Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

The advanced root and observation domain

The advanced root and observation domain · lines 1523–1553

Em01-F@3#eq:root

Let $P_{m} = □_{η} + m^{2}$ . Its advanced inverse is specified by

P_{m} E_{m}^{A} f = f, supp (E_{m}^{A} f) \subset J^{-} (supp f),

and vanishing future data. This is the equation-normalized inverse of $P_{m}$ , not the inverse of the scalar action Hessian $- P_{m}$ .

With

f (s, p) = \int e^{- i p \cdot x} f (s, x) d^{3} x, ω_{p} = ∣ p ∣^{2} + m^{2},

define

F (t, x) = E_{m}^{A} f (t, x) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot x} \int_{t}^{\infty} \frac{sin ( ω _{p} ( s - t ))}{ω _{p}} f (s, p) d s .

The massless zero-momentum quotient means $s - t$ .

Displayed formulas in this passage

Em01-F@3#eq:root · line 1542
$F (t, x) = E_{m}^{A} f (t, x) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot x} \int_{t}^{\infty} \frac{sin ( ω _{p} ( s - t ))}{ω _{p}} f (s, p) d s .$
unlabeled · line 1527
$P_{m} E_{m}^{A} f = f, supp (E_{m}^{A} f) \subset J^{-} (supp f),$
unlabeled · line 1535
$f (s, p) = \int e^{- i p \cdot x} f (s, x) d^{3} x, ω_{p} = ∣ p ∣^{2} + m^{2},$