What is Many Worlds? What isn't Many Worlds?

[Charlie!]

The author, however, might not know the origin of a probability if it came up and bit him in the face. For example section A. When agents in a classical multiverse flip a bunch of coins and follow some probability-updating procedure, the author faults the ones who flipped all heads for coming to the wrong conclusion!

Which section A? Also, keep in mind he is arguing mostly against arguments of the Deutsch/Wallace type- who argues about probabilities by talking about a rational observers expectations. I'm reasonably certain we can both agree that a good Bayesian will conclude after flipping hundreds of heads, that a coin is biased.

Ah, sorry. 4 A. I'll retcon that into my post. Agreed on what people will conclude (though of course "conclude" refers to a probability distribution, which never goes to 0 or infinity for any finite amount of information, otherwise being irredeemably wrong would be possible and would be a serious problem).

Yeah. Failing the Born rule isn't a property of any branch considered by itself, it's a property of the set of all branches from a given point, so any kind of branch pruning or mangling process can't just operate locally within a (potential) branch, it has to operate globally on all the branches at each branch point.

I would actually disagree with this too. Like how a fair coin is logically equivalent to "all heads" being a possible outcome, adherence to the Born rule requires the existence of those small-probability options. Sure, approximation is still possible by various complicated means, but implementing the realio, trulio (no subtext to that link, I just like saying "realio, trulio") Born rule really would mean breaking the symmetry of information about states with different measure, and not pruning anything out.

[SU3SU2U1]

I can't read the text of the discretization article

Its on the arxiv, it should be freely available?

Also, Charlie, I don't think the second quote above is from me. And we are in agreement- the problem is not the existence of the small probability outcome branches, the problem is their overabundance. No matter how strongly you bias your coin, the majority of many worlds observers will conclude after a finite number of measurements that the coin is probably fair. I think thats why the usual pop-science account of many worlds discusses 50/50 outcomes- it hides the problem.

[mfb]

\left<+-+...\right|\left.\psi\right>^2

And this quantity will be 0 for any infinite sequence of measurements that falls outside the Born probability. Look! We get born probabilities because those are the only amplitudes that exist!

With an infinite sequence of measurements, every sequence has the same probability, and every amplitude is 0.
100 heads in a row have the same probability as every (specific) arbitrary sequence of heads and tails: 1/2^100.
"50 heads within 100 trials" has a higher probability than 0 heads, but it consists of many different possible states.

>> That means that if we count observers MOST observers will have seen an equal number of spin up and spin downs. Very, very few observers will have observed born probabilities.
As long as there is no discrete structure behind the universe, counting is not a good idea.
Every interaction between two particles (and even a particle and the vacuum) has an infinite number of possible Feynman diagrams. There is no natural way to get probabilities by counting, if both options consist of infinite sets without any order. This is not MWI-specific. The usual way to calculate these collisions (with perturbation theory) is to calculate the dominant Feynman diagrams and add their complex amplitudes. Which is like a small MWI-world without significant decoherence. Or Copenhagen without a collapse within this system.

[SU3SU2U1]

With an infinite sequence of measurements, every sequence has the same probability, and every amplitude is 0.

Yes, but some are more 0 than others. We know that with infinite measurements, ONLY sequences that conform to Born probabilities will be measured. Each specific sequence is very improbable. Sequences that don't conform to Born probabilities should not be measured by anyone (remember, this isn't A LOT of measurements, its INFINITE measurements). Just like if someone could flip a fair coin infinite times, there are infinitely many highly improbably outcome,but all of them would have 50/50 heads-tails.

As long as there is no discrete structure behind the universe, counting is not a good idea.

For the situation being discussed, there is a discrete number of 'you's after entanglement.

Every interaction between two particles (and even a particle and the vacuum) has an infinite number of possible Feynman diagrams

That is totally irrelevant for several reasons. Remember that quantum mechanics and quantum field theory are different entities- interacting quantum field theories don't live in a Fokk space. Its also not a true statement- there are a small number of exactly solvable field theories where you can calculate Green functions exactly, without perturbation theory. And lastly- Feynman diagrams don't have anything to do with measurement, or measurement problems. You're trying to map two totally unrelated things on top of each other.

