Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

Reduced bosonic operator

Gauge fixing and the normally hyperbolic realization › Reduced bosonic operator · lines 788–814

XX-B@1#eq:Bbulk, XX-B@1#eq:Amap

The auxiliary equation is $B = - r F z$ . Eliminating $B$ gives

S_{2, red} = \frac{1}{2} (z, H z)_{0} - \frac{r}{2} (F z, F z)_{1} .

Its operator is

B = H - r F^{†} F .

Define the four-dimensional trace reversal

(T h)_{μν} = h_{μν} - \frac{1}{2} g ˉ_{μν} h^{ρ}_{ρ}, T^{2} = id,

and the parallel invertible fiber map

A = diag (\frac{M ^{2}}{4} T, id), A^{- 1} = diag (\frac{4}{M ^{2}} T, id) .

Displayed formulas in this passage

XX-B@1#eq:Bbulk · line 797
$B = H - r F^{†} F .$
XX-B@1#eq:Amap · line 808
$A = diag (\frac{M ^{2}}{4} T, id), A^{- 1} = diag (\frac{4}{M ^{2}} T, id) .$
unlabeled · line 792
$S_{2, red} = \frac{1}{2} (z, H z)_{0} - \frac{r}{2} (F z, F z)_{1} .$
unlabeled · line 802
$(T h)_{μν} = h_{μν} - \frac{1}{2} g ˉ_{μν} h^{ρ}_{ρ}, T^{2} = id,$