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July 24, 2015 at 12:54 am #2959Rod SutherlandMember
Hi Nathan,
Thanks for your further comments. It’s now clearer to me what your concerns are. And yes, I think you’ve summarised the situation correctly. In particular, for an ensemble, the distribution of final states in my model is simply assumed to satisfy the Born rule. I see this as a separate issue to be solved for most, if not all models including mine. And yes, you’re also correct that feeding in a “wrong” choice for the initial (hidden) position distribution in my model won’t necessarily lead to “wrong” statistics at a subsequent measurement if the distribution of final states is still the “right” one as given by the Born rule.
Concerning this rule, if you have a way of deriving it I’d be interested to see your method. In particular, I haven’t seen either the toy model or the comparison table which you’ve mentioned, so please let me know where to look.
Also, just in case you’re interested, I should mention that I proposed a way of deriving the Born rule some years ago in a paper published in Foundations of Physics Letters 13, 379 (2000). Any such derivation requires the introduction of extra structure though.July 19, 2015 at 2:07 am #2891Rod SutherlandMemberHi Aurelien,
Please feel free to keep in touch after the workshop if you have further thoughts on these matters. My email address is: [email protected].
By the way, I think on reflection that my comment labelled (ii) in my previous reply to you was inadequate, because it only covers the singleparticle case. To establish the consistency of my theory with the QM predictions in the manyparticle case, the proof given in Sec. 8 of my new paper is also needed, together with the usual notions of the Bohm Theory of Measurement.
Best wishes,
RodJuly 18, 2015 at 2:58 am #2850Rod SutherlandMemberHi Aurelien,
(i) You seem to be saying that the final state in my model should simply be “the evolving state of psi in”. But this is not the case. Even in standard QM, the final state is the collapsed psi after measurement. In both standard Bohm theory and in my retrocausal version, the final state is that branch of the initial wavefunction that the particle actually follows after this wavefunction is spatially split by the measurement interaction, with the empty branches being deleted. Hence, in every theory, the final wavefunction is quite independent of the initial one (i.e., it is a different vector in Hilbert space) and is determined by either the particle’s hidden position, or by conditions further in the future, or by nothing at all, depending on which theory we use.
(ii) The fact that my theory reduces back to the standard Bohm model in the usual situation where the future measurement result is not yet known was meant to be established by the short proof in Sec. 5 of my new paper (i.e., it is shown that the 4current density becomes identical and so the velocity and probability density will too). If you are not happy with this proof, please let me know.
(iii) Concerning Bell’s nonlocality, I thought it was generally accepted that this nonlocal effect cannot be explained within the standard Bohm model without using a preferred reference frame. So when I said “two boundary conditions are needed in this picture”, I was trying to say that it does not seem to be possible to achieve Lorentz invariance within a Bohmian framework without introducing a final boundary condition.
Best wishes,
RodJuly 18, 2015 at 1:20 am #2848Rod SutherlandMemberHi Dustin,
As a preliminary step towards answering your mathematical questions, I’d like to focus on your query about my Eq. (1). In particular, I’d like to point out that this equation is meant to be part of standard quantum mechanics, not just of my model. In this context, hopefully we can agree on the following points:
1. In the relativistic case, it’s possible to have a pair of particles which are in an entangled state, but which have essentially ceased interacting and are now far apart.
2. Once a measurement is performed on one of the particles, the other particle can then be described by its own, singleparticle state.
3. This new state can be deduced by combining the original twoparticle state and the measurement result together in some way.
4. Even though there is some ambiguity in the time at which the singleparticle state becomes available, points 2 and 3 remain valid.
My Eq. (1) was simply my attempt to express point 3 in mathematical form. So my question to you as a mathematician is: How would you describe this piece of standard quantum mechanics in equation form?
It might help here for me to comment on the entangled state psi(x,x’). It’s usual to express the initial boundary conditions of any theory on one particular hyperplane at some initial time ti, so Lorentz invariance is automatically broken in this sense. However, the situation at a later time t should then be Lorentz invariant. My entangled state at time t is defined to be something like:
psi(x,x’) = integral of [K(x,xi)K'(x’,xi’)psi(xi,xi’)] dxi dxi’
where K and K’ are propagators from ti to t and the coordinates xi and xi’ are assumed to be at the same time ti.
And yes, we’ll definitely need to continue after the workshop! I’ll need your email address.
