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>) 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. -- 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. 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