ}^{ρ}, L_{ab}^{†} = L_{ab} .

These symmetries have been proved directly; they are not used as an assumption replacing the reality calculation.

Displayed formulas in this passage

unlabeled · line 2009
$⟨ U^{†}, Φ ⟩ = ⟨ U, Φ \circ τ ⟩ .$
unlabeled · line 2015
$D_{ε} (z) = D_{ε} (- z) .$
unlabeled · line 2019
$(T_{ε}^{ρ})^{†} = T_{ε}^{ρ} .$
unlabeled · line 2024
$r^{†} = r, (u_{μ}^{ρ})^{†} = u_{μ}^{ρ}, L_{ab}^{†} = L_{ab} .$