Best wishes,
RodJuly 17, 2015 at 1:32 am #2817Rod SutherlandMemberHi Dustin,
Thanks for your feedback. First, I’m rereading Durr, Goldstein and Zanghi’s paper to see if I should modify my previous opinion on the conclusiveness of their argument. This will take a little while since it’s a long paper (75 pages), so I’ll get back to you on this. Second, I’m going through the equations in my paper to see how the notation could be improved and to check if anything deeper is wrong. I’ve deliberately tried to keep the notation neat and simple, but I may have sacrificed some clarity in doing this. I think I could answer some of your queries immediately but, again, I’ll wait until I’ve sorted through all of it before giving you a response. It’s good to have a mathematician scrutinizing my work, since the philosophers tend to have different concerns.
Best wishes,
RodJuly 15, 2015 at 12:54 am #2751Rod SutherlandMemberHi Nathan,
I’ve had trouble submitting a reply to your question electronically, but I seem to be managing now.
The short answer to your question is that my model is simply an “addon” to quantum mechanics and so just assumes the Born rule for probabilities as part of the preexisting formalism. Yes, I would certainly like to see a more fundamental derivation of this rule, but my personal opinion is that none of the interpretations of QM have succeeded in doing this in a way that is rigorous and generally accepted.
In the case of the standard Bohm model, all the maths seems to tell us is that if we start with the Born distribution then this distribution will persist through time. My understanding is that attempts have been made to show that other distributions will decay with time to the right one, but that these attempts have not been fully convincing. So it seems to me that the usual model is essentially just resorting to the rules of QM too. It is true that the Bohm theory of measurement is impressive and constitutes an advance (in my opinion), but again a similar version can be formulated for the my model (Sec. 13 in my 2008 paper). In particular, given the initial probability distribution for position provided by my model, the maths ensures that this distribution is maintained through time.
Finally, concerning nonstandard distributions, I would have thought that both models are on the same footing in being able to accommodate them.
Anyway, this time it’s my turn to ask if I’m understanding things correctly.
Best wishes,
RodJuly 15, 2015 at 12:51 am #2749Rod SutherlandMemberHi Aurelien,
Thanks for your question. For some reason I wasn’t able to submit a reply to your previous comment, but I’m in now.
The general rule for understanding the behaviour and propagation of the final wavefunction in my theory is to think about what happens with the usual, initial wavefunction and then do the same thing but in the opposite time direction. If you’re puzzled about something with the final psi, ask yourself if you would still be puzzled if you were just talking about the initial psi instead. For example, aspects of the usual Bohm Theory of Measurement are maintained in my model. In the standard theory, the measurement interaction (e.g., an externally applied potential) causes the initial wavefunction to spread so that the relevant eigenstates become spatially separate like the fingers of a hand. In spacetime, the fingers point in the forwards time direction. The same thing is assumed to happen with the final wavefunction as it goes through the interaction region coming from the future, except the fingers now point in the backwards time direction. So the final boundary condition is fixed in an analogous way to the state preparation of an initial psi. More details on this are in the measurement section of my 2008 paper.
Unlike classical physics, there is no doubt that two boundary conditions are needed in this picture, since otherwise Bell’s nonlocality could not be explained in a Lorentz invariant fashion via a spacetime zigzag. The reason for two wavefunctions is then simply to move the mathematical information about the two boundary conditions from both the earlier time ti and the later time tf to the present time t so we can calculate the effect on the present state (e.g., the particle’s velocity). Also, without both initial and final conditions, it is not possible to reduce the configuration space description usually required for the manyparticle case back to a description in real space, as explained in my present paper.
I hope I’ve understood your questions correctly.
Best wishes,
RodJuly 13, 2015 at 12:34 am #2673Rod SutherlandMemberHi Ruth,
Thanks for your question. As a physicist rather than a philosopher, I feel somewhat less qualified than you to argue on the issue of the block universe versus time flowing. Superficially, though, I still lean towards the block universe picture simply because it’s hard to see how a future boundary condition could influence what exists in the present if the future doesn’t exist yet. Perhaps the same thing could be argued about the transactional interpretation. I’m more apprehensive about expressing this viewpoint now, however, because last time I replied to you I hadn’t realised you weren’t a block universe supporter and I don’t want you to dismiss my model purely on this contentious philosophical point. The mathematical apparatus of the model is quite powerful and I would prefer you to embrace the maths and place your own interpretation on it. There is obviously retrocausality involved because choosing to measure a different observable at a later time changes the particle’s velocity at earlier times. Of course, an intuitive notion of free will is assumed here.
I hope this helps.
Rod 
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