[SU3SU2U1]

I used to think the MWI was cool, until I heard about the Transactional interpretation. I'd love to know what people here think about the TIQM, especially SU3SU2U1 & JWalker.

I've always found it very interesting, especially because the advanced Green function has been something thats bothered me for as long as I've been solving physics problems. Its always interesting to see someone try to use it for something.

Like all interpretations, though, it some issues. http://arxiv.org/abs/quant-ph/0408109 describes a thought experiment proposed by Maudlin that attempts to show the transactional interpretation is inconsistent (and has a rebuttal)

[Yakk]

Just like if someone could flip a fair coin infinite times, there are infinitely many highly improbably outcome,but all of them would have 50/50 heads-tails.

I'd like to see your proof of that.

[mfb]

With an infinite sequence of measurements, every sequence has the same probability, and every amplitude is 0.

Yes, but some are more 0 than others. We know that with infinite measurements, ONLY sequences that conform to Born probabilities will be measured. Each specific sequence is very improbable. Sequences that don't conform to Born probabilities should not be measured by anyone (remember, this isn't A LOT of measurements, its INFINITE measurements). Just like if someone could flip a fair coin infinite times, there are infinitely many highly improbably outcome,but all of them would have 50/50 heads-tails.

This is only true if you sum up states in a specific way (here: "States with Born probabilities" and "states without"). There is no difference between "heads all the time" and "head tails repeating every time", although the latter one gives 50% heads and the first does not.

For the situation being discussed, there is a discrete number of 'you's after entanglement.

Same here. You compress two sets of states to single states. Combine them in another way, and you'll get problems.


>> there are a small number of exactly solvable field theories where you can calculate Green functions exactly, without perturbation theory.
Too bad that this number is small. In addition, see below.

>> And lastly- Feynman diagrams don't have anything to do with measurement, or measurement problems.
Imagine something like this in a particle collider (pp, ppbar or whatever, the electron belongs to the detector material):


For significant momentum transfer of the photon, it has a tiny amplitude (Copenhagen: probability) - so many orders of magnitude below everything interesting that nobody cares about it.

But what about small momentum transfers? We know that charged particles can interact over a distance of 5mm. This is only one Feynman diagram, I could draw more of them for my whole life. There is no way to handle this analytically with all the detector components. Is that a measurement? If yes, why can you observe interference effects at all? If not, how can this change the state (more specific: the wave function) of the detector (which is certainly the case, as we have an interaction between detector and particles)?
In addition, we can compute a superposition of the detector with this Feynman diagram and the detector without. And we have an infinite amount of Feynman diagrams. You can express this in an analytic way, but you will still get a continuum of stuff. How do you count a continuum? You introduce a measure. Well... it has to be additive. Which quantity satisfies that and is non-trivial? The amplitude squared.

I hope this was not too compact.

[SU3SU2U1]

I'd like to see your proof of that.

Its an absolutely horrid exercise in defining measures on a probability space. I can motivate the plausibility by pointing out that its motivated by the frequentist definition of probability.

P(heads) = \lim_{n_{trials} -> \infty} n_{heads}/n_{trials}

This is only true if you sum up states in a specific way (here: "States with Born probabilities" and "states without")

Its actually implicit in the definition of frequentist probability, but its not worth getting into. I'm mostly arguing AGAINST the standard Everett/DeWitt argument anyway, and its irrelevant to me if the formal argument of Everett/DeWitt holds. The real world is finite measurements anyway, so the formal argument isn't applicable.

You compress two sets of states to single states.

If a detector becomes entangled with an electron in a superposition of spin up/spin down, then if we evolve by schroedinger we end up with a countably discrete (2 in this case) number of non-interacting branches of the wavefunction. If this isn't the case, the whole thing falls apart even more completely, because an experimentalist can enumerate the countably discrete outcomes of his experiment. I don't understand what you are trying to argue here? We seem to be wandering far afield.

Now, as to the Feynman diagrams, you have to be careful. You are making a muddle of various things. Feynman diagrams are just a tool to write down a peturbative expansion of the Green Function (or the S-matrix in your example). Part of the problem is that quantum field theories, while superficially like quantum mechanics are formally a nightmare.

