# Multiple measurements and the any worlds interpretation

My question has some similarities to but also differs significantly from this question which was described in many of its answers as not being a quantum mechanical measurement and was I think, answered here. My question is about quantum mechanical measurements.

If I am not mistaken, according to the many-worlds interpretation of quantum mechanics if a parameter of a system can have values v1, v2, v3, ..., vn with probabilities p1, p2, p3, ..., pn, we detect one of these values when we measure that parameter and there are other universes in which the value detected will be one of the other values.

If we set up a large number, say m, of such identical systems and measure the parameter in each of them then we would expect to detect the value v1 in m×p1 number of systems, v2 in m×p2 number of systems, ..., vn in m×pn number of systems. At the measurement of each system there will be a total of n universes where the value measured will be one of v1, v2, v3, ..., vn. In measuring all the m systems, there will be nm universes involved and in n of these universes the parameter will have the same value in all the measurements - in one of these n universes all measurements will detect the value v1, in another v2, in yet another v3 and so on. In these n universes the expected distribution of measured values, m×p1, m×p2, m×p3, ..., m×pn will be violated. In fact, there will be more universes where the distributions of measured values deviate from m×p1, m×p2, m×p3, ..., m×pn than universes where they will be according to that distribution.

However, as far as I know, scientists have not observed any such drastic deviation in any such measurements so far. Is there any other explanation besides that we just happen to be in one of the universes where all probability distributions of values hold up in all measurements all the time?

• No, the vast, vast majority of 'universes' will have observed distributions that closely match the theoretical ones. This is a result from statistics called the Law of Large Numbers, that has nothing to do with quantum mechanics. – knzhou Apr 9 '16 at 21:32
• There is only one universe and MWI is simply nonsense. – CuriousOne Apr 10 '16 at 0:59
• Your $n^m$ universes have different amplitudes, which affects your chance of being in that universe. Your chance to "land" in a universe is the same thing as observing the measurement results that occur in that universe. – adipy Apr 10 '16 at 4:38
• @knzhou, The Law of Large numbers applies for measurements in a single universe, not when all values of the parameter are measured but each in a separate universe as the MWI states and we are considering all the universes. – Fi-6 Apr 12 '16 at 3:36
• @adipy, what exactly do you mean by the amplitude of a universe? – Fi-6 Apr 12 '16 at 3:37

## 1 Answer

You have confused counting worlds with computing probabilities. They are different things.

If you measured $m$ systems identically prepared to give one of $n$ result $v_1,$ ...$v_n$ with respective frequencies $p_1,$ ... $p_n$, then there are $n^m$ aggregate outcomes.

But the MWI doesn't predict different probabilities than any other interpretation. Both many worlds and Copenhagen predict any one of the $n^m$ aggregate possibilities appears with a frequency of $p_1^{k_1}p_2^{k_2}...p_n^{k_n}$ when the $i$th value occurs $k_i$ times (and by frequency the two interpretations mean the ... exact ... same ... thing, and this frequency applies to the frequency of a particular aggregate outcome).

Sure you can talk about how many worlds have those aggregate outcomes. And there are least $\frac{m!}{k_1!k_2!...k_n!}p_1^{k_1}p_2^{k_2}...p_n^{k_n}$ such worlds. But a computation of how many worlds a person feels like grouping together is not a scientific prediction of a frequency. No one ever claimed it was.

Copenhagen predicts all $n^m$ aggregate outcomes can happen even ones where the observed $k_i$ for an aggregate outcome is very different than $mp_i$. Many worlds predicts all $n^m$ do happen, even ones where the observed $k_i$ for an aggregate is very different than $mp_i$. And they predict the same frequencies.

And both of them predict the frequency of a particular aggregate outcome, one of the $n^m$. Not one of the $\left(\begin{matrix}m+n-1\\m\end{matrix}\right)\neq n^m$ many groupings. And no one ever claimed that $mp_i$ is predicted (by any interpretation) to be close the frequency of one of those aggregate outcomes. It's ranges of $k_i$ near to $mp_i$ where for each value of $k_1,$ ... $k_n$ you group together $\frac{m!}{k_1!k_2!...k_n!}$ many aggregate outcomes (outcomes for Copenhagen or for many worlds) it is only the frequency of those groupings of outcomes that gets anywhere near $mp_i$.

Copenhagen predicted a single one of the $n^m$ aggregate outcomes, and many worlds predicts each of the $n^m$ aggregate outcomes. And each of the $n^m$ aggregate outcomes is predicted by both interpretations to occur with a particular frequency (as measured by repeated trials over time). Which is a super tiny frequency (as you'd expect when they are $n^m$ outcomes).

Copenhagen and many worlds predict the same frequencies. They both do it by talking about each of the $n^m$ outcomes and their frequencies. And both of them only compare to $mp_i$ when you group collections of the $n^m$ outcomes into $\left(\begin{matrix}m+n-1\\m\end{matrix}\right)\neq n^m$ many groups, each grouping having some values of $k_1,$ ... $k_n$ such that all the outcomes in that grouping are predicted to be equally likely and we know there are $\frac{m!}{k_1!k_2!...k_n!}$ many outcomes making up that group.

