Sunday, June 25, 2017

Guest Post defending MWI


As promised, here is a guest post from Per Arve. I am not interjecting my opinion in the main text but I will ask questions in the comments section.

Due to the popularity of this post I am delaying the next post for a week.

The reason to abandon the orthodox interpretation of quantum mechanics is its incompleteness. Bohr and Heisenberg refused the possibility to describe the measurement process as a physical process. This is encoded in Bohr's claim that the quantum world cannot be understood. Such an attitude served to avoid endless discussions about the weirdness of quantum mechanics and divert attention to the description of microscopic physics with quantum mechanics. Well done! A limited theory is better than no theory.

But, we should always try to find theories that in a unified way describes the larger set of processes. The work by Everett and the later development of decoherence theory by Zeh, Zurek and others have given us elements to describe also the measurement process as a quantum mechanical process. Their analysis of the measurement process implies that the unitary quantum evolution leads to the emergence of separate new "worlds". The appearance of separate "worlds" can only be avoided if there is some mechanism that breaks unitarity.

The most well-known problem of Everett's interpretation is that of the derivation of the Born rule. I describe the solution of that problem here. (You can also check my article on the arxiv [1603.01625] Postulates for and measurements in Everett's quantum mechanics)

The main point is to prove that physicists experience the Born rule. That is by taking an outside view of the parallel worlds created in a measurement situation. The question, what probability is from the perspective of an observer inside a particular branch, is more a matter of philosophy than of science.

The natural way to find out where something is located is to test with some force and find out where we find resistance. The force should not be so strong that it modifies the system we want to probe. This corresponds to the first order perturbation of the energy due to the external potential U(x),

\(\Delta E =\int d^3 x {|\psi (x)|}^2 U(x)\)  (1)

This shows that \({|\psi(x)|}^2\) gives where the system is located. (Here, spin and similar indexes are omitted.)

The argumentation for the Born rule relies on that one may ignore the presence of the system in regions, where integrated value of the wave function absolute square is very small.

In order to have a well defined starting point I have formulated two postulates for Everett's quantum mechanics.

EQM1 The state is a complex function of positions and a discrete index j for spin etc,

\(\Psi = \psi_j (t, x_1, x_2, ...) \)  (2)

Its basic interpretation is given by that the density 

\(\rho_j (t, x_1, x_2,...) = {|\psi_j (t, x_1, x_2, ...)|}^2 \)  (3)

answers where the system is in position, spin, etc.

It is absolute square integrable normalized to one 

\( \int \int···dx_1dx_2 ··· \sum_j {|\psi_j (t, x_1, x_2, ...)|}^2 = 1\)  (4)

This requirement signifies that the system has to be somewhere, not everywhere. If the value of the integral is zero, the system doesn’t exist anywhere.

EQM2 There is a unitary time development of the state, e.g.,

\(i \partial_t \Psi = H\Psi \),

where H is the hermitian Hamiltonian. The term unitary signifies that the value of the left hand side in (4) is constant for any state (2).

Consider the typical measurement where something happens in a reaction and what comes out is collected in an array of detectors, for instance the Stern-Gerlach experiment. Each detector will catch particles that have a certain value of the quantity B we want measure.

Write the state that enter the array of detectors as sum of components that enter the individual detectors, \(|\psi \rangle = \sum c_b |b\rangle\), where b is one of the possible values of B. When that state has entered the detectors we can ask, where is it? The answer is that it is distributed over the individual detectors. The distribution is 

\(\rho_b = {|c_b|}^2 \)  (5)

This derived by integrate the density (3) over the detector using that the states \(|b\rangle\) have support only inside its own detector. 

The interaction between \(|\psi \rangle\) and the detector array will cause decoherence. The total system of detector array and \(|\psi \rangle\) splits into separate "worlds" such that the different values b of the quantity B will belong to separate "worlds".

After repeating the measurement N times, the distribution that answer how many times have the value \(b=u\) been measured is

\(\rho(m:N | u)= b(N,m) {(\rho_u)}^m{(\rho_{¬u})}^{N−m} \)  (6)

where \(b(N,m)\) is the binomial coefficient \(N\) over \(m\) and \(\rho_{¬u}\) is the sum over all \(Ļ_b\) except \(b=u\).

