On Oct 29, 1:53 pm, Stathis Papaioannou <stath...@gmail.com> wrote: > On Oct 26, 2011, at 10:00 AM, Nick Prince <nickmag.pri...@googlemail.com> > 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. > > > 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. > > > 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>) (4)
> > > where |a|^2 + |b|^2 + |c|^2 =1. Also > > > |a|^2, |c|^2 are very small whilst |b|^2 is nearly = 1 > >> > > 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. > > Consider the first person psychology. The cat dies but there is another cat > in another universe with the same mental state which does not die, because in > that universe things are different; for example, there is no actual > experiment because the cat is dreaming. The consciousness of the cat > continues in the version that doesn't die. > > It is a mistake to assume that the cat is fixed in the cohort of cats that > have, as it were, travelled together up to that point. We have to consider > all the cats in the multiverse that hear the glass break. > > -- Stathis Papaioannou Hi Stathis Maybe you are thinking of Tegmark level 1 or level 2 type multiverses here, in which case I agree. What I was doing in my analysis was thinking about QM type 3 multiverses only. Let's pretend that these are the only variety for the moment, then my analysis does indicate that cul de sacs arise only if the unitary development during interactions follow the ideal measurement prescription. You can see this because in the times between the action of operator Mdev and Mc the cat is alive in both branches but destined to die in one of them. This has to be true for both ist and 3person points of view because there is nowhere for the consciousness to go. If you are going to include the other types of multiverse then yes, all sorts of possibilities open up. Indeed dreaming cats would be included too. Moreover, it seems to me from Bruno's Sane papers that ist person indeterminacy is non local in space and in time so, I guess in principle it's possible according to that reasoning, that the cat could find a contiuation of its consciousness in some other cat far off in the future in some universe. If we restrict ourselves to level 3 type QM branching of fungible universes then perfect functioning flask gassing mechanisms would provide cul de sacs. I was hoping that this might give a start to some form of extra support (although not a proof )of the no cul de sac conjecture because in the limit as the number of degrees of freedom in the devices introduce more and more branches due to evolutions of the form (4) (which could possibly be infinite linear combinations), then perhaps once the environment was included as well, the limit would ensure that the cul de sacs were avoided. If we factor in other level 1 and 2 type universes then this only helps the argument. Best wishes Nick -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