The S-matrix isn't a measure for a sum over feynman diagrams. With your detector example, you are summing up the final scattering states of the S-matrix. The measure is the standard spherical measure (you are integrating over a fixed total momentum). To see this, note that if we had feynman diagrams for scattering into a countable number of bound states, we wouldn't have a continuum of processes and we would just sum up the final states.

Also, your photon example is an odd one to pick- soft photon scattering is going to be formally divergent. The problem is that you are using bad asymptotic states- having a electron and no photons in infinite past or future is impossible- the proper asymptotic states are electrons+their field. So your questions regarding the measurements are the standard quantum-optics questions of what it means to measure an electric field.

But most importantly, we are ranging too far afield- I don't see what complicating the issue with a discussion of field theories does for us?

[mfb]

>> Also, your photon example is an odd one to pick- soft photon scattering is going to be formally divergent.
That is a reason why I like it. We know that it exists, and treating everything as wave function will give some result for this process (even if it is too hard to calculate), without any measurement issue.

Ok, let's make things easier, use the "classical" QM as the main point is similar there:

We know that charged particles interact over a distance of 5mm. There is no way to handle this analytically with all the detector components. Is that a measurement? If yes, why can you observe interference effects at all? If not, how can this change the state (more specific: the wave function) of the detector (which is certainly the case, as we have an interaction between detector and particles)?
In addition, the wave function of the scattering and the wave function of the detector get entangled. You can express the equations for that in an analytic way, but you will still get a continuum of stuff. How do you count a continuum? You introduce a measure. Well... it has to be additive. Which quantity satisfies that and is non-trivial? The amplitude squared.

>> If this isn't the case, the whole thing falls apart even more completely, because an experimentalist can enumerate the countably discrete outcomes of his experiment.
Outcomes which he can distinguish by the (probably digital) output of the detector readout. But what about outcomes in the sense of a superposition of different stuff?

[SU3SU2U1]

That is a reason why I like it. We know that it exists, and treating everything as wave function will give some result for this process (even if it is too hard to calculate), without any measurement issue.

Its not impossible to calculate in QED, you just have to be careful about using gauge invariant asymptotic states.

Also, if you calculate an S-matrix, you explicitly aren't taking everything on equal footing in either of these cases as you are asymptotically defining particles in a Fock space, which don't actually exist in the interacting theory.

We know that charged particles interact over a distance of 5mm. There is no way to handle this analytically with all the detector components...

This is just a measurement problem- its conceptually no different than the spin-up electron, spin-down electron. What do we add by changing from spin-up/spin-down electron to the electron position? (I'm using magnitude of the field to be a measure of electron position).

How do you count a continuum? You introduce a measure. Well... it has to be additive. Which quantity satisfies that and is non-trivial? The amplitude squared.

What continuum are you counting? If you are talking about a super-position of electron positions, your continuum is just the position, and your measure is just the volume (really the volume course grained by detector resolution). The count of the number of distinct branches must always be equal to the number of discernable outcomes.

Outcomes which he can distinguish by the (probably digital) output of the detector readout. But what about outcomes in the sense of a superposition of different stuff?

They are still all tangled up, so they haven't separated into distinct branches thats how unitary evolution works. What point are you trying to make here?

Can we agree- if we prepare and measure a large number of a super-position of spin-up/spin-down electrons, more distinct branches of the wave function will measure an equal number of spin-up/spin-down then will measure any other outcome, regardless of what we choose the amplitudes to be?

[Yakk]

Just like if someone could flip a fair coin infinite times, there are infinitely many highly improbably outcome,but all of them would have 50/50 heads-tails.

I'd like to see your proof of that.

Its an absolutely horrid exercise in defining measures on a probability space.

No, not so far as I'm aware? It isn't anything beyond undergrad analysis. You can pretty much sum it up as "A probability space is a measure of norm 1." Am I missing something?

I would still like a proof of your confident, blanket assertion. If you cannot produce one, I'd take a pointer towards someone who did prove it. I cannot even think of a good approach to prove it with any level of rigor.

