Manuscript XX-F@1 · XX-F.tex · section · de0bd24f

The same free operator and its causal kernel

Frozen geometric and free-field data › The same free operator and its causal kernel · lines 414–501

XX-F@1#eq:free, XX-F@1#eq:E, XX-F@1#eq:W

Let $V = S^{2} T^{*} M \oplus R$ . The integrated field pairing is

(z, w)_{0} = \int (h_{μν} j^{μν} + φ ω) d V_{g ˉ} .

It is nondegenerate, not positive definite.

At the background define

H z = (- \frac{1}{2} δ E_{μν} [z], δ E_{ϕ} [z]),

where $E_{μν} = M^{2} G_{μν} - T_{μν}$ and $E_{ϕ} = □_{g} ϕ - V^{'} (ϕ)$ . Thus $S_{2} (z) = \frac{1}{2} (z, H z)_{0}$ .

Retain

K ξ F_{ν} z r A = (L_{ξ} g ˉ, L_{ξ} ϕ ˉ), = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h - M^{- 2} \partial_{ν} ϕ ˉ φ, = M^{2} /2, = diag (\frac{M ^{2}}{4} T, id), A_{ξ} z Q L = (L_{ξ} h, L_{ξ} φ), = F K, = A^{- 1} B . B = H - r F^{†} F,

Here $T$ is four-dimensional trace reversal.

The second-derivative metric part of $H$ is

\frac{M ^{2}}{4} T □_{g ˉ} + \frac{M ^{2}}{2} D^{†} D, D h = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h .

The $r F^{†} F$ term cancels the second summand. Mixed metric–scalar entries have at most one derivative. Consequently

L = □_{g ˉ} id + smooth lower-order terms .

Also

H K = 0, B K = - r F^{†} Q, Q ξ = □_{g ˉ} ξ + R ˉ \cdot ξ - M^{- 2} d ϕ ˉ (d ϕ ˉ \cdot ξ) .

The slab is globally hyperbolic: causal curves obey $∣ d x / d t ∣ \leq a (t)^{- 1}$ , so an inextendible curve cannot terminate at an interior time. Normally hyperbolic Cauchy and Green theory therefore applies BGP. The causal action-normalized kernel is

E = G_{B}^{ret} - G_{B}^{adv}, G_{B}^{ret, adv} = G_{L}^{ret, adv} A^{- 1} .

When used in a comparison, $W$ denotes a bidistribution satisfying

B W = W B = 0, W^{*} = W, W - W^{T} = i E, WF^{'} (W) \subset N_{+} .

No selected physical state or native infrared estimate is inferred from these relations.

Displayed formulas in this passage

XX-F@1#eq:free · line 435
$K ξ F_{ν} z r A = (L_{ξ} g ˉ, L_{ξ} ϕ ˉ), = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h - M^{- 2} \partial_{ν} ϕ ˉ φ, = M^{2} /2, = diag (\frac{M ^{2}}{4} T, id), A_{ξ} z Q L = (L_{ξ} h, L_{ξ} φ), = F K, = A^{- 1} B . B = H - r F^{†} F,$
XX-F@1#eq:E · line 483
$E = G_{B}^{ret} - G_{B}^{adv}, G_{B}^{ret, adv} = G_{L}^{ret, adv} A^{- 1} .$
XX-F@1#eq:W · line 492
$B W = W B = 0, W^{*} = W, W - W^{T} = i E, WF^{'} (W) \subset N_{+} .$
unlabeled · line 419
$(z, w)_{0} = \int (h_{μν} j^{μν} + φ ω) d V_{g ˉ} .$
unlabeled · line 426
$H z = (- \frac{1}{2} δ E_{μν} [z], δ E_{ϕ} [z]),$
unlabeled · line 456
$\frac{M ^{2}}{4} T □_{g ˉ} + \frac{M ^{2}}{2} D^{†} D, D h = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h .$
unlabeled · line 466
$L = □_{g ˉ} id + smooth lower-order terms .$
unlabeled · line 470
$H K = 0, B K = - r F^{†} Q, Q ξ = □_{g ˉ} ξ + R ˉ \cdot ξ - M^{- 2} d ϕ ˉ (d ϕ ˉ \cdot ξ) .$