The relative frequency \(z=m/N\) is then given by

\(\rho(z|u) \approx \sqrt{(N/(2\pi \rho_u \rho_{¬u}))} exp( −N{(z−\rho_u)}^2/(2\rho_u \rho_{¬u}) ) \)  (7)

This approaches a Dirac delta \(\delta(z − \rho_u)\). If the tails of (7) with low integrated value are ignored, we are left with a distribution with \(z \approx u\). This shows that the observer experiences a relative frequency close to the Born value. Reasonably, the observer will therefore believe in the Born rule.

The palpability of the densities (6) and (7) may be seen by replacing the detectors by a mechanism that captures and holds the system at the different locations. Then, we can measure to what extent the system is at the different locations (4) using an external perturbation (1). In principle, also the distribution from N measurements is directly measurable if we consider N parallel experiments. The relative frequency distribution (7) is then also in principle a directly measurable quantity.

A physicist that believes in the Born rule will use that for statistical inference in quantum experiments. According to the analysis above, it will work just as well as we expect it to do using the Born rule in a single world theory.

A physicist who believes in a single world will view the Born rule as a law about probabilities. A many-worlder may view it as a rule that can be used for inference about quantum states as if the Born rule is about probabilities.

With my postulates, Everett's quantum mechanics describe the world as we see it. That is what should be discussed. Not whether it pleases anybody or not.

If the reader is interested what to do in a quantum russian roulette situation, I have not much to offer. How to decide your future seems to be a philosophical and psychological question. As a physicist, I don't feel obliged to help you with that.

Per Arve, Stockholm June 24, 2017

34 comments:

  1. Per, let me break the ice on the comments.

    Imagine you have prepare the spin of many particles say up on the z direction. The you have 2 SG devices one measuring the spin on x axis, and one measuring the spin on an axis very close to z (say within a few degrees).

    Say you perform 5 and 5 measurements for the two devices (to keep the math simple). What do you get? Can you have a side by side comparison between the two cases?

    Thanks,
    Florin

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    Replies
    1. I think you can get the answer from my guest blog contribution here. In the case you test x-direction, the universe will have worlds for any possible outcome. Equation 6 gives where the universe is with respect to how many times spin in the positive x-direction was found as the following table shows
      m šœŒ
      0 0.03125
      1 0.15625
      2 0.3125
      3 0.3125
      4 0.15625
      5 0.03125

      In the case of measuring in an angle close to z-axis, say 5°.
      m šœŒ
      0 0.172977594
      1 0.363574479
      2 0.30567289
      3 0.128496251
      4 0.027008098
      5 0.002270688

      I know that some believe that only one thing happen, though that breaks unitary evolution, which have no reason to believe happens. Such people take the rho-numbers as being probabilities, but that is up them to defend. :)

      Per

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    2. Per, I don't understand where you get the real numbers like: 0.363574479

      The experiment produces only up or down outcomes. I expect to see something like:

      1: up
      2: up
      ...
      5: down

      1: down
      2: up
      ...
      5: down

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    3. Florin, m is the number of times up is measured. Equation 6 shows how the "universe" is distributed with respect to the variable m.

      Delete
    4. Per, my question was to work out what happens when I perform the measurement 5 times. Let's make things even simpler: I perform the measurement only 3 times. I expect something like this:

      Experiment 1: run 1-up run 2-up, run 3-down
      Experiment 2: run 1-down, run 2-down, run 3-up

      Then I want to see how you apply your equations comparing the two experimental setups. I do have a point on why I am asking this.

      Delete
    5. Florin, I have answered you and I have clarified my answer! You cannot dictate how the answer should be. Perhaps you should clarify what your intended follow up question is, so that I can help you ask the question or answer you why your intended question is not appropriate in the many-world theory.

      You write a sequence of measurement results. Yes there is a branch for that sequence. Is your question, how can the observer infer the absolute values of amplitudes?

      Per

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    6. Per, no, you did not answer my question - so far you presented only handwavings. Peer review implies you have to convince your critics and address their questions to their satisfaction - the burden of proof is on you.

      I think your derivation of Born rule is circular. Prove me wrong by working out step by step through your equations. So here is my question again: start by imagining the experimental outcomes of the two experiments for 3 runs of the experiment. Then I'll ask question from there based on the equations from your guest post. I will not ask anything you did not claim so far; I will be strictly following your text above.

