Manuscript XIV-D@1 · XIV-D.tex · section · 1a7f173d

The unchanged entries

D0: the unchanged contract › The unchanged entries · lines 443–524

XIV-D@1#eq:Tpolar, XIV-D@1#eq:first

Define the polarized stress tensor by

T^{μν} (ψ_{1}, ψ_{2}) = \frac{1}{2} (\partial^{μ} ψ_{1} \partial^{ν} ψ_{2} + \partial^{μ} ψ_{2} \partial^{ν} ψ_{1}) - \frac{1}{2} η^{μν} (\partial_{ρ} ψ_{1} \partial^{ρ} ψ_{2} - m^{2} ψ_{1} ψ_{2}) .

Thus $T (ϕ, ϕ) = T_{0} [ϕ]$ , and

V_{g} = - \frac{1}{2} \int g γ_{μν} T^{μν} (ϕ, ϕ) d^{4} x .

Put

a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r,

b_{2} [γ] = \frac{1}{8} (tr_{η} γ)^{2} - \frac{1}{4} γ_{ρ σ} γ^{ρ σ},

a_{2}^{μν} [γ] = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} (tr_{η} γ) γ^{μν} + b_{2} [γ] η^{μν} .

The first-response operators are

X^{μ} [r] V_{1, g} [r] ψ p [r; ψ] A_{1} (r; ψ) = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ, = - E_{m}^{R} (V_{1, g} [r] ψ), = \int f (p [r; ψ] - X^{μ} [r] \partial_{μ} ψ) d^{4} x .

Arguments $(f, g)$ on $A_{1}$ are suppressed when fixed.

The inherited filled slots are

r_{D} [γ] = P E_{0}^{R} J_{D} [γ], r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T (ϕ, ϕ)) .

These are the already admitted classical slots, not new quantum products with $J_{D}$ as an argument.

For a finite intrinsic description of the fixed source, write $q (ρ) = η + ρ γ$ and

ℓ_{3} [γ] = - 2 [ρ^{3}] (- det q (ρ) R [q (ρ)]), J_{D} [γ] = EL_{γ} (σ_{ε} ℓ_{3} [γ]) .

The curvature representative is the unintegrated one of XII-C, with

R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .

Coefficient extraction is finite at this order. It fixes the same local quadratic source, including its first and second ramp derivatives; it changes neither the representative nor the action.

Displayed formulas in this passage

XIV-D@1#eq:Tpolar · line 446
$T^{μν} (ψ_{1}, ψ_{2}) = \frac{1}{2} (\partial^{μ} ψ_{1} \partial^{ν} ψ_{2} + \partial^{μ} ψ_{2} \partial^{ν} ψ_{1}) - \frac{1}{2} η^{μν} (\partial_{ρ} ψ_{1} \partial^{ρ} ψ_{2} - m^{2} ψ_{1} ψ_{2}) .$
XIV-D@1#eq:first · line 482
$X^{μ} [r] V_{1, g} [r] ψ p [r; ψ] A_{1} (r; ψ) = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ, = - E_{m}^{R} (V_{1, g} [r] ψ), = \int f (p [r; ψ] - X^{μ} [r] \partial_{μ} ψ) d^{4} x .$
unlabeled · line 458
$V_{g} = - \frac{1}{2} \int g γ_{μν} T^{μν} (ϕ, ϕ) d^{4} x .$
unlabeled · line 464
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r,$
unlabeled · line 470
$b_{2} [γ] = \frac{1}{8} (tr_{η} γ)^{2} - \frac{1}{4} γ_{ρ σ} γ^{ρ σ},$
unlabeled · line 475
$a_{2}^{μν} [γ] = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} (tr_{η} γ) γ^{μν} + b_{2} [γ] η^{μν} .$
unlabeled · line 496
$r_{D} [γ] = P E_{0}^{R} J_{D} [γ], r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T (ϕ, ϕ)) .$
unlabeled · line 506
$ℓ_{3} [γ] = - 2 [ρ^{3}] (- det q (ρ) R [q (ρ)]), J_{D} [γ] = EL_{γ} (σ_{ε} ℓ_{3} [γ]) .$
unlabeled · line 514
$R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .$