Manuscript VII-F-ST@1 · VII-F-ST.tex · proposition · 9eee7a44

Proposition: Conditioning becomes evolution only with covariance

Finite destructive clock and an exact evaluation table › Optional relational conditioning, with its actual hypotheses · lines 1399–1437 · proof included or linked

VII-F-ST@1#prop:relational

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Let finite Hermitian $H_{C}, H_{S}$ and a state $ρ_{C S}$ satisfy

[H_{C} \otimes I + I \otimes H_{S}, ρ_{C S}] = 0.

Suppose a complete orthonormal clock basis satisfies, at its declared readings $t_{a}$ ,

P_{a} = U_{C} (t_{a}) P_{0} U_{C} (t_{a})^{†}, U_{C} (t) = e^{- i H_{C} t /ℏ} .

Then

σ_{S, a} = U_{S} (t_{a}) σ_{S, 0} U_{S} (t_{a})^{†}, U_{S} (t) = e^{- i H_{S} t /ℏ},

where $σ_{S, a} = D_{a} ρ D_{a}^{†}$ . All reading probabilities equal $1/ D$ .

The same evolution statement on $S \otimes E$ requires stationarity of the full extension, not merely its $C S$ marginal.

Proof. Stationarity gives

(U_{C}^{†} \otimes I) ρ (U_{C} \otimes I) = (I \otimes U_{S}) ρ (I \otimes U_{S}^{†}) .

Insert the covariant $P_{a}$ into the partial-trace expression for $σ_{S, a}$ and use invariance of partial trace under clock conjugation. This gives the displayed equality. Its traces are equal and sum to one. The extended proof is the same precisely when the extended stationarity hypothesis holds.

Displayed formulas in this passage

unlabeled · line 1403
$[H_{C} \otimes I + I \otimes H_{S}, ρ_{C S}] = 0.$
unlabeled · line 1408
$P_{a} = U_{C} (t_{a}) P_{0} U_{C} (t_{a})^{†}, U_{C} (t) = e^{- i H_{C} t /ℏ} .$
unlabeled · line 1413
$σ_{S, a} = U_{S} (t_{a}) σ_{S, 0} U_{S} (t_{a})^{†}, U_{S} (t) = e^{- i H_{S} t /ℏ},$
unlabeled · line 1427
$(U_{C}^{†} \otimes I) ρ (U_{C} \otimes I) = (I \otimes U_{S}) ρ (I \otimes U_{S}^{†}) .$

Proposition: Conditioning becomes evolution only with covariance

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