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

Signature and operator ledger

The prescribed host and its causal geometry › Signature and operator ledger · lines 219–255

I-B-G@2#eq:signature, I-B-G@2#eq:P

Let

η = d t^{2} - d X^{2} - d Y^{2} - d z^{2}

on the analytical completion $M = R^{1 + 3}$ . On a supported developing region the map

dev (t, r, φ, z) = (t, r cos (α φ), r sin (α φ), z)

satisfies

dev^{*} η = - g .

Equivalently, if $j = dev^{- 1}$ is the chart embedding from the developed laboratory to the host, then $j^{*} g = - η$ . This is the signature ledger used in X-A-ST and its successors CheungXAST (:signature-ledger, :host) CheungXBST,CheungXCST,CheungXDST (:host).

We use the scalar equation operator

P_{m} = - □_{g} + m^{2} = \partial_{t}^{2} - Δ_{q} + m^{2} .

On the completion it is $□_{η} + m^{2}$ . If a companion uses the action Hessian $- P_{m}$ , its inverse is $- G_{\pm}$ , not $G_{\pm}$ . This distinction preserves the normalization of X-B-ST and XI-B CheungXBST,CheungXIB.

Displayed formulas in this passage

I-B-G@2#eq:signature · line 232
$dev^{*} η = - g .$
I-B-G@2#eq:P · line 245
$P_{m} = - □_{g} + m^{2} = \partial_{t}^{2} - Δ_{q} + m^{2} .$
unlabeled · line 222
$η = d t^{2} - d X^{2} - d Y^{2} - d z^{2}$
unlabeled · line 227
$dev (t, r, φ, z) = (t, r cos (α φ), r sin (α φ), z)$

Nearby in this section

remark: Remark: Deficit convention