Retrocausal Bohm Model

This topic contains 23 replies, has 5 voices, and was last updated by Avatar Nathan Argaman 4 years, 1 month ago.

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  • #2646

    By adding retrocausality into the usual Bohm model, it is possible to “improve” on the model in a number of ways, these improvements being that:

    1. The model is easily set in Lorentz invariant form
    2. A general form of the model can be formulated which is applicable for any wave equation
    3. The model becomes local from a spacetime viewpoint
    4. A configuration space description is avoided, with the many-particle case remaining in 4-dimensional spacetime
    5. A separate velocity expression can be provided for each of the n particles in an entangled state, rather than just a single, overall velocity defined in 3n-dimensional configuration space
    6. The correct statistical correlations can be maintained while keeping the picture of reality within 4 dimensions
    7. Action – reaction is restored
    8. Energy and momentum conservation is restored
    9. A Lagrangian formulation becomes possible
    10. A source is provided for the quantum potential

    A paper is attached with details.

    So what is the price to pay in exchange for these advantages? Well, apart from the retrocausality itself, it also becomes necessary to allow the covariant world lines of the particles to have some spacelike segments. This is easily shown to be compatible with special relativity since the theory automatically restricts these segments to times other than measurement. It is then a matter of personal taste which version of the Bohm model is preferred.

    #2652
    Ruth Kastner
    Ruth Kastner
    Participant

    Hi Rod,

    Have you thought any more about the block universe question? It seems to me that with the initial and final BC, all spacetime events are ‘set’ and therefore we must have a block world here. I’d be curious to know whether you agree.

    Thanks,
    Ruth

    #2673

    Hi 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

    #2674
    Ruth Kastner
    Ruth Kastner
    Participant

    Thanks Rod,

    I certainly would not dismiss a model simply because it implied a block universe. I have had extensive discussions with the Relational Block World folks (Mark and Michael), and while that’s not my favored approach, it’s not because it’s a block world.

    My concern about the model you’re proposing is that while it describes dynamical entities (guiding waves etc.) it seems to me that these kinds of dynamics end up being superfluous if there is a future BC in addition to a past BC. It seems to me you get a ‘superdetermined’ situation. You talk above about someone choosing to measure a different observable. But if there are future BC in place with the constraints implied by the dependence of particle trajectories on the quantum states, then it seems to me that all those particle trajectories must be determinate in the block world. This seems to mean that what measurements we make are ‘already’ there in the future, so in what sense could we make different measurements than those dictated by the past and future BC and the states involved–the ones already there in spacetime? Thus the only form of ‘free will’ that seems to be available in this picture is a compatibilist sort–an account that must define free will as consistent with the physical determination of all our choices “before” we get to them.

    Regardless of whether or not one is concerned about free will, the other issue is that the dynamics seems superfluous: events/trajectories are what they are, and nothing is being ‘guided’ in any dynamical sense to where it will end up–because it’s already there.

    In contrast, PTI allows for emergence of spacetime from a quantum substratum in a growing universe picture, so there is no future BC to constrain future events in this way. What I prefer about PTI is that the quantum dynamics is actually doing something-giving rise to spacetime events. Also, it has an open future that provides for a robust form of free will.

    In any case, though I should emphasize that I am not prima facie opposed to a block world. My main concern is consistency–and it seems to me that imposing past and future BC gives us a block world in which talk of causation, either forward or backward, is superfluous or even inconsistent, Everything is just there in the block world so it does not need any causation. It’s a static picture. The only sense in which causation can be applied is in terms of static relations between objects or events–thus rendering the idea of ‘retrocausation’ empty.

    The same concern applies to other ‘retrocausal’ models such as Huw Price’s time-symmetric hidden variables and Ken Wharton’s version of that idea. Everything is just there in the block world and the given dynamics don’t seem to have anything dynamically to do. The quantum states just become epistemic measures of our ignorance of where all the hidden variables actually are. Again, a static block world picture.

    Best
    Ruth

    #2689

    Hello Rod,

    So far, I only skimmed through your present article, but I did read the 2008 (or rather, the 2006) version thoroughly at the time. My understanding was that it gives a very interesting description of what happens between a preparation and a measurement, but it does not give a definite prescription for calculating the probabilities for different outcomes of the said measurement (for this reason, I did not include it in the comparison table in my article on Bell’s theorem, as noted there). It seemed to me that, implicitly, one is to calculate these probabilities by the usual rules of QM. That is quite distinct from Bohmian mechanics, where there is a clear independent prescription for evaluating probabilities for measurement results (and, in fact, if the original density distribution is taken as non-standard, one may obtain non-QM results).

