Manuscript XIII-A@1 · XIII-A.tex · section · f56e91c7

The inherited product and ordering

A0: prerequisite sheet and inherited conventions › The inherited product and ordering · lines 474–516

XIII-A@1#eq:carrier, XIII-A@1#eq:causal-blocks, XIII-A@1#eq:ordering-map

XII-A supplies

E_{H} = (F_{mc, pol} [[ℏ, κ]], ⋆_{H}) / N_{sol} [[ℏ, κ]], N_{sol} = {F : F ∣_{C_{sol}} = 0},

where $N_{sol} [[ℏ, κ]]$ is a two-sided ideal and

F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .

For linear fields the causal relation is

[Φ_{A} (u), Φ_{B} (v)]_{gr} = i ℏ ⟨ u, Δ_{A B} v ⟩ 1 .

The relevant even blocks are

Δ_{γ γ} = - P E_{0}, Δ_{γ b} = R E_{0}, Δ_{bγ} = - D P E_{0}, Δ_{bb} = 0,

with $E_{0} = E_{0}^{R} - E_{0}^{A}$ the ambient causal wave operator.

An admissible ordering difference is a smooth graded-symmetric bisolution $d$ , respecting parity and ghost number. Its map is

α_{d} = exp (\frac{ℏ}{2} ⟨ d, \frac{δ ^{2}}{δ Φ ^{2}} ⟩), H^{'} = H + d .

It acts on functionals, preserves the full-shell ideal, and descends to an isomorphism $α ˉ_{d} : E_{H} \to E_{H^{'}}$ XIIA. No preservation of the new vanishing space is included in that input.

Displayed formulas in this passage

XIII-A@1#eq:carrier · line 477
$E_{H} = (F_{mc, pol} [[ℏ, κ]], ⋆_{H}) / N_{sol} [[ℏ, κ]], N_{sol} = {F : F ∣_{C_{sol}} = 0},$
XIII-A@1#eq:causal-blocks · line 495
$Δ_{γ γ} = - P E_{0}, Δ_{γ b} = R E_{0}, Δ_{bγ} = - D P E_{0}, Δ_{bb} = 0,$
XIII-A@1#eq:ordering-map · line 506
$α_{d} = exp (\frac{ℏ}{2} ⟨ d, \frac{δ ^{2}}{δ Φ ^{2}} ⟩), H^{'} = H + d .$
unlabeled · line 486
$F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .$
unlabeled · line 490
$[Φ_{A} (u), Φ_{B} (v)]_{gr} = i ℏ ⟨ u, Δ_{A B} v ⟩ 1 .$