      Delete
    7. Does my answer to Andrei also answer your question?

      Delete
    8. Florin

      There is also burdon on the critics to point specific points that are objectionable. If you are not specific in your criticism, I will have to redo the full argumentation which will then be a repetition of my article. Not knowing what is unclear in my previous presentations, I have no idea where to improve.

      Regards
      Per

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    9. Just realized that the šœŒ numbers for 5° I gave in my first reply corresponds to measuring spin down.

      Delete
    10. Per, you cannot have it both ways: on one hand complaining (like you did at the conference) that the critics dismiss MWI, and on the other not answering questions. The equations you wrote (5), (6), and (7) contain a lot of information and I want to unpack them in a simple case. The hardest questions are always the simplest ones. Let's unpack them in the case an observer makes 3 measurements in the two different scenarios mentioned above and see how in the large measurement limit they will converge to different limits.

      Delete
    11. The unitary time development given by the quantum equations all alternatives have to remain after the measurement has finished. According to postulate EQM1, where the system is during the measurement is given by the (density) distribution (3). In a SG experiment the density of the particle is divided into to two separate parts located at the spin up and spin down detectors, respectively. Where the particle is with respect to the spin value is given by the two values (5) that is given by integrating the density either over the spin up or the spin down detectors.

      Case A, measurement in x-direction will give Ļ↑ = 0.5; Ļ↓= 0.5.
      Case B, measurement in 5° direction Ļ↑ = (cos 2.5°)^2 = 0.998097349; Ļ↓= (sin 2.5°)^2 = 0.001902651

      Three measurements
      Case A: Measured spin
      up up up Ļ↑↑↑= Ļ↑Ļ↑Ļ↑ = 0.5^3
      up up down Ļ↑↑↓ = Ļ↑Ļ↑Ļ↓ = 0.5^3
      up down up Ļ↑↓↑= Ļ↑Ļ↑ Ļ↓= 0.5^3
      up down down Ļ↑↓↓ = Ļ↑Ļ↓Ļ↓ = 0.5^3
      down up up Ļ↓↑↑= Ļ↓Ļ↑Ļ↑ = 0.5^3
      down up down Ļ↓↑↓ = Ļ↓Ļ↑Ļ↓ = 0.5^3
      down down up Ļ↓↓↑= Ļ↓Ļ↑ Ļ↓= 0.5^3
      down down down Ļ↓↓↓ = Ļ↓ Ļ↓Ļ↓ = 0.5^3


      Case B
      up up up Ļ↑↑↑= Ļ↑Ļ↑Ļ↑ = 0.998097349^3
      up up down Ļ↑↑↓ = Ļ↑Ļ↑Ļ↓ = 0.998097349^2*0.001902651
      up down up Ļ↑↓↑= Ļ↑Ļ↑ Ļ↓= 0.998097349^2*0.001902651
      up down down Ļ↑↓↓ = Ļ↑Ļ↓Ļ↓ = 0.998097349*0.001902651^2
      down up up Ļ↓↑↑= Ļ↓Ļ↑Ļ↑ = 0.998097349^2*0.001902651
      down up down Ļ↓↑↓ = Ļ↓Ļ↑Ļ↓ = 0.998097349*0.001902651^2
      down down up Ļ↓↓↑= Ļ↓Ļ↑ Ļ↓= 0.998097349*0.001902651^2
      down down down Ļ↓↓↓ = Ļ↓ Ļ↓Ļ↓ = 0.0019026513^3

      The total density for measure m up (6) you get by summing over all cases with m up
      Case B
      m Ļ
      3 0.5^3 = 0.125
      2 3* 0.5^3 = 0.375
      1 3* 0.5^3 = 0.375
      0 0.5^3 = 0.125

      Case B
      m Ļ
      3 0.998097349^3 = 0.9943029
      2 3* 0.998097349^2*0.001902651 = 0.005686253
      1 3* 0.998097349*0.001902651^2 = 1.08396E-05
      0 0.001902651^3 = 6.88775E-09

      Equation (7) applies only for (very) large N. But, divide the absolute frequency m with the total number of measurements N = 3.

