On Sun, Jan 5, 2025 at 6:36 PM Bruce Kellett <[email protected]> wrote:
> On Mon, Jan 6, 2025 at 10:21 AM Jesse Mazer <[email protected]> wrote: > >> On Sun, Jan 5, 2025 at 5:35 PM Bruce Kellett <[email protected]> >> wrote: >> >>> On Mon, Jan 6, 2025 at 9:14 AM Jesse Mazer <[email protected]> wrote: >>> >>>> On Sun, Jan 5, 2025 at 12:44 AM Bruce Kellett <[email protected]> >>>> wrote: >>>> >>>>> On Sun, Jan 5, 2025 at 7:46 AM John Clark <[email protected]> >>>>> wrote: >>>>> >>>>>> >>>>>> *About a month ago Sean Carroll uploaded a very good video explaining >>>>>> the Many Worlds theory, but it's over an hour long so I know there's >>>>>> about >>>>>> as much chance of a dilettante such as yourself of actually watching it >>>>>> is >>>>>> there is of you reading a post of mine if it's longer than about 100 >>>>>> words. >>>>>> * >>>>>> >>>>>> *The Many Worlds of Quantum Mechanics | Dr. Sean Carroll >>>>>> <https://www.youtube.com/watch?v=FTmxIUz21bo&t=8s> * >>>>>> >>>>> >>>>> I watched this video, but it is not as comprehensive as Carroll's book >>>>> "Something Deeply Hidden". >>>>> >>>>> However, something came up in the question period that might warrant a >>>>> comment. Talking about the Born rule, Carroll justifies it by saying that >>>>> if you measure the spin of 1000 unpolarized particles, you get 2^1000 >>>>> different UP-DOWN sequences. However, the vast majority of these sequences >>>>> will show proportions of UP vs DOWN close to the Born rule prediction of >>>>> 50/50. In the limit, if such a limit makes sense, the proportion of >>>>> sequences that show marked deviations from the Born Rule proportions will >>>>> form a set of measure zero, and can be ignored. >>>>> >>>>> That is just the law of large numbers at work, and is all very well if >>>>> the amplitudes are such that the Born probabilities are equal to 0.5. But >>>>> it is easy to rotate your S-G magnets so that the Born probabilities are >>>>> quite different, say, 0.9-Up to 0.1-DOWN. Now take 1000 trials again. >>>>> According to Everett, you necessarily get the same 2^1000 sequences of >>>>> UP-DOWN that you had before. The law of large numbers will then tell you >>>>> that the majority of these will have approximately a 50/50 UP/DOWN split, >>>>> which is grossly in violation of the Born rule result of a 90/10 split. In >>>>> other words, MWI. or Everettian QM. has a problem reproducing the Born >>>>> rule. It works in the simple case of equal probabilities, but fails >>>>> miserably once one departs substantially from equal probabilities. >>>>> >>>>> Bruce >>>>> >>>> >>>> David Z Albert mentions that if you define a measurement operator that >>>> just tells you the *fraction* of spin-up vs. spin-down in a large sequence >>>> of identical measurements, then even without any collapse assumption, in >>>> the limit as # measurements goes to infinity the wavefunction will approach >>>> an eigenstate of this operator that matches the probability that would be >>>> predicted by the Born rule. See his comments on p. 238 of The Cosmos of >>>> Science at >>>> https://books.google.com/books?id=_HgF3wfADJIC&lpg=PP1&pg=PA238#v=onepage&q&f=false >>>> >>>> "Then, even though there will actually be no matter of fact about what >>>> h takes the outcomes of any of those measurements to be, nonetheless, as >>>> the number of those measurements which have already been carried out goes >>>> to infinity, the state of the world will approach (not as a merely >>>> probabilistic limit, but as a well-defined mathematical >>>> epsilon-and-delta-type limit) a state in which the reports of h about the >>>> statistical frequency of any particular outcome of those measurements will >>>> be perfectly definite, and also perfectly in accord with the standard >>>> quantum mechanical predictions about what the frequency out to be." >>>> >>> >>> But then Albert goes on to say that there are all sorts of reasons why >>> this simple theory cannot be the answer to the origin of the Born rule. I >>> have pointed out one of the most cogent of these. If you perform similar >>> measurements on N identically prepared systems (say z-spin measurements on >>> systems prepared in an x-spin-left state), then according to Everett, you >>> get all 2^N possible sequences of UP/DOWN spins. This exhausts the >>> possibilities for the outcome of N trials, and, significantly, you