Manuscript Em03-H@8 · Em03-H-v8.tex · section · 91e9e3bd

Declaration and physical boundary

Restricted continuum experiment › Declaration and physical boundary · lines 1068–1138

Nearly- $A d S_{2}$ , Schwarzian dynamics, and probe matter motivate the model MSY,EMV,MTV. The following tensor-product reduction is declared, not derived with a controlled full-gravity remainder.

The motivating disk parameters are

L = 1, 8 π G = 1, S_{0} = 0, C = ϕ_{r} = 100, β = 2 π .

A motivating action is

I_{JT} = - S_{0} χ (M) - \frac{1}{16 π G} [\int_{M} g ϕ (R + 2) + 2 \int_{\partial M} h ϕ (K - 1)],

with $h_{τ τ} = ϵ^{- 2}$ , $ϕ ∣_{\partial M} = ϕ_{r} / ϵ$ . Eight complex massless Dirac species have action

I_{D} = \frac{1}{2} j = 1 \sum^{8} \int_{M} g [ψ_{j} γ^{μ} \nabla_{μ} ψ_{j} - (\nabla_{μ} ψ_{j}) γ^{μ} ψ_{j}] .

A disk-extending spin structure and homogeneous reflecting boundary condition may be declared using

B = i γ_{*} γ_{n}, B^{2} = I, {B, γ_{n}} = 0, P_{\pm} = (I \pm B) /2,

with $P_{-} ψ = ψ P_{-} = 0$ , including variations. The boundary variation vanishes because $P_{+} γ_{n} P_{+} = 0$ .

The exterior metric

d s^{2} = - (r^{2} - 1) d t^{2} + \frac{d r ^{2}}{r ^{2} - 1}, r > 1,

motivates the reflected positive-frequency one-body model. The exact analysis begins with the directly declared multiplication operator

(h f) (ω) = ω f (ω), D (h) = {f \in L^{2} (R_{+}) : ω f \in L^{2} (R_{+})},

and normal-ordered matter Hamiltonian

H_{D} = j \sum \int_{0}^{\infty} ω [a_{j}^{†} (ω) a_{j} (ω) + d_{j}^{†} (ω) d_{j} (ω)] d ω .

Declare

H_{Sch} = L^{2} (R_{+}, ρ_{C} (E) d E), ρ_{C} (E) = sinh (2 π 2 C E), H_{Sch} ϕ = E ϕ,

H_{red} = H_{Sch} \otimes F_{f} (h_{D}), H_{store, red} = H_{Sch} \otimes I + I \otimes H_{D} .

The spectral normalization is part of the declaration. Backreaction, fluctuating dressing, and loop corrections are neither included nor bounded. The static Fock vacuum is not assumed to be a thermal or horizon-regular Hartle–Hawking state.

Displayed formulas in this passage

unlabeled · line 1075
$L = 1, 8 π G = 1, S_{0} = 0, C = ϕ_{r} = 100, β = 2 π .$
unlabeled · line 1080
$I_{JT} = - S_{0} χ (M) - \frac{1}{16 π G} [\int_{M} g ϕ (R + 2) + 2 \int_{\partial M} h ϕ (K - 1)],$
unlabeled · line 1089
$I_{D} = \frac{1}{2} j = 1 \sum^{8} \int_{M} g [ψ_{j} γ^{μ} \nabla_{μ} ψ_{j} - (\nabla_{μ} ψ_{j}) γ^{μ} ψ_{j}] .$
unlabeled · line 1096
$B = i γ_{*} γ_{n}, B^{2} = I, {B, γ_{n}} = 0, P_{\pm} = (I \pm B) /2,$
unlabeled · line 1105
$d s^{2} = - (r^{2} - 1) d t^{2} + \frac{d r ^{2}}{r ^{2} - 1}, r > 1,$
unlabeled · line 1111
$(h f) (ω) = ω f (ω), D (h) = {f \in L^{2} (R_{+}) : ω f \in L^{2} (R_{+})},$
unlabeled · line 1116
$H_{D} = j \sum \int_{0}^{\infty} ω [a_{j}^{†} (ω) a_{j} (ω) + d_{j}^{†} (ω) d_{j} (ω)] d ω .$
unlabeled · line 1123
$H_{Sch} = L^{2} (R_{+}, ρ_{C} (E) d E), ρ_{C} (E) = sinh (2 π 2 C E), H_{Sch} ϕ = E ϕ,$
unlabeled · line 1129
$H_{red} = H_{Sch} \otimes F_{f} (h_{D}), H_{store, red} = H_{Sch} \otimes I + I \otimes H_{D} .$