On 10/25/2011 7:00 PM, Nick Prince wrote:
QTI, Cul de sacs and differentiation

I’m trying to get a  picture of how David Deutsch’s idea of
differentiation works – especially in relation to QTI.  With a
standard treatment it looks as if there might be cul de sacs for  a
dying cat.  However I think I can see why this conclusion could be
wrong.  Maybe someone could check my reasoning for this and tell me if
there are any flaws.

I’ve entered this on both the FOR and the Everything list because I
hope it is relevant to both Forums.

Firstly I am adopting the position that consciousness supervenes on
all identical  worldlines and where the multiverse differentiates, the
first person experience is indeterminate. Secondly I assume local
causality applies. Thirdly (to begin with anyway) I assume that all
“measuring” systems function as they should do to obtain  “correct
measurements/outcomes”  (I’ll drop this part later though).   Now
suppose a SINGLE electron is prepared so that its spin is aligned in
the x - right direction (  |Xr>  – for x spin in the right direction)
is sent through a SG device and that, whether the electron comes out
spinning up in the z direction or down determines the triggering of a
device which breaks the flask of gas – (let’s say it is the electron
with spin up and moving upwards which is the lethal combination) which
kills the cat. On the other hand, an electron emerging with spin down,
and moving down in the z direction leaves the measuring device
triggered in the down state but this does nothing to the flask)  So
there is a 50% chance of the cat being killed for each electron fired
through. This means there would be interaction Hamiltonians which
would make up unitary evolution operators of the form

  M = exp(-iHt/hbar) which would act on states as follows:

(Mc) (Mdev) ( Msg) |Ca>  |Dn>  |moves to right>  |Xr>

Msg = stern gerlach interaction evolution operator
Mdev = triggering and flask breaking evolution operator
Mc = cat /poisonous gas evolution operator.
|Dn>  = neutral detector state
|Ca>  = alive cat state etc.

Now standard QM gives |Xr>  = (1/sqrt2)(|Zu>  +|Zd>)

Notice how I have put operators in time order so that the rightmost
operator is implied to operate earlier than those to the left. The
order of the state vectors reflects this too.
Hi,

Are we sure that this ordering, at the level of the state vectors, really matters? We are, after all, only considering observables that mutually commute and thus ordering should be irrelevant.


Msg is the unitary operator which causes an evolution from |moves to
right>  to either moves up |moves up>   or moves down |moves down>.
Mdev allows evolution of the detector device plus flask smashing
mechanism which, if it causes evolution to  |Du>,  breaks open the
flask of poisonous gas.  |Dd>  leaves the flask intact.   Finally the
interaction of the gas with the cat due to the evolution operator Mc
leaves the cat either dead |Cd>  or alive |Ca>.

Now, assuming a causally functioning system, then we can write:

(Mc) (Mdev) ( Msg) |Ca>  |Dn>  |moves to right>  |Xr>

=(Mc) |Ca>   (Mdev) |Dn>  ( Msg) |moves to right>  (1/sqrt2)(|Zu>  +|Zd>)

=(Mc) |Ca>   (Mdev) |Dn>(1/sqrt2) [|moves up>|Zu>+|moves down>|Zd>]

=(Mc) |Ca>   (1/sqrt2) [|Du>|moves up>|Zu>+Dd>|moves down>|Zd>]

= (1/sqrt2) [|Cd>|Du>|moves up>|Zu>+|Ca>|Dd>|moves down>|Zd>]

All this is just the standard Schrödinger’s cat problem.  However note
that if one thinks in terms of a differentiated multiverse we begin
with an infinite number of identical experiments in identical
universes.  By the end of the first evolution due to Msg, the infinite
bundle of universes has partitioned into two bundles i.e. one bundle
of universes that have a Z spin up electron moving upwards with a
neutral detector reading and an alive cat, and another bundle of
universes  that have a Z spin down electron moving downwards with a
neutral detector reading and an alive cat.  As time progresses the
partition is communicated through the system and by the end of the
period during which operator Md operates we have two bundles that are
even more differentiated because the strands (universes) now are those
that have either a Z spin up electron which moved upwards causing a
detector to  smash the flask but with an, (as yet) still alive cat in
one of them and another bundle of universes  that had a  Z spin down
electron which moved downwards, which triggered the detector in a way
which did not smash the flask and thus, as yet still also has an alive
cat. Finally by the time the last (Mc) operator’s effect has finished
its evolutionary action,  the cat lays dead (or alive) and thus the
final bundles are now partitioned into either a Z spin up electron
that moved upwards with a detector that  smashed the flask that killed
the cat; or, a Z spin down electron that moved downwards with a
detector reading that triggered no flask smashing and left the cat
alive.


There’s nothing new in any of this. It is just the standard paradox
normally used to illustrate the problem of collapse.  However I have
highlighted the fact that the experiment takes time to untangle the
different strands of the differentiated multiverse such that the cat
can discover which type of strand he is in. Now suppose the cat
discovers that he is in the wrong bundle – i.e.  he hears the flask
smash!  Now because we have assumed all devices – including flask
smashers and poisons work as they ideally should do, then the cat must
now be in a cul de sac!  He cannot be in any other branch than one in
which he dies because everything by definition is working properly and
against his survival from either 1st person or 3rd person viewpoints.
Indeed the cat was doomed to die (by cul de sac) at the first
partitioning when it unwittingly ended up in the bundle of universes
wherein the electron was moved upwards by the SG device rather than
down.