Maybe in an extension of standard measure theory of some kind you could prove something like that? (where you have some kind of infinitesimal measure to distinguish some kind of zeros from others?) However, I'd struggle to imagine how that non-standard measure theory would differ from standard measure theory in its description of what goes on in cases we could actually experience, which would make any proof from it questionable.

I could be wrong, which is why I'm asking for a proof. I'd like to see it.

[SU3SU2U1]

I would still like a proof of your confident, blanket assertion. If you cannot produce one, I'd take a pointer towards someone who did prove it.

Sure, remember that this started because I was briefly summarizing the standard many-worlds result of DeWitt (which I did in admittedly a handwavey/'physics math' sort of way). The paper is DeWitt and Graham, and I think it was Graham's phd thesis. The language is many worlds quantum mechanics, but its really a proof about frequentist probability. For some reason, I had always assumed it was uncontroversial result of frequentist probability that the two had rediscovered.

[Charlie!]

Consider the amount of probability within some distance 'a' of the middle. First you take the limit as the number of trials -> infinity (always 1). Then you take the limit a-> 0. Seems like my calculus 2 section on delta-epsilon limits all over again

However, if you reverse the order of these simple limits you get 0, so there was probably some mumbo jumbo about lebesgue measures or normalization or something.

Ooh! Or maybe you could do a direct proof that the limit is a delta function by taking the integral as the number of trials goes to infinity and imposing normalization. That sounds like it would actually be convincing.

EDIT: Oh, right, or just take the limit of 2N choose N, as N goes to infinity, and notice how it becomes infinitely large relative to everything else.

[mfb]

We know that charged particles interact over a distance of 5mm. There is no way to handle this analytically with all the detector components...

This is just a measurement problem- its conceptually no different than the spin-up electron, spin-down electron. What do we add by changing from spin-up/spin-down electron to the electron position? (I'm using magnitude of the field to be a measure of electron position).

An electron position is not binary (like "up" and "down").
A volume is not a measure, but you can define one in it. But that is not the interesting part. You still need a way to integrate over all possible states, regardless whether you look at them by an electron position or any other variable. The important question is which function (which measure) you use for this.

Outcomes which he can distinguish by the (probably digital) output of the detector readout. But what about outcomes in the sense of a superposition of different stuff?

They are still all tangled up, so they haven't separated into distinct branches thats how unitary evolution works. What point are you trying to make here?

They might be separated. The detector was just not sensitive enough to see a difference.
There is no single point in time where you can say "see! Now the world splits up!". They just get separated more and more, up to a point where it is possible to treat the different parts as separate independent branches.
For Copenhagen: The detector gets tangled up with the reaction, but without collapse? When does this collapse happen in that case? With an interaction which is violent enough. But where is the threshold?

Can we agree- if we prepare and measure a large number of a super-position of spin-up/spin-down electrons, more distinct branches of the wave function will measure an equal number of spin-up/spin-down then will measure any other outcome, regardless of what we choose the amplitudes to be?

As long as you compare "equal number" to "difference of n" for each n: Of course we can.
But this is just driven by the fact that (n choose n/2) is much larger than (n choose k) with |k-n/2|>>1 for large n (assuming even n here).
Can we agree that for each specific series of measurements (like "up,up,down,up, down") the amplitude/probability is the same (2^(-n)), if the superposition is symmetric in the two states?

[SU3SU2U1]

An electron position is not binary (like "up" and "down").
A volume is not a measure, but you can define one in it. But that is not the interesting part. You still need a way to integrate over all possible states, regardless whether you look at them by an electron position or any other variable. The important question is which function (which measure) you use for this.

Volume is the Lesbague measure on R^3. If we want to count the distinct worlds that come out of a measurement of electron position, its the measure we use (although really, its volume course-grained by detector resolution).

I think what you are trying to back into is that IF we want to define a probability measure on our space, there is only the usual amplitude squared (this is essentially Gleason's theorem). The problem is, there is exactly one observer per branch, and we can count branches. Why do we need a probability measure? What can it possibly mean?

There is no single point in time where you can say "see! Now the world splits up!"

According to Everett, DeWitt, etc, a branch has "fully split" when the observer states are orthogonal. When you count worlds, you are counting the number of orthogonal observers.