    Let me ask: Have I understood correctly? Is the current updated version different in this sense?

    Of course, even if the answer is negative, your work does accomplish a lot, and is quite impressive. And also, it is in good company – the two-time formalism of Aharonov et al shares this attribute – it provides no way of predicting the outcome probabilities of the final measurement (other than using standard QM).

    Thanks, Nathan.

    #2730
    Avatar
    Aurelien Drezet
    Participant

    Dear Rod, Like Nathan Argaman I was impressed by your first manuscript. It is indeed a bohmian version of the two-time formalism of Aharonov et al. I have however several problems with this idea. I will only discuss briefly one here. My question is what is fixing your final boundary condition? In classical physics we have some freedom for choosing a Green function retarded advanced symmetric etc.. all of them are strictly equivalent if we use the good boundary conditions. For example you can write a field E like Ein +Eret= Eout + Eadv. Now in your theory you need two wave functions and this seems for me representing a breaking of this equivalence. What does it mean really to use such a future amplitude going backward in time? I have the feeling that your theory still needs a deeper justification.

    #2749

    Hi 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 many-particle case back to a description in real space, as explained in my present paper.
    I hope I’ve understood your questions correctly.
    Best wishes,
    Rod

    #2751

    Hi 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 “add-on” to quantum mechanics and so just assumes the Born rule for probabilities as part of the pre-existing 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 non-standard 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,
    Rod

    #2754
    Dustin Lazarovici
    Dustin Lazarovici
    Participant

    Dear Rod,

    I am very interested in your model. I hope I’ll be able to study it in more detail and ask more qualified questions before the workshop is over. Until then, I’d like to address your previous post and point out that in standard Bohmian mechanics, the Born rule CAN be derived from first principles in a rigorous way.

    This was done in a seminal paper by Dürr, Goldstein and Zanghí titled “Quantum Equilibrium and the Origin of Absolute Uncertainty”. You can find the online version here: http://arxiv.org/abs/quant-ph/0308039v1

    The argument is very similar to Boltzmann’s analysis for classical statistical mechanics. It is shown that Born’s rule is true in typical Bohmian universes, i.e. in quantum equilibrium. More precisely, it is shown that for typical initial configurations (of all the particles in the universe), the particle positions in an ensemble of subsystems with effective wave-function psi are distributed according to |psi|^2.

    I would expect that the quantum equilibrium analysis doesn not – without further ado – carry over to the time-symmetric version of the theory, where you have two boundary conditions. However, a colleague of mine is currently working on a “timeless” extension – using a “history measure”, so to speak – that might.

    Best wishes,
    Dustin

    #2757
    Avatar
    Aurelien Drezet
    Participant

    Thanks a lot Rod,
    You understood my question quite well . Any way on such a subject anbiguities can come easily and this is certainly my fault not yours.

    I will go back to your work soon I will recontact you with pleasure since I am currently writing a paper on the same subject with a different strategy.

    #2811
    Dustin Lazarovici
    Dustin Lazarovici
    Participant

    Dear Rod,

    I was trying to study your paper. I think your motivations are exactly right and the paper contains a lot of exciting ideas. I must admit that I have great difficulties understanding the model, though. In part, it may just be an issue of (bad) notations, but many things don’t even make sense to me on the formal level.

    For instance, at the very beginning, you introduce x and x’ as space-time coordinates. Hence, Psi(x,x’) is a multi-time wave-function of two particles. I don’t even know then what you mean when you talk about the wave-functions “before” and “after” measurement, since all the wave-functions seem to be defined on the entire space-time (or multiple copies thereof).

    In the following, I already don’t understand equation (1). The right-hand-side of the equation seems to depend on the time-coordinate of particle 2, whereas the LHS doesn’t. And even if you fixed a particle time t’, the expression would be strikingly not Lorentz invariant, since you integrate over one particular spacelike hypersurface in one particular frame.

    My difficulties with the presentation continue from there. For instance, I’m not sure what <x|i> means if |i> is a two-particle wave-function.

    I’m not even sure what you mean by the expression <f|i>. Is this a scalar product on 4-dimensional space-time or on a 3-dimensional hypersurface? In the first case, I’m not sure if it defines a transition amplitude. In the second case, the expression (and hence the modified 4-density) is not Lorentz-invariant, since initial and final states are prescribed on certain hypersurfaces.