      How equation (6) leads to equation (7) is a math exercise that is well explained in statistics books in there discussion of the Binomial distribution and its relation to the standard distribution. It is slightly too involved to reproduce here.

      At Hammerfest in north of Norway,
      Per

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    12. Thank you Per, but why rho for case B is different than rho for case A? It is here where you inject Born's rule and have circularity. Empirically one has only the following cases:

      uuu
      uud
      udu
      duu
      ddu
      dud
      udd
      ddd

      and those are the experimental outcomes which are the same in case A and case B. Rho does not follow from the experimental outcomes.

      This is not the case in the frequentist approach to probability: the probability is a limit case of a long series of repeated experiments, and it arises out of experimental outcomes only.

      Delete
    13. In my reply June 27, the table that is given for a measurement angle of 5° is actually for 20°. The error was caused by me multiplying the angle with two when I should have divided with two.

      Delete
    14. Florin,

      I don't inject the Born rule. I only use my postulates. I have a quantum description of the measurement situation. The postulates tell you to use rho to evaluate where the system total system is with respect to different variables. Where the observer is with respect to the measurement result is what I have given my tables. I simply use that the absolute values of the amplitudes of the eigenstates states to the operator being measured are cos(angle/2) and sin(angle/2), angle = 0 or 5°. Ļ↑ or Ļ↓ are the square of the amplitudes.

      If you argue that there is a circularity, you have to either argue that there is a flaw in the mathematical calculation, or that the Born rule is already present in the postulates. The latter cannot be true, because probabilities are never appearing in the theory, only the perception of probability. There is no collapse of the wave function, only a perception of collapse.

      It seems me that, you are not addressing my theory, but like Andrei you are addressing some general many-worlds theory. That is not a possible endeavor. Most of the proposals of many-worlds theories are ill-defined, e.g. Everett's and DeWitt's theories.

      Per

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  2. Hi Per,
    Apart from derivation of Born rule, the splitting of the worlds seems to be very arbitrary, generating infinite number of branches if you take it seriously. Suppose you ask your student to do quantum experiment tomorrow morning. If he gets up early, goes to the lab and does the experiment the world including him and his multiple copies, splits. If he gets up late and does not do the experiment, his world does not split. Do you want to give that much power to people and especially to graduate students?!! If the splitting is already made in heavens before, it is a metaphysics worse than any religion. If the splitting takes place in observer’s mind it may be slightly more palatable. But even then, the actual meaning may not be clear until we understand consciousness.
    Anyway, cosmological many worlds may be more acceptable than quantum many worlds. In the former case at least it is an act of God rather than people on a measly little planet in the whole big universe! So it seems MWI does not explain anything and increases unnecessary metaphysical baggage!
    kashyap

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    Replies
    1. Kashyap,
      Have you actually read my contribution or are you giving me your preconception about MWI?

      I wrote "The appearance of separate 'worlds' can only be avoided if there is some mechanism that breaks unitarity."
      Physicist very much trust that the rule of unitarity applies when they investigate any physical system.

      The splitting into different worlds is by no means arbitrary. We have found that it is a result from the equations of quantum physics applied to realistic measurement situations. It is not at all about philosophical reasoning. It is all about accepting our logical/mathematical analysis of the equations, in particular the process of decoherence. Disbelief in many-worlds ought to be accompanied with some argument about what is wrong in our analysis.

      Your statement that an infinite number of branches are created is probably not correct. At least not in a finite universe in a finite time.

      Per

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    2. I should clarify that in Everett's quantum mechanics the measurement process is described within the same theory as we in all interpretation describe the process between preparation and measurement. The achievements are a unification and added consistency. The latter part refers to that in Everett's quantum mechanics there is a unification between how the experimenter says his detectors actually work and how the interpretation describes the measurement process.

      Per

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  3. Dear Per,

    Please explain me where I get wrong with my understanding of MWI.

    Assume we have a collection of electrons in a state that gives you 1/3 probability to get spin +1/2 and 2/3 probability to get spin -1/2 along X axis.

    First electron is measured. You get two branches, one corresponding to +1/2 and another one to -1/2.

    The second electron is measured. The observer in first branch splits again in two (one for +1/2 and one for -1/2) and the same happens for the observer in the second branch. As the experiment continues the number of branches with +1/2 seems to remain equal to the -1/2 ones. I just don't see where the 1/3 and 2/3 probabilities could appear from.