For Copenhagen...

For Copenhagen, or objective collapse theories collapse occurs when the detector spits out "this is the electron position." Its not particularly nice physically, but Copenhagen has its own problems.

If you measure a superposition biased so that 4/5 of the measurements should be spin up according to born probabilities, can you at least see why it might be a problem that there are more observers who measure half up/half down then there are observers who measure 4/5 up.

[mfb]

I think what you are trying to back into is that IF we want to define a probability measure on our space, there is only the usual amplitude squared (this is essentially Gleason's theorem). The problem is, there is exactly one observer per branch, and we can count branches. Why do we need a probability measure? What can it possibly mean?

No, you count branches .

>> According to Everett, DeWitt, etc, a branch has "fully split" when the observer states are orthogonal.
And what happens before?

>> When you count worlds, you are counting the number of orthogonal observers.
Without looking at actual calculations, I doubt that the observers are ever perfectly orthogonal.

>> Its not particularly nice physically, but Copenhagen has its own problems.
Indeed.

If you measure a superposition biased so that 4/5 of the measurements should be spin up according to born probabilities, can you at least see why it might be a problem that there are more observers who measure half up/half down then there are observers who measure 4/5 up.

As long as you count something: Of course. Counting can only work if there is something which destroys stuff with small amplitudes or does something else to get more 4/5-like observers.

[SU3SU2U1]

Without looking at actual calculations, I doubt that the observers are ever perfectly orthogonal.

See Zureck's work on entanglement.

No, you count branches

The point is, we have a fully deterministic system. Why introduce a probability measure? What does that probability mean? Probability of what?

As soon as you try and say "its the probability that 'you' end up in branch x" you have to invoke some structure external to the wavefunction, generally something like many minds.

As long as you count something: Of course

So thats the core issue with many worlds- most observers can't be expected to measure Born probabilities.

Counting can only work if there is something which destroys stuff with small amplitudes or does something else to get more 4/5-like observers.

Which requires adding something to the formalism. You can add non-linear terms to Schroedinger like the Weissman paper above, or you can add a structure to space like the Hsu/Zee paper above. Either way- its no longer true that evolution is always and everywhere governed by Schroedinger.

[Yakk]

No, you count branches

The point is, we have a fully deterministic system. Why introduce a probability measure? What does that probability mean? Probability of what?

As soon as you try and say "its the probability that 'you' end up in branch x" you have to invoke some structure external to the wavefunction, generally something like many minds.

As long as you count something: Of course

So thats the core issue with many worlds- most observers can't be expected to measure Born probabilities.

I don't get it.

Count, or do not count. Counting possibilities and comparing their size is introducing a probability measure. If you object to a probability measure, then you should object to counting and comparing sizes. Is it just that the particular probability measure (that corresponds to counting in some cases) is one that you prefer for philosophical reasons? And when it doesn't pan out, you give up?

Which requires adding something to the formalism. You can add non-linear terms to Schroedinger like the Weissman paper above, or you can add a structure to space like the Hsu/Zee paper above. Either way- its no longer true that evolution is always and everywhere governed by Schroedinger.

Yes, for your counting of worlds to work, you need to add something to the formalism to make counting make sense, otherwise you don't have anything to count, as there aren't orthogonal worlds. There are worlds that get close to being orthogonal, but never completely so.

If you do force worlds to be orthogonal (really), and you count, you don't get something that lines up with observations (the probabilities from your forced-orthogonalized count-based probability measure don't line up well with our observations). To me, this looks like a sign that count-based forced-orthogonalization probability measure isn't a good approach -- it introduces additional overhead (to enable the counting), and the probability measure that it generates doesn't have predictive power.

So don't count.

[SU3SU2U1]

Counting possibilities and comparing their size is introducing a probability measure. If you object to a probability measure, then you should object to counting and comparing sizes.

Maybe I didn't express this well. Many worlds is deterministic- the wavefunction exists, and its all there is. One property is the amplitude of each non-interacting section of the wavefunction. Another property of the wavefunction is that there are is a countable number of orthogonal copies of 'me' in the wavefunction. Thats built in to the wavefunction, its not any additional structure that has been added.