must get >>> exactly the same 2^N sequences whatever the amplitudes of the initial >>> superposition might be. So you get these 2^N sequences if the amplitudes >>> are equal, and also if the amplitudes are in the ratio 0.9/0.1. This >>> behaviour is incompatible with the Born rule, and hence with ordinary >>> quantum mechanics. >>> >> >> You do get all these sequences but this tells us nothing about what their >> relative probabilities/frequencies are. I assume as an extension of his >> analysis, if we did repeated experiments where on each trial we performed >> exactly N measurements and this was repeated over many trials (approaching >> infinity), then you could define a measurement operator that would tell you >> the fraction with any specific N-sequence (for example, for N=3 there would >> be an operator giving the fraction of trials with result 000, likewise >> other operators for 001 and 010 and 011 and 100 and 101 and 110 and 111). >> If you had a setup where the relative probability of these sequences was >> not uniform according to the Born rule, then if the number of trials with >> that setup goes to infinity, it will presumably likewise be true that the >> state approaches the eigenstate of this operator with the frequency >> predicted by the Born rule, without ever actually invoking the Born rule. >> >> Albert would presumably say that this still doesn't resolve the >> measurement problem because it doesn't give an outcome on any particular >> trial, only a sort of aggregate over many trials, but this is different >> from the criticism you are making. Even if we do use the Born rule in the >> above scenario, it's still true that each of the specific outcomes that are >> possible for a given trial with N measurements (eg the outcomes 000, 001, >> 010, 011, 100, 101, 110, and 111) will occur in the long term, but that >> doesn't mean they are equiprobable. >> > > The trouble that I have pointed to is that if every possible outcome > occurs for each measurement, then the sequences are all present whatever > the amplitudes in the wavefunction. so the sequences > 000...,001...,010...,100...,011...,... etc are all equiprobable, whatever > the wave function. > How do you get from "all present" to "equiprobable"? If you flip a pair of coins enough times you will surely get all the sequences HH, HT, TH and TT, but you could easily be using weighted coins that don't have 50/50 chances of landing heads and tails, in which case those 4 sequences won't occur equally often. > Thus, the operator that Albert talks about does not give the relative > probabilities for each sequence. > But he's talking very generally about any sort of operator that gives relative frequencies of different results of quantum experiments, so do you disagree that it's plausible this could be generalized to an operator that gives frequencies of different multi-measurement sequences in the way I described? Making N measurements in a row can itself be considered a type of repeatable experiment that has 2^N possible outcomes each time you perform it. And each of those 2^N outcomes would be assigned some probability by the Born rule, so you should be able to design an operator that gives you the frequency of any one of those 2^N outcomes, which may not be equiprobable depending on the experimental setup. Albert's statement about the wavefunction converging to the correct eigenfunction of such an operator wasn't limited to cases where all outcomes are equiprobable, I can't say for sure but I'd expect his statement would cover this sort of case as well. > Actually, Albert is talking about the case where there is only one outcome > for each measurement. > He doesn't specify that each "outcome" has to be a single spin measurement though. There's also another presentation of the same idea (apparently called 'Mittelstaedt’s theorem') starting on p. 13 at https://www.academia.edu/6975159/Quantum_dispositions_and_the_notion_of_measurement and it seems to be stated in a very general way, talking about an operator that gives "the relative frequency of the outcome a_k in a given sequence of N outcomes" without placing any conditions on an "outcome" only involving a single particle measurement. Jesse -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/CAPCWU3KA_eot8RQ5%2BkhtXZz_%2BZL3gYy1ygvFm9So3k%2BjQK9MVw%40mail.gmail.com.
