Manuscript X-B-ST@1 · X-B-ST.tex · section · d03c150c

Green operators and topology

The completion free field and its linear sector › Green operators and topology · lines 562–610

X-B-ST@1#eq:Green, X-B-ST@1#eq:signs

Set

P_{m} = □ + m^{2}, □ = \partial_{t}^{2} - Δ_{x}, D = C_{c}^{\infty} (M; C) .

The test space has its standard LF topology; on a fixed compact $K$ , $D_{K}$ has all smooth seminorms.

Let

E_{R}, E_{A} : D ⟶ C^{\infty} (M)

be the positive Klein–Gordon retarded and advanced Green operators. They satisfy

P_{m} \ERc = E_{R} P_{m} c = c, P_{m} \EAc = E_{A} P_{m} c = c, supp \ERc \subset J_{η}^{+} (supp c), supp \EAc \subset J_{η}^{-} (supp c), E_{R}^{T} = E_{A} .

Existence, smoothness, finite propagation, and continuity are the standard Green-operator facts on the globally hyperbolic completion BGP.

For example, after spatial Fourier transformation,

(\ERc) (t) = \int_{- \infty}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} c (s) d s, Ω_{m} = (- Δ_{x} + m^{2})^{1/2},

and the advanced formula is the transposed one. The quotient at zero frequency has its continuous value. These equations prescribe Green operators, not quantum Cauchy data.

Put

Δ_{S} = E_{A} - E_{R}, Δ_{S}^{R} = - E_{R}, Δ_{S}^{A} = - E_{A} .

Then $Δ_{S}^{T} = - Δ_{S}$ .

Displayed formulas in this passage

X-B-ST@1#eq:Green · line 579
$P_{m} \ERc = E_{R} P_{m} c = c, P_{m} \EAc = E_{A} P_{m} c = c, supp \ERc \subset J_{η}^{+} (supp c), supp \EAc \subset J_{η}^{-} (supp c), E_{R}^{T} = E_{A} .$
X-B-ST@1#eq:signs · line 603
$Δ_{S} = E_{A} - E_{R}, Δ_{S}^{R} = - E_{R}, Δ_{S}^{A} = - E_{A} .$
unlabeled · line 565
$P_{m} = □ + m^{2}, □ = \partial_{t}^{2} - Δ_{x}, D = C_{c}^{\infty} (M; C) .$
unlabeled · line 574
$E_{R}, E_{A} : D ⟶ C^{\infty} (M)$
unlabeled · line 592
$(\ERc) (t) = \int_{- \infty}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} c (s) d s, Ω_{m} = (- Δ_{x} + m^{2})^{1/2},$