charters said:
But the boolean algebra of a un-made/unitarily reversed event can't be appropriate, right?
Sorry, I forgot to respond to this even though I said I would.
I got some interesting (and hopefully correct) results. For simplicity, I'll represent Wigner's friend, his device, and his lab all with ##F##, and Wigner's own lab including himself with ##W##. I'll also ignore the coin toss (which I think is not relevant for this new question).
CH let's us use unitary evolution or collapse, depending what properties we want to discuss. A fully unitary evolution of the whole system (particle ##\psi##, friend ##F##, and Wigner ##W##) is
\begin{eqnarray*}
U(t_0,t_1)|\psi\rangle|F_\Omega\rangle|W_\Omega\rangle &=& |1_\mathcal{X}\rangle|W_\Omega\rangle\\
U(t_1,t_2)|1_\mathcal{X}\rangle|W_\Omega\rangle &=& |1_\mathcal{X}\rangle|W_{1}\rangle
\end{eqnarray*}
In the histories formalism, this unitary evolution implies the family of histories containing only one history with nonzero probability:
$$[\psi,F_\Omega,W_\Omega]_{t_0}\odot[1_\mathcal{X},W_\Omega]_{t_1}\odot[1_\mathcal{X},W_1]_{t_2}$$
We have a history with probability ##1## that contains Wigner's measurement outcome as the property ##[W_1]##.
If, on the other hand, we insert a collapse for Wigner's friend's measurement, we get evolution that looks like (e.g. for a measurement outcome ##F_\uparrow##)
\begin{eqnarray*}
U(t_0,t_1)|\psi\rangle|F_\Omega\rangle|W_\Omega\rangle &=& |1_\mathcal{X}\rangle|W_\Omega\rangle\\
|F_\uparrow\rangle\langle F_\uparrow|1_\mathcal{X}\rangle|W_\Omega\rangle &=& \frac{1}{\sqrt{2}}|\uparrow\rangle|F_\uparrow\rangle|W_\Omega\rangle\\
U(t_1,t_2)\frac{1}{\sqrt{2}}|\uparrow\rangle|F_\uparrow\rangle|W_\Omega\rangle &=& \frac{1}{2}|\uparrow\rangle|F_\uparrow\rangle(|W_{0}\rangle-|W_{1}\rangle) + \frac{1}{2}|\downarrow\rangle|F_\downarrow\rangle(|W_{0}\rangle+|W_{1}\rangle)\end{eqnarray*}
This implies the possible histories
\begin{eqnarray*}
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\uparrow,F_\uparrow,I_W]_{t_2}\\
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\downarrow,F_\downarrow,I_W]_{t_2}\\
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\downarrow,F_\downarrow,W_\Omega]_{t_1}&\odot&[\uparrow,F_\uparrow,I_W]_{t_2}\\
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\downarrow,F_\downarrow,W_\Omega]_{t_1}&\odot&[\downarrow,F_\downarrow,I_W]_{t_2}
\end{eqnarray*}
But this family is inconsistent! The 1st + 3rd family don't decohere. Nor do the 2nd + 4th. This is due to the dynamics ##U(t_1,t_2)## in our evolution. If Wigner left the system alone, they would decohere. But they don't.
This means we cannot reason about the correlations between the property ##F_\uparrow## (i.e. the friend recording ##\uparrow##) after Wigner's measurement, with respect to the property before Wigner's measurement. But we need this kind of reasoning to make statements like "Wigner's friend still remembers his measurement, even after Wigner's measurement". We cannot identify a record of Wigner's friend's measurement at any time after Wigner's measurement. Erasure! I suspect (but haven't proved) that this is true for any of Wigner's friend's experiences since they should all fail to commute with Wigner's measurement.
I also looked at a special case where the particle is prepared with the property ##\uparrow##. I ended up with the consistent family
\begin{eqnarray*}
[\uparrow,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\uparrow,F_\uparrow,I_W]_{t_2}\\
[\uparrow,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\downarrow,F_\downarrow,I_W]_{t_2}
\end{eqnarray*}
with a probability of ##0.5## for each. This implies ##P([F_\uparrow]_{t_1} \vert [F_\uparrow]_{t_2}) = P([F_\uparrow]_{t_1}\vert [F_\downarrow]_{t_2}) = 0.5##. I.e. Zero correlation. So even though we can now construct histories referencing the friend's memory both before and after Wigner's measurement, there is no correlation with past memories. I.e. Still no record.
[edit] - What's also weird is it implies Wigner's friend is still alive! Wigner's precise interactions, with its exotic dynamics, precisely tampers with his friend's memories, but keeps him healthy.
All of the above comes with the caveat that we are hardly talking about a realistic scenario, as discussed by DarMM. So none of the above should imply any of it is realizeable.
[edit]- I also put this out there for criticism. There might be all sorts of issues with my reasoning.
[edit 2] - PS my conclusion that Wigner's friend is alive hinges on Wigner perfoming a non-destructive measurement of ##\mathcal{X}##.
