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Manuscript IV@1 · IV.tex · figure · 351cb2d7

Four-leg quadratic reduction

Model 2: dense vacuum TT form and the actual cutoffs › The exact mask family · lines 1656–1686

IV@1#fig:four-leg-quadratic

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gravitationalscalaroperator / instrumenterror / restrictionsupport / scope
scalar aϕa_\phiscalar aϕ†a_\phi^\dagger
aga_g
H++agaϕH^{++}a_g a_\phi
pair annihilation
H+−agaϕ†H^{+-}a_g a_\phi^\dagger
number-preserving
ag†a_g^\dagger
H−+ag†aϕH^{-+}a_g^\dagger a_\phi
number-preserving
H−−ag†aϕ†H^{--}a_g^\dagger a_\phi^\dagger
pair creation
LR=HR−+,PR=HR−−L_R=H_R^{-+},\quad P_R=H_R^{--}
TR=(0LRLR∗0)T_R=\begin{pmatrix}0&L_R\\ L_R^*&0\end{pmatrix}
symmetrized cross-species pair tensor KRK_R
∥TR∥=∥LR∥,∥KR∥=2∥PR∥HS\lVert T_R\rVert=\lVert L_R\rVert,\quad \lVert K_R\rVert=\sqrt{2}\lVert P_R\rVert_{\mathrm{HS}}
BR=dΓ(TR)+12[a†(KR)+a(KR)]B_R=\mathrm{d}\Gamma(T_R)+\tfrac12\bigl[a^\dagger(K_R)+a(K_R)\bigr]

No constant term: the two species commute, so reordering produces no contraction.

Four-leg reduction of the dense-TT bilinear. The mixed legs form the off-diagonal number coefficient, and the pair legs form a symmetric cross-species tensor. Their quadratic reduction has no constant term because reordering the distinct species produces no contraction.

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