Now I actually think this conclusion is wrong.  This is because the
whole system is idealised and I have tried to show why, by some
reasoning which I hope is correct, However, I would welcome some
confirmation that my treatment is not erroneous.
It seems to me that we have to take the environment of the system into account, so we have to have a {environment> in the equation, no? From what I can tell, cul de sac's would have 3p consequences that would have an effect on the distribution of branches. Maybe we should consider what effect the 'rest of the universe' has on the 1p of the cat.

Onward!

Stephen


If we just go back to a simple measurement example and modify it to
account for alternative possiblities in the bundles then the analysis
might go as follows:

Let S be the state space  of a quantum system and A be that of the
experimental apparatus ( also considered as a quantum system)used for
measuring the system S

Consider the development of the combined system S cross A (tensor
product) and let |s1>,|s2>  belonging to S  be two eigenstates of the
object corresponding to two different results of the experiment.
These results must leave the apparatus in different states |a1>  and |
a2>  (describing say, different positions of a pointer.).  Suppose the
apparatus is initially in another (“neutral ”) eigenstate   |a0>. The
experiment therefore consists of allowing the object and the apparatus
to interact in such a way that if the object state is |s1>, then after
the experiment, the object will still be in the state |s1>    and the
apparatus will record the appropriate result, i.e. will be in the
state |a1>. A similar argument holding for  |s2>  and |a2>.   This is
what I believe to represent an ideal measurement where the measuring
device “works properly”.

Thus during the experiment the Hamiltonian H in exp(-iHt/hbar) must be
such that:

exp(-iHt/hbar) (|s1>|a0>)=|s1>|a1>

exp(-iHt/hbar) (|s2>|a0>)=|s2>|a1>          (1)

Where t is the time taken for the experiment to yield a definite
result.

Now, if before the experiment the system was in the state

|s0>  = c1|s1>  + c2|s2>                               (2)

Where |c1|^2 + |c2|^2 = 1

Then after it, the system and the apparatus together will be in the
state



exp(-iHt/hbar) (|s0>|a0>)

= exp(-iHt/hbar) (c1|s1>|a0>  + c2|s2>|a0>)  (3)

This is the simplified  ideal case again.

Now let’s consider how things would be in real non ideal measurements
where say the apparatus (and lots of other things for that matter) was
less reliable.  Being a macroscopic device with many degrees of
freedom means there are many ways one could imagine from the classical
point of view that false readings might occur – i.e. the pointer
sticks for some reason etc.  From the quantum microscopic point of
view errors could occur simply because it may not be possible to make
systems with Hamiltonians that are exactly appropriate – i.e does the
moving electron interact with some field fluctuation, air molecule
etc?  I have to admit I’m not sure about things at this level so if
anyone can give me valid reasons then I would be grateful.  Anyway to
account for non ideal functioning of an apparatus/device/process I am
GUESSING that during the experiment we might instead have a
Hamiltonian which is different to (1) and  is such that we have
something more like:




exp(-iHt/hbar)(|s1>|a0>)=|s1>(a|a0>  + b|a1>  + c|a2>)

exp(-iHt/hbar)(|s2>|a0>)=|s2>(a|a0>  + c|a1>  + b|a2>)

where |a|^2 + |b|^2 + |c|^2 =1.  Also

|a|^2, |c|^2 are very small whilst |b|^2 is nearly = 1
(4)



Now,  following the original argument, if before the experiment the
system was in the state
|s0>  = c1|s1>  + c2|s2>               (5)

Then after it, the system and the apparatus together will be in the
state


exp(-iHt/hbar)(|s0>|a0>) =  exp(-iHt/hbar) (c1|s1>|a0>) + c2|s2>|a0>)

= c1|s1>(a|a0>  + b|a1>  + c|a2>)+ c2|s2>(a|a0>  + c|a1>  + b|a2>)

= b c1|s1>|a1>  + b c2|s2>|a2>  + c c1|s1>|a2>  + a c2|s2>|a0>  +

c c2|s2>|a1>+ a c1|s1>|a0>           (6)

The first two terms are “almost those” of those on the RHS of equation
(3), but now there are new terms.  However, the probability amplitudes
of these new terms are far less dominant because of the relative sizes
of the
|a|^2, |c|^2 and |b|^2 terms.

I am guessing that I can interpret this then as an experiment where
the measuring apparatus usually works correctly but on rare
occasions:

(i)     Does not detect anything when in fact it should have done - e.g.
the presence of the |s1>|a0>   and |s2>|a0>   terms.

(ii)    Very occasionally records the eigenvalue of one state  |s1>  when
it should have recorded  the eigenvalue of the other state |s2>  and
vice versa – e.g. the presence of the|s1>|a2>,  and  |s2>|a1>  terms.

Going back to the cat experiment it does mean that if things work this
way then now the cat can always expect some probability that an upward
moving electron and/or a smashed flask may not be the end.  Indeed
there are many more bundles available for the consciousness to access
even at any late stage, thereby avoiding any cul de sacs and hence,
all the usual QTI argument carry over again.

I’m not sure whether I can use this approach (particularly the step
(4) above) and would like comments and criticisms as to whether my
analysis can be considered valid.  For example if the effect is due to
macroscopic failure (like the pointer sticking or a fuse blows and
therefore the reading is in the |a0>  state irrespective of the system
state) then can this be a possible valid interpretation of (6)? Or
microscopically say, for example, the direction of motion of a
particle gets redirected because it recoils from a rogue air molecule
and goes the wrong way in the detector chamber.

Many thanks for anyone who takes the trouble to answer my query.




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