    Maybe it’s just me, as a mathematician, being unfamiliar with the notation or being to picky und unflexible about formalities. But if all of this makes sense, I think your presentation would benefit a lot form being more precise and explicit about these things since they matter in this context.

    On a more conceptual level, I don’t understand what your equations of motions are supposed to describe. As far as I can see, your Lagrangian involves only field degrees of freedom. What you call the particle 4-velocity u is actually a velocity field. In so far as your theory is supposed to be inspired by (or similar to) Bohmian mechanics, you seem to confuse the guiding field with the actual velocity of Bohmian particles and/or the variable in the wave-function with the actual position of Bohmian particles.

    I’m sorry if I criticize your paper out of mere ignorance, but so far, I wasn’t able to understand what you’re doing and I’d really like to, since if your theory achieved what you claim it does, it would be nothing short of brilliant.

    Best, Dustin

    #2817

    Hi Dustin,
    Thanks for your feedback. First, I’m re-reading 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,
    Rod

    #2823
    Avatar
    Aurelien Drezet
    Participant

    Dear Rod, I realized That I didnt make justice to all you comments concerning my questions. Here a small list of points:
    –You wrote :’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.’
    This is an important point it reminds me the double-standard objection of Huw Price. But what is puzzling for me is not that you are using the final psi or the initial psi but that you are using both. I am not convinced that this will not introduce some contradictions if you want to prove the equivalence with the usual Bohm formalism: that is with QM it self. Actually, I have the same problem with the two-time formalism of Aharonov et al..

    — You wrote: ‘So the final boundary condition is fixed in an analogous way to the state preparation of an initial psi. ‘
    This is also mysterious. The final state should equivalent to what is given by orthodox QM and represents therefore the evoving state of Psi in. How can we be sure that we dont introduce too many degrees of freedom in the model? Feynman used a trick like yours in QED but we can show that this is formally equivalent to usual Quantul Field with only one initial boundary condition. The equivalence is not obvious in your model.

    — You wrote: ‘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. ‘
    That’s not obvious. Even if a time symmetric theory is better for explaining Bell results and nonlocality (on this I agree completely) this is not a proof that more usual way will not work. Again, this is the story of Feynman theory. May be the usual wave function contains already retrocausality like in some model by Nikolic (which I dont want to comment here since I do not agree with the details).
    Any way your approach is one of most interesting presented to solve the difficulties of covariance. This is much better that the usual preferred foliation used by Durr et al.

    #2825
    Dustin Lazarovici
    Dustin Lazarovici
    Participant

    Dear Rod,
    thanks for your patience. If you can help be better understand your model or if maybe even I could help you with some of the formalities, I’d be happy to stay in touch even after the workshop.

    I’m also happy to answer questions about the DGZ-paper. It took me a while to fully appreciate the result, but it’s worth it. I think it’s fair to say that it’s accepted by the majority of the Bohmian community as a proof/derivation of Born’s rule, although there is no consensus. But then again, there isn’t even consensus about the foundations of classical statistical mechanics.

    Best, Dustin

    #2826
    Avatar
    Aurelien Drezet
    Participant

    Dear Dustin, The DGZ paper is not the only way to see Born’s rule (Chaos, H theorem etc…) and I think that it is fair to say that the derivation didnt convince every one including myself. It will be too long to discuss that today but the problem is still waiting for a good answer.
    best Aurelien

    PS: the foundation of classical statistical mechanics is much better 🙂

    #2848

    Hi 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, single-particle state.
    3. This new state can be deduced by combining the original two-particle state and the measurement result together in some way.
    4. Even though there is some ambiguity in the time at which the single-particle 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,
    Rod

    #2850

    Hi 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 4-current 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,
    Rod

    #2851
    Avatar
    Aurelien Drezet
    Participant

    Thank you very much rod, I have still problems with the two-time boundary condition but I have to solve that myself
    best wishes
    Aurélien

    #2884
    Dustin Lazarovici
    Dustin Lazarovici
    Participant

    Hi Rod,
    thanks for your answer. I understand that eq. (1) is supposed to be a projection. I’m pretty sure that, if psi(x,x’) is really a multitime wave-function, the expression, as it stands, is not correct. But I’m mostly concerned about the fact that “after an interaction has occured” or “two wave-functions have ceased interacting” is ambiguous in a relativistic setting. That’s why I think it’s important to be precise about these things, to make sure that the relevant expressions are actually well-defined and Lorentz-invariant.

    I don’t want to get caught up in equation (1), though. There are several points in the model where I’m not sure if I just don’t understand the mathematical formulation, or if the notation is actually deceptive in that it sweeps serious problems under the rug.

