Manuscript En06-B@1 · En06-B.tex · section · ee886ae8

Point-labelled local observable algebras

The finite spacetime observable net › Point-labelled local observable algebras · lines 1571–1633

En06-B@1#eq:leftpre, En06-B@1#eq:rightpre, En06-B@1#eq:leftpost, En06-B@1#eq:rightpost

All observables are represented on the common pre-source Hilbert space $H_{0}$ . Dormant apparatus factors are not assigned a local observable algebra before their allocation events.

At $A$ , assign the algebra

A_{A} = B (Q_{L} Q_{R} W) \otimes I_{rest} .

Before the wing measurements, assign along the left path

A_{L}^{pre} = Ad_{U_{B}^{†}} (B (Q_{L} W) \otimes I_{rest}),

and along the right path

A_{R}^{pre} = Ad_{U_{B}^{†}} (B (Q_{R}) \otimes I_{rest}) .

Let

H_{L} = Q_{L} \otimes W \otimes S_{L} \otimes R_{L} \otimes E_{L} \otimes F_{L}, H_{R} = Q_{R} \otimes S_{R} \otimes R_{R} \otimes E_{R} \otimes F_{R} .

At and after $B_{L}$ , the full local dilation algebra is

A_{L}^{post} = Ad_{U_{B}^{†}} (B (H_{L}) \otimes I_{R}),

and analogously

A_{R}^{post} = Ad_{U_{B}^{†}} (I_{L} \otimes B (H_{R})) .

The source unitary is understood as acting trivially on all other factors.

These expressions include the full input and output algebra at a local measurement event. Conjugating a full wing algebra by $U_{L}$ or $U_{R}$ leaves that full wing algebra unchanged. Specific postmeasurement observables nevertheless have their specific Heisenberg representatives. For example, an output pointer projection is represented by

U^{†} Π_{a}^{R_{L}} U .

It belongs to $A_{L}^{post}$ .

Displayed formulas in this passage

unlabeled · line 1578
$A_{A} = B (Q_{L} Q_{R} W) \otimes I_{rest} .$
En06-B@1#eq:leftpre · line 1582
$A_{L}^{pre} = Ad_{U_{B}^{†}} (B (Q_{L} W) \otimes I_{rest}),$
En06-B@1#eq:rightpre · line 1590
$A_{R}^{pre} = Ad_{U_{B}^{†}} (B (Q_{R}) \otimes I_{rest}) .$
unlabeled · line 1599
$H_{L} = Q_{L} \otimes W \otimes S_{L} \otimes R_{L} \otimes E_{L} \otimes F_{L}, H_{R} = Q_{R} \otimes S_{R} \otimes R_{R} \otimes E_{R} \otimes F_{R} .$
En06-B@1#eq:leftpost · line 1605
$A_{L}^{post} = Ad_{U_{B}^{†}} (B (H_{L}) \otimes I_{R}),$
En06-B@1#eq:rightpost · line 1613
$A_{R}^{post} = Ad_{U_{B}^{†}} (I_{L} \otimes B (H_{R})) .$
unlabeled · line 1629
$U^{†} Π_{a}^{R_{L}} U .$

Point-labelled local observable algebras

Displayed formulas in this passage

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