Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

An explicit two-point coefficient

E4: the isolated matter-induced tree › An explicit two-point coefficient · lines 1421–1490

X-E-ST@1#eq:D, X-E-ST@1#eq:twotree, X-E-ST@1#eq:t

Let

F = E_{m}^{A} f, a^{μν, α β} = \frac{1}{2} η^{μν} η^{α β} - \frac{1}{2} (η^{μα} η^{ν β} + η^{μ β} η^{ν α}) .

Then $a_{1}^{μν} [l] = a^{μν, α β} l_{α β}$ . The mixed test kernel is

d_{1} [l] (x) D^{α β} (x, z) = Λ^{μ, α β} (x, z) = = \int D^{α β} (x, z) l_{α β} (z) d^{4} z, \partial_{x^{μ}} (f (x) Λ^{μ, α β} (x, z)) - \partial_{x^{μ}} (g (x) a^{μν, α β} \partial_{ν} F (x) δ (x - z)) - \frac{1}{2} m^{2} g (x) F (x) η^{α β} δ (x - z), - σ (z) a^{ν μ, α β} \partial_{x^{ν}} R (x - z), R = E_{0}^{R} .

Here $R (x - z)$ denotes the retarded kernel, not a pointwise value at its singular support. This is the mixed first-coefficient formula of X-E XE (Kernel and support), with its scalar-response part retained.

Let $R_{2} = R * R$ , the retarded fundamental solution of $□^{2}$ . The convolution is defined: on the product of two future cones, addition has compact fibres over compact sets. Equivalently, it is the twice-iterated zero-past solution operator. Translation invariance implies

WF (R_{j} (x - y)) \subset {(x, p; y, - p) : p \neq = 0}, j = 1, 2.

Since $g (y) \neq = 0$ implies $y^{0} > 2 δ$ , every retarded chain $y ⪯ z ⪯ x$ contributing to the tree has $σ (z) = 1$ . Consequently

τ t^{α β} (x, y) = = \iint t^{α β} (x, y) : ϕ (x) (P_{tr} Θ)_{α β} (y) : d^{4} x d^{4} y, 2 g (y) \partial_{x^{μ}} (f (x) a^{ν μ, α β} \partial_{x^{ν}} R_{2} (x - y)) + 2 g (y) \partial_{x^{μ}} (g (x) a^{μν, α β} \partial_{ν} F (x) R (x - y)) + m^{2} g (y) g (x) F (x) η^{α β} R (x - y) .

Displayed formulas in this passage

X-E-ST@1#eq:D · line 1434
$d_{1} [l] (x) D^{α β} (x, z) = Λ^{μ, α β} (x, z) = = \int D^{α β} (x, z) l_{α β} (z) d^{4} z, \partial_{x^{μ}} (f (x) Λ^{μ, α β} (x, z)) - \partial_{x^{μ}} (g (x) a^{μν, α β} \partial_{ν} F (x) δ (x - z)) - \frac{1}{2} m^{2} g (x) F (x) η^{α β} δ (x - z), - σ (z) a^{ν μ, α β} \partial_{x^{ν}} R (x - z), R = E_{0}^{R} .$
X-E-ST@1#eq:twotree · line 1472
$τ t^{α β} (x, y) = = \iint t^{α β} (x, y) : ϕ (x) (P_{tr} Θ)_{α β} (y) : d^{4} x d^{4} y, 2 g (y) \partial_{x^{μ}} (f (x) a^{ν μ, α β} \partial_{x^{ν}} R_{2} (x - y)) + 2 g (y) \partial_{x^{μ}} (g (x) a^{μν, α β} \partial_{ν} F (x) R (x - y)) + m^{2} g (y) g (x) F (x) η^{α β} R (x - y) .$
X-E-ST@1#eq:t · line 1472
$τ t^{α β} (x, y) = = \iint t^{α β} (x, y) : ϕ (x) (P_{tr} Θ)_{α β} (y) : d^{4} x d^{4} y, 2 g (y) \partial_{x^{μ}} (f (x) a^{ν μ, α β} \partial_{x^{ν}} R_{2} (x - y)) + 2 g (y) \partial_{x^{μ}} (g (x) a^{μν, α β} \partial_{ν} F (x) R (x - y)) + m^{2} g (y) g (x) F (x) η^{α β} R (x - y) .$
unlabeled · line 1424
$F = E_{m}^{A} f, a^{μν, α β} = \frac{1}{2} η^{μν} η^{α β} - \frac{1}{2} (η^{μα} η^{ν β} + η^{μ β} η^{ν α}) .$
unlabeled · line 1464
$WF (R_{j} (x - y)) \subset {(x, p; y, - p) : p \neq = 0}, j = 1, 2.$

An explicit two-point coefficient

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