    Anyway, I’m happy to discuss this in more detail. You can email me at: lazarovici@math.lmu.de . Or, I just get in touch with you.

    Best, Dustin

    #2891

    Hi Aurelien,
    Please feel free to keep in touch after the workshop if you have further thoughts on these matters. My email address is: rod.sutherland@sydney.edu.au.
    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 single-particle case. To establish the consistency of my theory with the QM predictions in the many-particle 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,
    Rod

    #2956

    Hello again Rod,

    I’m sorry I didn’t respond to your reply at the time, but better late than never. First I want to thank you for it, but then I want to clarify what I meant.

    I wanted to find out in what sense you claim that your model explains the phenomena Bell’s theorem identifies as perplexing. In my mind, the first thing a model must do to achieve that is to give formulae which generate the correct probabilities for the outcomes. Standard QM does that, and Bohmian mechanics does that in a different way, provided you assume the “equilibrium” distribution for the initial positions. I mentioned the possibility of other distributions only as a reminder to this – if you choose a “wrong” distribution, you can even get wrong results! You say that your model can also accommodate “wrong” distributions, but if the final boundary conditions are supplied in the “usual” manner, i.e., with the same probabilities, a “wrong” distribution won’t lead to “wrong” results, will it?

    In my work, I provided a retrocausal toy model which gives the “correct” results in a different way, and I wanted to compare this with other publications discussing zig-zag causation, but I couldn’t make a meaningful comparison with your work. I think that’s not a surprise, because as you say your work is an “add-on,” and uses the standard QM formulae to get the outcome probabilities.

    In this sense, I think it’s quite different from Bohmian Mechanics (BM). True, you have the particle path going down the corresponding “finger” in your measurement device, as in BM, but you’ve supplied the corresponding final boundary condition, so the probability for this or that outcome is predetermined, unlike BM. And if I want to compare and ask how the probabilities are determined, I’m back to comparing with standard QM.

    Am I right?

    Thanks, Nathan.

    #2958
    Avatar
    Aurelien Drezet
    Participant

    Dear Nathan, I think that the model of Rod is very similar to the one proposed by Miller with negative probabilities in his Phys lett A 222 31-36 (1996) [butwith the real part in addition to get only real probability and not complex like in Miller’s work]. Of course Miller didn’t use hidden variables but the theory of Rod is really the cournterpart of Miller’s work in the same way as Bohm is the couterpart of usual QM. The main ingredient in the paper of Miller (like in a different paperof 2005 by Aharonov and Gruss) is to define a sum rule for going back to the usual QM probabilities. this is done by introducing a requirement acting like the postulate used by Bohm for justifying born rule. I think that the theory is self consisten if we accept this rule. However this also means that we have a statistical ensemble of systems with different final states.
    with best regards
    Aurélien

    #2959

    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.

    #2966

    Hello Aurelian and Rod,

    Thanks very much for your replies.

    I will need to take a look at the papers of Miller and of Aharonov and Gruss.

    The way I see it, it is completely OK for the Born rule to be stipulated, rather than derived. Newton also stipulated the universal law of gravitation, even though he disliked the idea of action at a distance. Of course, having a derivation from some “basic” physical principles is nice, but it is not strictly required.

    What I was looking for a few years ago, as Ken Wharton also was and still is, is a reformulation of QM in terms of exclusively local beables. We know from Bell’s theorem that such a formulation must be retrocausal. The non-local wavefunction can then be understood as an epistemic tool, and it cannot affect the “paths of the particles” or whatever the ontic variables may be. Needless to say, such a formulation has not been found yet.

    My paper on this was published in 2010, and is available in
    http://scitation.aip.org/content/aapt/journal/ajp/78/10/10.1119/1.3456564
    and in
    http://arxiv.org/abs/0807.2041
    (I thought you would see that I linked to it in my contribution to this conference). I was able to include a simplistic retrocausal toy model in it, designed to be as simplistic as the nonlocal toy model which Bell himself included in his original paper. You will not find a derivation of Born’s rule there, or anything as general as that, but it seems that it is the earliest place in which one can find an explicit retrocausal model which reproduces Bell-type correlations in terms of only local beables.

    In the discussion, I wanted to compare with other retrocausal models, and I cited your model but did not include it in the comparison because it did not provide a different route to obtaining the probabilities of the measurement outcomes. In this sense it is similar not only to Aharonov et al, but also to the Transactional Interpretation, which uses the same math, but nevertheless introduces the novel concept of retrocausality.

    Thanks again, Nathan.

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