    Thanks,

    Andrei

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    Replies
    1. Andrei

      I have seen no compelling arguments for branch counting. If the magnitudes of the amplitudes are not the same, then you have no symmetry argument. Thus no reason to think that probabilities should be the same. Anyhow, branch counting is an impossible idea as branch number is not a well defined concept. If we look closer, branch is only an approximate concept. What we have is an extremely small possibility for interference between terms corresponding to the different measurement values.

      Per

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    2. Andrei

      Beware that the classic single world probability concept is not fully applicable.
      What I prove is that after many repeated measurements the typical "observer experiences a relative frequency close to the Born value. Reasonably, the observer will therefore believe in the Born rule."

      This implies that it is appropriate to use ordinary statistical methods to deduce the magnitude of amplitudes from experimental frequencies.

      Note that "typicality" is also assumed in any classical statistical analysis.

      If we are to fruitfully discuss Everett's quantum mechanics, you have to be able to distance yourself from views about quantum physics that you might have acquired assuming a single world interpretation.

      I feel only obliged to explain how my analysis explains the empirical facts that we have, e.g. that we can use the Born rule to deduce the magnitude of amplitudes.

      Delete
  4. Dear future Commenter,

    I hope you have read my post and that your comment is about my starting point or my reasoning. If both are correct then the conclusions ought to be correct. I look forward to discussions and questions about those two.

    Per

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  5. Per,

    Can you give me a detailed explanation of how a typical observer performing spin measurements according my example above is going to experience the 1/3 probability for +1/2 result and 2/3 probability for the -1/2 result? I know that the amplitudes are not the same but I fail to understand how is this "translated" to observer's experience in your interpretation.

    Thanks,

    Andrei

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    Replies
    1. Dear Andrei,

      I don't want to repeat the whole blog (which is already condensed version of my article). Please tell me if there is any particular statement that you don't agree with or understand, starting with the postulates.

      However, I note that didn't use the word 'typical' in your recent reply. It is only in a typical branch the observer will find the Born rule. Typical corresponds to an observer that is located close to the peak of the relative frequency distribution, which for large number of repeated measurements N is given by (7).

      I insist that any reader should mathematically analyze the expressions I have given, including inserting specific numbers, if that is of interest.

      Per

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  6. I realize that the questions that concern measuring the spin of a particle has not been addressed well in my blog. My article has more detail which can be helpful in this case. I address particles that not only have a spin state but also a (configuration) space dependent wave function. When you measure spin, you actually measure position. Thus, where the observer is in measured relative frequency, can be rephrased into where the particles are/were in configuration space. In this way, everything is referred back to EQM1.

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  7. Per,

    At page 5 in your article you write:

    "Looking at the many “worlds” from the outside the question: What reading did the observer get? is equivalent to What is the distribution of observer readings?

    The answer is given by the distribution
    Ļb (17)."

    I do not understand why it has to be so.

    Going back to my example, an observer performing a spin measurement on a system in a superposition as described above will find itself either in a world corresponding to +1/2 result or in one corresponding to -1/2 result. Repeating the experiment would lead again to the same result. I just don't understand why repeating the experiment would produce the required distribution of observer readings (1/3 for +1/2 and 2/3 for -1/2) instead of 1/2 for both results. The number of eigenstates is 2 and each experiment would produce 2 worlds, regardless of density, right? For example, for 2 experiments you get 4 worlds:

    1. The observer gets +1/2 in the first run and +1/2 in the second one.

    2. The observer gets +1/2 in the first run and -1/2 in the second one.

    3. The observer gets -1/2 in the first run and +1/2 in the second one.

    4. The observer gets -1/2 in the first run and -1/2 in the second one.

    As you continue increasing the number of experiments you get closer to 1/2 probability. Where am I wrong?

    Andrei

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    Replies
    1. Andrei,

      I am glad to try to answer your question about the distribution of observer readings.

      After the observer has registered the measurement result, the total state is represented by the final state of expression (19). What value the observer has measured, is a variant of the question, where is the system in EQM1. The argument in the article corresponds to that your integrate the total density over all other degrees of freedom, but the observer state. You are then left with the density \rho_b signifying where the observer is in the variable 'what value has been observed'.

