>>Nobody knows for sure whether
>>quantum gravity eliminates singularities. If it does not, and
>>things really "fall off the edge of the world", then pure states
>>probably become mixed, even if one takes the correlation between
>>gravitation and particles into account.
>I think what worries me most about this prospect is that it prohibits
>use of the Heisenberg picture. The Schrodinger picture requires a
>time parameter which seems awkward in this context - not unmanageable,
>I suppose, just awkward. (It seems entirely possible that a working
>theory of quantum gravity will be a quite different model that won't
>make this distinction. In this case it might not make a clear
>distinction between pure and mixed states either.)
I have no trouble using the Heisenberg picture in this situation --
and you'll be happy to note that my Heisenberg state is (or can be) pure!
If time t is before the object has fallen off the edge of the world
and time t' is after the object has fallen off the edge of the world,
then I believe that a pure Schroedinger state at time t
can very well evolve into a mixed Schroedinger state at time t'.
But that's because the Schroedinger state at time t' is *really*
a restriction of the (true) Heisenberg state to some subsystem.
The time t defines a hypersurface S (the set of all spacetime points
whose time coordinate is t); let us assume that S is a Cauchy surface.
The time t' defines a hypersurface S', which is *not* a Cauchy surface
because its past lacks the region behind the event horizon
which hides the singularity where the object falls off the edge of the world.
If A is the algebra of Heisenberg observables,
consisting of such things as O(t), O(t'), O'(t), O'(t'), and so on,
then let A_S be those observables which can be measured on S,
which includes O(t) and O'(t) but not (a priori) O(t') and O'(t'),
and let A_S' be those observables which can be measured on S',
which includes O(t') and O'(t') but not (a priori) O(t) and O'(t).
To get the Schroedinger state at time t,
we look only at those observables in A_S;
to get the Schroedinger state at time t',
we look only at those observables in A_S'.
Now, since S is a Cauchy surface, A_S actually equals A;
O(t') can be expressed in terms of O(t), O'(t), and so on.
Therefore, the Schroedinger state at time t is
simply another way of looking at the Heisenberg state.
However, since S' is not a Cauchy surface, A_S' is not all of A;
perhaps O(t) can be expressed in terms of A_S' but O'(t) can't.
In particular, details about the object that fell off the edge of the world
will be described by observables which don't belong to A_S'.
Therefore, the Schroedinger state at time t' is
a *restriction* of the Heisenberg state to a proper subsystem,
and there's no reason to be surprised if it turns out to be mixed.
As a Bayesian, I use a mixed Heisenberg state to reflect the fact
that I not only don't know everything (which is impossible thanks to QM)
but don't even know as much as it is possible to know.
Nevertheless, the ultimate TOE doubtless allows pure Heisenberg states.
However, if Psi is a pure Heisenberg state that says that
something falls off the edge of the world,
then the Schroedinger states Psi(t) corresponding to Psi
will be mixed if the time t is sufficiently late.
Similarly, Psi(t) for t sufficiently early
will be mixed if anything falls *on* the edge of the world
(such as happens when something appears inside a white hole).
I could even imagine a pure Psi so full of black and white holes
in such a way that Psi(t) is pure for *no* time t,
no matter how you parametrise time in Psi's spacetime.
(I'm assuming that Psi is an eigenstate of spacetime geometry.)
But that wouldn't invalidate the Heisenberg picture
or change the fact that the Heisenberg state in question is pure.