Manuscript I-B-G@2 · I-B-G-v2.tex · definition · 3b4d700f

Definition: Retarded, advanced, and causal operators

Green operators on an extension-dependent spectral domain · lines 746–765

I-B-G@2#def:Green, I-B-G@2#eq:Green

For $f \in D_{tc}_{B}$ , define

(G_{+}^{B} f) (t) (G_{-}^{B} f) (t) E_{B} = \int_{- \infty}^{t} W_{B} (t - s) f (s) d s, = - \int_{t}^{\infty} W_{B} (t - s) f (s) d s = \int_{t}^{\infty} W_{B} (s - t) f (s) d s, = G_{+}^{B} - G_{-}^{B} .

Thus, in time-convolution notation,

G_{\pm}^{B} (t) = \pm θ (\pm t) W_{B} (t) .

The minus sign in the first advanced integral is essential.

Displayed formulas in this passage

I-B-G@2#eq:Green · line 749
$(G_{+}^{B} f) (t) (G_{-}^{B} f) (t) E_{B} = \int_{- \infty}^{t} W_{B} (t - s) f (s) d s, = - \int_{t}^{\infty} W_{B} (t - s) f (s) d s = \int_{t}^{\infty} W_{B} (s - t) f (s) d s, = G_{+}^{B} - G_{-}^{B} .$
unlabeled · line 761
$G_{\pm}^{B} (t) = \pm θ (\pm t) W_{B} (t) .$