I say "but wait, there is only one me." Many worlds says "thats ok, they don't interact so each 'me' knows nothing about the other 'me's."

But now, I can say "but wait, I must have a very special life experience- most of those other 'me's don't see anything like what I've seen- their rules of quantum mechanics are very different!" And thats the problem- we can count up copies of me that exist in the wavefunction, and all of them exist. The majority of them have different laws of measurement. We could fix the issue with an anthropic-non-answer,if I weren't in a Born branch, I wouldn't be sitting here talking about Born probabilities, but that is unsatisfactory.

Edit: To see that observers in different branches live in orthogonal branches, consider again what happens when you measure an electron spin. Once you become entangled with the electron you "branch into" either spin up or spin down. If you take the inner product of those two worlds, you'll be taking an inner product of spin up and spin down electron states, which is 0.

[Yakk]

One property is the amplitude of each non-interacting section of the wavefunction. Another property of the wavefunction is that there are is a countable number of orthogonal copies of 'me' in the wavefunction.

What process in QM leads to actual orthogonality? (Not "close enough", but actual orthogonality)

Where do you end, and the orthogonal "copy" begin? The counting mechanism relies on some method to distinguish between "orthogonal copies" that says "well, this is close enough to orthogonal for our purposes", if I understand what is going on. It isn't actually orthogonal. We are transforming a continuous wavefunction into discrete worlds, and that transformation is questionable.

After doing that questionable transformation, if we then use the resulting discrete worlds as the basis of our counting probability measure, it has been found that the resulting probabilities do not have predictive power.

At this point, I'd go back and question the transformation that discretized the wavefunction, or using said transformation as the basis for our choice of probability measure.

[SU3SU2U1]

What process in QM leads to actual orthogonality? (Not "close enough", but actual orthogonality)

Ah, I just edited in above, but it bears repeating. If entanglement/decoherence didn't give us orthogonal worlds as the wavefunction evolved (actual, honest to god orthogonal) , there is no way you could ever recover anything like what we see from it- I know when I measure an electron spin I get one of two answers. I never get an in-between answer. At the start, many worlds has to provide a spin-up world and a spin-down world or it can't work.

The actual process is entanglement/decoherence. See Zureck "Decoherence, einselection, and the quantum origins of the classical" Its important to remember- if the many-worlds wavefunction couldn't evolve into orthogonal states, no one would be trying to make it work, because there would be no way it would model the collapse interpretations that (while somewhat ill defined) do make valid predictions.

[Yakk]

That is true if your entanglement is perfect, right?

And if your entanglement is nearly perfect, it is nearly true? Ie, when we experimentally entangle two particles, we cannot do it perfectly -- we can only do it nearly perfectly. The measurements we'd get will be nearly exactly what you'd get with a perfect entanglement -- possibly any difference would be below the level of sensitivity / the error rate of our tools.

But that doesn't mean we actually had perfect entanglement.

You are claiming we have "honest to god" orthogonality, and not "almost perfect, but not quite" orthogonality. Suppose you where almost, but not quite, orthogonal to the electron being in the spin up state. How would you distinguish this from being "honest to god orthogonal" experimentally?

(Ie, <a,b> is really small means "almost perfect, but not quite, orthogonality", while <a,b> being exactly 0 is "honest to god orthogonal".)

---

Zureck's paper, if I understand it at all (and I probably don't!), deals with classical mechanics emerging from quantum mechanics in the limit. We don't actually reach the limit, we just get exceedingly good at approximating it -- but the speed at which we approach the limit is ridiculously fast (the decay occurs really fast).

And your counting probability corresponds to counting Zureck's pointer states (the Einselected "yous")? After presuming that each such state has an equal probability of occurring, that is.

[SU3SU2U1]

I had a long post typed up about the issue of measurements and measurement bases, but it was eaten and then I realized it isn't necessary.

You can rephrase the whole argument but instead of saying 'orthogonal observers' you can say instead Zureck's 'pointer states'. The same argument still holds.

The point is this- many world tells me we have a wavefunction, and Zureck tells me that the wavefunction looks like a superposition of 'me' in a countable number of pointer states. All the 'me's exist- very few of the 'me's have observed Born probabilities for their historical measurements.

