Manuscript En05-C@1 · En05-C.tex · proposition · aa213ea6

Proposition: Regional expectation versus gauge averaging

Stage D: a finite constrained Z2 model › Expectations in the demonstrator · lines 1165–1202 · proof included or linked

En05-C@1#prop:thetaexpect, En05-C@1#eq:thetaexpect

The regional conditional expectation onto $A ({1, 2})$ is

E_{12} (a) = e_{+} a e_{+} + e_{-} a e_{-} = \frac{1}{2} (a + F a F) .

It preserves the ordinary trace. Gauge averaging by $G$ acts as the identity on the physical algebra and is therefore not $E_{12}$ .

There are faithful physical states for which no state-preserving conditional expectation onto $A ({1, 2})$ exists.

Proof. Equation (eq:thetaexpect) is block pinching onto the two full matrix blocks, hence a conditional expectation by Proposition prop:finiteexpect. The constraint $G$ is the identity on $H_{phys}$ .

Any conditional expectation onto these full blocks is forced by bimodularity to kill every off-diagonal block $e_{+} a e_{-}$ and $e_{-} a e_{+}$ , and to fix every diagonal block. It must therefore equal $E_{12}$ .

Choose

ρ_{ε} = \frac{1}{4} 1 + ε W, 0 < ε < \frac{1}{4} .

This state is faithful, and

Tr (ρ_{ε} W) = 4 ε, E_{12} (W) = 0.

The unique expectation does not preserve it.

Displayed formulas in this passage

En05-C@1#eq:thetaexpect · line 1168
$E_{12} (a) = e_{+} a e_{+} + e_{-} a e_{-} = \frac{1}{2} (a + F a F) .$
unlabeled · line 1192
$ρ_{ε} = \frac{1}{4} 1 + ε W, 0 < ε < \frac{1}{4} .$
unlabeled · line 1197
$Tr (ρ_{ε} W) = 4 ε, E_{12} (W) = 0.$