Manuscript IX-D@3 · IX-D-v3.tex · section · 3ad4586e

D0: the constructed IX-Free-2 sector

D0: the constructed IX-Free-2 sector · lines 467–531

IX-D@3#eq:Q, IX-D@3#eq:QR, IX-D@3#eq:Dfirst, IX-D@3#eq:Dsecond, IX-D@3#eq:Dold, IX-D@3#eq:Ddecl

Define the actual spaces

B = span {B_{f}}, W = span {W_{g}},

L_{4} = span {B_{f} W_{g}}, L_{2} = span {Q_{f, g}},

R_{4} = span {W_{g} B_{f^{'}}}, R_{2} = span {Q^{R}_{g, f^{'}}},

and

L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2} .

The last sums are direct by actual field degree, four and two.

The locked contractions are

Q_{f, g} Q^{R}_{g, f^{'}} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f^{'} (x^{'}) K (x^{'}, y), h_{00} (y) \partial_{t^{'}} ϕ (x^{'}) ⟩ .

The algebraic presentation maps are

p_{L} (f \otimes g) = B_{f} W_{g}, q_{L} (f \otimes g) = Q_{f, g},

p_{R} (g \otimes f^{'}) = W_{g} B_{f^{'}}, q_{R} (g \otimes f^{'}) = Q^{R}_{g, f^{'}} .

Inside $P_{Σ} \otimes_{C} P_{Σ}$ , put

D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, D_{old} = P_{mc} \otimes P_{mc}, = D_{new} + D_{old} .

All these tensor products are algebraic.

Let

V_{Σ}^{[3]} = P_{Σ} \otimes_{C} (C [ℏ] / (ℏ^{4})) .

Displayed formulas in this passage

IX-D@3#eq:Q · line 491
$Q_{f, g} Q^{R}_{g, f^{'}} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f^{'} (x^{'}) K (x^{'}, y), h_{00} (y) \partial_{t^{'}} ϕ (x^{'}) ⟩ .$
IX-D@3#eq:QR · line 491
$Q_{f, g} Q^{R}_{g, f^{'}} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f^{'} (x^{'}) K (x^{'}, y), h_{00} (y) \partial_{t^{'}} ϕ (x^{'}) ⟩ .$
IX-D@3#eq:Dfirst · line 512
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, D_{old} = P_{mc} \otimes P_{mc}, = D_{new} + D_{old} .$
IX-D@3#eq:Dsecond · line 512
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, D_{old} = P_{mc} \otimes P_{mc}, = D_{new} + D_{old} .$
IX-D@3#eq:Dold · line 512
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, D_{old} = P_{mc} \otimes P_{mc}, = D_{new} + D_{old} .$
IX-D@3#eq:Ddecl · line 512
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, D_{old} = P_{mc} \otimes P_{mc}, = D_{new} + D_{old} .$
unlabeled · line 471
$B = span {B_{f}}, W = span {W_{g}},$
unlabeled · line 475
$L_{4} = span {B_{f} W_{g}}, L_{2} = span {Q_{f, g}},$
unlabeled · line 479
$R_{4} = span {W_{g} B_{f^{'}}}, R_{2} = span {Q^{R}_{g, f^{'}}},$
unlabeled · line 484
$L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2} .$
unlabeled · line 502
$p_{L} (f \otimes g) = B_{f} W_{g}, q_{L} (f \otimes g) = Q_{f, g},$
unlabeled · line 506
$p_{R} (g \otimes f^{'}) = W_{g} B_{f^{'}}, q_{R} (g \otimes f^{'}) = Q^{R}_{g, f^{'}} .$
unlabeled · line 528
$V_{Σ}^{[3]} = P_{Σ} \otimes_{C} (C [ℏ] / (ℏ^{4})) .$

D0: the constructed IX-Free-2 sector

Displayed formulas in this passage

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