Also, to be clear- these aren't my arguments,these are in the literature I've discussed.

[Yakk]

You can rephrase the whole argument but instead of saying 'orthogonal observers' you can say instead Zureck's 'pointer states'. The same argument still holds.

First, pointer states are only reached in the limit. You get arbitrarily close to a pointer state at a very short time scale, but the pointer state itself is just an attractor point.

Second, lets look at a less weird eigenstate.

We play a game of flipping coins. We have 10 coins. We flip 1 coin every second in a circular order.

Whenever we get a tail, we modify the previous coin that flipped to have tails on both sides.
Whenever we get two heads in a row, we modify the previous coin that was flipped to have heads on both sides.

There are two converging states in this system -- the coins become all heads, or the coins become all tails. In this case, a simple argument can show that the chance of converging to all tails is much, much higher than all heads.

At a scale of years, the convergence of this system to one of those two states is going to be nearly instantaneous. Treating the two states as being "of equal probability" because they are the only two states that can practically occur (on the scale of years) doesn't work.

So, why in the world would you even think to use a counting probability measure on pointer states in a continuous version would make any sense at all?

Those states aren't even reached -- they are just clusters of microstates that we identify with a single macro state that the actual microstate converges to. On top of that, in simpler cases counting actual states that a system converges and applying a counting probability measure doesn't produce anything useful. I don't get why you'd even try? I guess one could get ridiculously lucky!

/shrug. And on top of that, if we apply a pretty simple quantization procedure to QM itself, and make it ridiculously ridiculously fine, we get a Born rule falling out... right?

[SU3SU2U1]

I think you are missing the forest for the trees. In many worlds, literally all we have is a wavefunction, and an evolution equation.

To make any predictions at all, we need a way to define 'observers' consistently in the wavefunction. You are arguing about how this is handled, and I agree its a subtle problem, and there is a fair amount of literature on this, that goes under the heading of "the problem of a preferred basis." Zurek's pointer states are as good a procedure for counting observers as simply projecting on to the measurement basis (which is what I was doing when talking about orthogonal observers). The point of Zurek's work is that the wavefunction quickly seperates into a bunch of big blobs in phase space, widely separated and evolving mostly independently from each other in time. Any not-perfect-convergence represents how much the different blobs effect each other as they evolve forward in time. Roughly speaking, we count the blobs to count observers (there are ways to formalize this).

Once we have solved this problem (I believe it has been solved, others disagree), we have the further problem that all the observers exist. When we ask about probability, we have to ask about 'probability of what?' This is the problem I believe is unsolved. BUT, if you can't solve the first problem, you can't even begin to talk about the second. If we switch into many-minds for a second, if you can't count observers, you can't ask 'what is the weight of minds associated with this observer?'

And on top of that, if we apply a pretty simple quantization procedure to QM itself, and make it ridiculously ridiculously fine, we get a Born rule falling out... right?

I don't follow your meaning here.

Edit: To summarize my post, think about it like this. Many worlds says the only object that exists is a huge multi-dimensional function on phase space \psi_U where the U is for universe. We have a rule that tells us how it evolves in time.

Now, to make a prediction, we have to define a measurement basis. After we define a measurement basis and evolve forward in time at a decoherence point according to Zurek:

\psi_U = A \psi_{UA} \left|+\right> + B\psi_{UB} \left|-\right>

Here A and B are complex amplitudes. UA is universe A where + is measured and UB is universe B where - happens. We can ALWAYS do that projection, its simply changing bases. + is orthogonal to - as they are eigenfunctions of the non-degenerate spin operator. What Zurek tells us is that A and B quickly separate. i.e. the time evolution of A has little-to-no bearing on the evolution of B and visa versa. This is super important for many worlds- otherwise you could detect the worlds and empirically we do not.

Now, we have one universal wavefunction with two sub-components that evolve independently in time. Why should I interpret those amplitudes as probabilities? Probabilities of what?

