Manuscript XVII-A@1 · XVII-A.tex · section · 711d7ee4

Declared bulk field and named observable

Objects fixed before the reconstruction map › Declared bulk field and named observable · lines 259–286

XVII-A@1#eq:bulk-field, XVII-A@1#eq:named-observable

Define the self-adjoint bulk matrix field

Φ (t, x) = n = 1 \sum^{K} \frac{sin ( n ( x + π /2 ) )}{π n} (a_{n} e^{- in t} + a_{n}^{†} e^{in t}) .

All sums and all operator domains are finite. Set

Q_{bulk} = Φ (p_{*}) = Φ (0, 0), λ_{*} = \frac{π ^{3/2}}{6}, O_{bulk} = \frac{1}{2} (1 + sin (λ_{*} Q_{bulk})) .

The sine is the ordinary functional calculus of a self-adjoint finite matrix. Consequently

0 \leq O_{bulk} \leq 1 .

This bounded effect is the one named bulk observable attempted in XVII-A.

Displayed formulas in this passage

XVII-A@1#eq:bulk-field · line 262
$Φ (t, x) = n = 1 \sum^{K} \frac{sin ( n ( x + π /2 ) )}{π n} (a_{n} e^{- in t} + a_{n}^{†} e^{in t}) .$
XVII-A@1#eq:named-observable · line 270
$Q_{bulk} = Φ (p_{*}) = Φ (0, 0), λ_{*} = \frac{π ^{3/2}}{6}, O_{bulk} = \frac{1}{2} (1 + sin (λ_{*} Q_{bulk})) .$
unlabeled · line 281
$0 \leq O_{bulk} \leq 1 .$