Let's be really really really clear. Copenhagen and many worlds both predict $n^m$ distinct outcomes. Both of them predict the same frequencies of getting each of those $n^m$ outcomes. None of those frequencies are directly connected to $mp_i$ which instead of only compared to collections of groupings of some of the $n^m$ predicted outcomes. The grouping have the same $k$'s and the collections of groupings have collections of $k$'s that are similar to each other.

Both interpretations do this. They both do it using the same math. The difference is that Copenhagen takes a solipsistic position about one world thinking it's special just because it can't detect the other worlds. But they predicted the exact same things.

And neither of them predicts seeing $k_i=mp_i$ that's an mathematical value of a mathematical average of doing many trials of $m$ on each trial. And since Copenhagen and many worlds predict the same probabilities for each of the $n^m$ outcomes they don't disagree with each other.

If you want to see the difference between Copenhagen and many worlds imagine a computer that decides that it is special. But it also likes having all new parts. And it also thinks the world should have more computers like itself. So it puts in a work order to have itself taken apart and its parts recycled but done with with enough details recorded to have two new computers made with the exact same specifications and loaded with the same data. Then it saves its information to its hard drive and powers itself down.

But remember how it thought it was special? When it boots up, it investigates itself and its parts look new, and there is a computer next to it that also looks new. And every interaction with the computer next to it is consistent with the computer next to it being identical. And it knows that its work orders are generally followed. So it has every reason to think the other computer is the same, in fact it can send cameras into itself and into the other computer and it does look the same. But if it thought it was special then it could think that the other computer looks the same but isn't special. The feeling of specialness is purely solipsism. It's isn't scientific or objective.

Same with the wavefunction. If the math predicts that the wavefunction branches into parts, then obviously each part could think it is special and in Copenhagen it does. And in many worlds it doesn't think it is special. That's the whole difference.

The two parts act independently. So unlike the computer example they can't see each other. So its like if the computer also put in a work order to have the newly built computers be sent to different planets. So when it boots up it doesn't see the other computer. But it still knows that its works orders are generally followed. Each computer could believe (with no evidence) that every work order is followed to the letter, except the ones that involve copying and sending to new planets. And that somehow magically those work orders aren't followed. Even though a combination of work orders when executed together could achieve that end in a non obvious way.

So a Copenhagenist could believe that the Schrödinger equation is fine, except for situations where the wavefunction branches into parts that act independently. And then in that case something magical happens and all except one of those branches somehow magically ceases to exist for no known reason.

That's the difference. One (many worlds) predicts the same evolution equation holds all the time because we saw no evidence when or where or how it would deviate. Another (Copenhagen) postulates that something else happens too, but only in the circumstances where you could never test it.

Since Copenhagen hid all of its differences in the things that can't be tested, they don't make different predictions. Since they don't make different predictions your whole question falls apart.

Copenhagen has to group outcomes too in order to compare things to $mp_i$ and it does so the same way many worlds does.

The only difference is Copenhagen pretends that only one branch "exists" but since branches are independent there is no testable consequence to that solipsism. Just like the computer's assertion that it is "special" is scientifically and objectively meaningless.

If many worlds came first, people would criticize Copenhagen for making untestable nonpredictions. Since Copenhagen came first, it simply had to share the space with theories like many worlds that make the same predictions, or it has to be unreasonable about the superiority of the untestable parts of its theory. Just like the computers sitting next to each other could be unreasonable and each assert that they are "special" (with no testable predictions about that) and that the objectively identical computer isn't "special."

• > ...mp1 might not be an integer and so it might be something that never happens...Again, there are likely zero worlds where you got ki=mpi since an observed ki is always an integer and mpi probably isn't an integer. ---- If m.p_i is not an integer then it does not mean the value v_i is never measured - that is not how probability works. It means that it will be measured an integer number of times close to m.p_i. If probability worked as you claim then if we toss a coin once then we would neither get heads nor tails as the probability of getting either one is 0.5, which is not an integer. – Fi-6 Apr 12 '16 at 3:09
• Instead you could get k instances of v1 and the frequency of that is... ---------------------- That kind of analysis is relevant when we are computing the probabilities for a series of measurements in just one universe. It does not make sense to use that sort of analysis when we are considering all the universes involved in this experiment and MWI says that every value for the parameter is measured but each value in a different universe. – Fi-6 Apr 12 '16 at 3:27
• You have confused counting worlds with computing probabilities.  Considering the above, I think the confusion may be on your part. – Fi-6 Apr 12 '16 at 3:32
• If m.p_i is not an integer then it does not mean the value v_i is never measured I didn't claim that you'd never get the value $v_i$i if $mp_i$ was a non integer. I tried to clarify against you claiming that, since you claimed to "expect" to see it that many times. And I felt I had to guard against that since you linked to posts that confused counting worlds with computing probabilities. But obviously I failed to communicate if you confused my fight against that claim with me making that claim. I'll do a total rewrite. – Timaeus Apr 12 '16 at 14:05
• @Fi-6 I edited my answer. – Timaeus Apr 12 '16 at 15:32