Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

Transport construction

Native local subtraction parametrix › Transport construction · lines 863–961

XX-C@1#eq:Utransport, XX-C@1#eq:V0, XX-C@1#eq:Vrecursion, XX-C@1#eq:Hplus, XX-C@1#eq:HF, XX-C@1#eq:Hnormalized

Define

U (x, y) = Δ_{vV} (x, y)^{1/2} P (x, y),

where $P$ is $D$ -parallel transport from $y$ to $x$ . Equivalently,

2 σ^{; μ} D_{μ} U + (□_{g ˉ} σ - 4) U = 0, U (y, y) = id .

The logarithmic coefficients are determined by

2 σ^{; μ} D_{μ} V_{0} + (□_{g ˉ} σ - 2) V_{0} 2 (n + 1) σ^{; μ} D_{μ} V_{n + 1} + (n + 1) (□_{g ˉ} σ + 2 n) V_{n + 1} = - L_{x} U, = - L_{x} V_{n}, n \geq 0.

These are regular transport equations along the unique geodesic in $O$ . Their smooth solutions are fixed by regularity at coincidence. In particular,

V_{0} (y, y) = - \frac{1}{2} (L_{x} U) (y, y) .

Let

V \sim n = 0 \sum^{\infty} V_{n} σ^{n}

be a smooth asymptotic summation with cutoffs in $σ$ , chosen so that the residual of the transport construction vanishes to all orders on the local characteristic set. Fix one such summation. It is not claimed unique modulo smooth terms.

For the positive-frequency boundary value, use

σ_{+} (x, y) = ϵ ↓ 0 lim [σ (x, y) + i ϵ (t_{x} - t_{y}) + ϵ^{2}] .

The length $ℓ > 0$ is a declared subtraction scale. Set

H_{L}^{+} (x, y) H_{L}^{F} (x, y) = \frac{1}{8 π ^{2}} [\frac{U ( x , y )}{σ _{+} ( x , y )} + V (x, y) lo g \frac{σ _{+} ( x , y )}{ℓ ^{2}}], = - \frac{i}{8 π ^{2}} [\frac{U ( x , y )}{σ ( x , y ) + i 0} + V (x, y) lo g \frac{σ ( x , y ) + i 0}{ℓ ^{2}}] .

A cutoff equal to one near the diagonal may be added. Only its near-diagonal germ is used.

The native action-Hessian subtraction is

H_{B}^{+} = H_{L}^{+} A_{y}^{- 1}, H_{B}^{F} = H_{L}^{F} A_{y}^{- 1} .

The multiplication in the source slot is essential.

The sign in (eq:HF) matches the XX-B convention

W_{B} = i (F_{B} - G_{B}^{adv}) up to the stated Hermitian normalization .

For the flat scalar action operator $□ = - \partial_{t}^{2} + Δ$ , the corresponding massless positive-frequency singularity is

\frac{1}{4 π ^{2}} \frac{1}{∣ x - y ∣ ^{2} - ( t _{x} - t _{y} - i 0 ) ^{2}} .

This fixes the factor and boundary-value convention.

Displayed formulas in this passage

XX-C@1#eq:Utransport · line 873
$2 σ^{; μ} D_{μ} U + (□_{g ˉ} σ - 4) U = 0, U (y, y) = id .$
XX-C@1#eq:V0 · line 881
$2 σ^{; μ} D_{μ} V_{0} + (□_{g ˉ} σ - 2) V_{0} 2 (n + 1) σ^{; μ} D_{μ} V_{n + 1} + (n + 1) (□_{g ˉ} σ + 2 n) V_{n + 1} = - L_{x} U, = - L_{x} V_{n}, n \geq 0.$
XX-C@1#eq:Vrecursion · line 881
$2 σ^{; μ} D_{μ} V_{0} + (□_{g ˉ} σ - 2) V_{0} 2 (n + 1) σ^{; μ} D_{μ} V_{n + 1} + (n + 1) (□_{g ˉ} σ + 2 n) V_{n + 1} = - L_{x} U, = - L_{x} V_{n}, n \geq 0.$
XX-C@1#eq:Hplus · line 919
$H_{L}^{+} (x, y) H_{L}^{F} (x, y) = \frac{1}{8 π ^{2}} [\frac{U ( x , y )}{σ _{+} ( x , y )} + V (x, y) lo g \frac{σ _{+} ( x , y )}{ℓ ^{2}}], = - \frac{i}{8 π ^{2}} [\frac{U ( x , y )}{σ ( x , y ) + i 0} + V (x, y) lo g \frac{σ ( x , y ) + i 0}{ℓ ^{2}}] .$
XX-C@1#eq:HF · line 919
$H_{L}^{+} (x, y) H_{L}^{F} (x, y) = \frac{1}{8 π ^{2}} [\frac{U ( x , y )}{σ _{+} ( x , y )} + V (x, y) lo g \frac{σ _{+} ( x , y )}{ℓ ^{2}}], = - \frac{i}{8 π ^{2}} [\frac{U ( x , y )}{σ ( x , y ) + i 0} + V (x, y) lo g \frac{σ ( x , y ) + i 0}{ℓ ^{2}}] .$
XX-C@1#eq:Hnormalized · line 939
$H_{B}^{+} = H_{L}^{+} A_{y}^{- 1}, H_{B}^{F} = H_{L}^{F} A_{y}^{- 1} .$
unlabeled · line 866
$U (x, y) = Δ_{vV} (x, y)^{1/2} P (x, y),$
unlabeled · line 896
$V_{0} (y, y) = - \frac{1}{2} (L_{x} U) (y, y) .$
unlabeled · line 901
$V \sim n = 0 \sum^{\infty} V_{n} σ^{n}$
unlabeled · line 911
$σ_{+} (x, y) = ϵ ↓ 0 lim [σ (x, y) + i ϵ (t_{x} - t_{y}) + ϵ^{2}] .$
unlabeled · line 948
$W_{B} = i (F_{B} - G_{B}^{adv}) up to the stated Hermitian normalization .$
unlabeled · line 955
$\frac{1}{4 π ^{2}} \frac{1}{∣ x - y ∣ ^{2} - ( t _{x} - t _{y} - i 0 ) ^{2}} .$

Transport construction

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