      Perhaps it is more convincing to use that the density of the measured system with respect which detector it has landed in \rho_b doesn't change during the interaction with the detector or the observation. The density of which detector reacted or the density for the which value the observer noted must be the same as the one for which detector the system landed in.

      Your example

      The perspective is unclear. If you want to consider the whole universe with all its worlds, the notion of probability doesn't apply. You have a state from which you can calculate various quantities, notably densities that answer where the total system is with respect to some interesting variable as relative frequency.

      After many measurement, the distribution of where the system is in relative frequency is peaked at 1/3. The way that comes about is analogous to how you find that the probability of the values of the relative frequency is peaked at 1/3.

      The densities of the different worlds that you listed:
      1. 1/3*1/3
      2. 1/3*2/3
      3. 2/3*1/3
      4. 2/3*2/3

      The distribution of the universe with respect to the variable 'the relative frequency z of the value +1/2' is
      z \rho_z

      0 4/9
      0.5 4/9
      1 1/9

      When I get questions about my theory, it is possible for me to write an answer that is not tens of pages long. Thank you for willingness to engage. Hope to learn your further thoughts and questions about this.

      Per

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    2. Andrei

      What I have shown in my article is that physicists believe in the Born rule. It works as way to experimentally determine the magnitudes of amplitudes. In such a case, you look at past events in your own branch.

      The concept of probability seems me only to apply with a single branch. Classical probability theory is not concerned with events causing branching and is as applicable there as it has always been.

      Per

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    3. Per,

      I still do not understand how different densities in different worlds are reflected in observer's measurements.

      You seem to agree with how I said the worlds split (4 worlds for 2 measurements). If you continue the experiment, say 1000 times most of the worlds will have observer witnessing a similar number of +1/2 and -1/2 results. In my naive view this would be the "typical observer", the observer that finds itself in most worlds. True, the worlds with around 2/3 results of -1/2 will have a higher density, but why should I consider them containg the typical observer?

      Andrei

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  8. Andrei,

    The short answer: The postulates EQM1 and 2.

    The answer with explanation:
    As with most physical theories, we have a (set of) equation(s) and we have a a physical interpretation (prescription) of what observable phenomena the symbols corresponds to.

    A theory is then judged useful ("correct") if this leads to agreement with observation.

    Quantum theory, if applied to everything, has to have an interpretation that relate the theory to observable macroscopic objects in a similar way to classical mechanics, which relates to the position of objects and properties derived from that like velocity and acceleration.

    EQM1 relates the quantum state to the position of objects. For macroscopic objects with relatively narrow position distribution we get a description close to the classical theory. We get a description of the position of detectors and other objects as well as a description of microscopic systems.

    The postulates EQM1 and EQM2 are the only elements we shall use to describe the "universe", e.g. the situation of the observer. Every objects position with respect some variabel is given by the density that can be derived from the density (3).

    Do you argue that the derivation of the observer position in measured relative frequency lacks logic. If so could you help me with a more precise analysis that shows how it fails.

    For example, your 'naiv view' based on naiv value branch counting need to be derived from my postulates, if that is to be the correct procedure within the theory that I have defined.

    May be you find branch counting more natural, but that can only serve as an inspiration for your attempts to find flaws in my derivation and cannot be an argument by itself. Perhaps you like to consider another set of postulates that would lead to your branch counting result, but that theory would disagree with observations.

    If there are no logical flaws, you should accept that this theory describes the world as we observe it. If there are flaws, I will try to improve my argumentation or concede that many-worlds theories fail.

    From Hammerfest in the north of Norway,
    Per

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    1. Dear Per,
      What are your thoughts on the Mangled Worlds derivation of the born rule by hanson? Do you think it uses any circular arguments?
      Nick

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    2. Dear Nick,

      I had another look at the Mangled Worlds idea. I don't think it is circular, but I think it is flawed for several reasons.

      Firstly, he has not defined the theory he is working with because he has not attached any meaning to the quantum state.

      Secondly, he gives no rational for using world counting to get the Born rule.

      Thirdly, the logics of world mangling is sloppy. He is using unproven conjectures. Conclusions he tries to draw from some of the math is disputable.

      Per

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