[lightvector]

Now, to make a prediction, we have to define a measurement basis. After we define a measurement basis and evolve forward in time at a decoherence point according to Zurek:

\psi_U = A \psi_{UA} \left|+\right> + B\psi_{UB} \left|-\right>

Here A and B are complex amplitudes. UA is universe A where + is measured and UB is universe B where - happens. We can ALWAYS do that projection, its simply changing bases. + is orthogonal to - as they are eigenfunctions of the non-degenerate spin operator. What Zurek tells us is that A and B quickly separate. i.e. the time evolution of A has little-to-no bearing on the evolution of B and visa versa. This is super important for many worlds- otherwise you could detect the worlds and empirically we do not.

All of what's quoted above seems fairly uncontroversial.

Now, we have one universal wavefunction with two sub-components that evolve independently in time. Why should I interpret those amplitudes as probabilities? Probabilities of what?

I understand why you are hostile to the idea of probability here. So let's try to answer one of the objections you had, avoiding the terminology and language of probability, and just talking about measure.

You want to argue that many-worlds is a problem because all the worlds exist, and so observers in "most" of the worlds will not observe the born rule. The problem is that "most" is not well-defined by itself, because the space is continuous. So to specify what you mean by "most", you need to introduce a measure on the space.

Remember, the two components A and B are not actually independent. They are only approximately independent. It's not as if the "blobs" of amplitude compsing A and B are sharply defined, so trying to count A and B as "two" worlds (or more, if A and B are individually also fragmented into separate amplitude blobs, but let's assume not), is misguided. Indeed, during the transition where A and B separate out from one interacting blob of amplitude into two, at what point do you transition from counting "one" to counting "two"? Do you count "1.5" at some point? What about cases where a component splits off that has so little amplitude relative to other things that it's barely distinguishable from any "background" amplitude? What about when components temporarily separate but then later rejoin and interact, as with interference? It's all a bunch of smooth interacting ripples and blobs of amplitude, which indicates that it's is terribly arbitrary and unnatural to attempt to "count" in any discrete fashion here.

So who cares if "most" observers don't see the correct statistics under some arbitrary unnatural measure based on discrete counting? There is a much more natural measure to use here. There are continuum-many worlds in the space, each with continuously varying amplitude, so the natural thing to do is to integrate over the space in a continuous fashion. And since the evolution is unitary, the correct measure is obvious, it's the one that corresponds to the conserved quantity. Under this measure, the problem goes away.

[SU3SU2U1]

You want to argue that many-worlds is a problem because observers in "most" of the worlds will not observe the born rule. The problem is that "most" is not well-defined by itself, because the space is continuous.

These aren't my arguments (although I agree with them). I've been summarizing the standard problem of many worlds, as pointed out by even Deutsch, Wallace, (hence their need to use game theory to try and develop the probabilities) probably the two most pre-eminent many worlders. The same problem is presented in the Zee and Hsu paper I linked to, and the paper by Weismann. I could dig up dozens more. Have you read any of the literature I've linked to?

Also- you should always consider that if there is a fair amount of scientific literature on something you consider trivial, and experts disagree on whether a question is settled, its probably not as trivial as you think it is.

It's not as if the "blobs" of amplitude compsing A and B are sharply defined

Yes, they are sharply defined. If they aren't, many worlds can't make any predictions. You need measuring A to be sharply defined from measuring B or you can't make predictions that match Copenhagen. This is related to the basis issue- for measurable operators, the outcomes are eigenvectors of the operators, which are countable.

Indeed, during the transition where A and B separate out from one interacting blob of amplitude into two, at what point do you transition from counting "one" to counting "two"?

See Zurek's work for timescales of decoherence. There is a timescale of measurement, its a solid prediction of decoherence work.

What about when components temporarily separate but then later rejoin and interact, as with interference?

It doesn't happen after decoherence. If it did, many worlds COULD NEVER match Copenhagen! After a measurement, things no longer interfere -> after decoherence, blobs of wavefunction cannot come back together.

There are continuum-many worlds

No, there aren't. Go back to Copenhagen for a second- are the outcomes of a given experiment countable?

And since the evolution is unitary, the correct measure is obvious, it's the one that corresponds to the conserved quantity.

Gleason's theorem only works if you decide to impose a PROBABILITY measure. There are many measures on the space.

[mfb]

You want to argue that many-worlds is a problem because all the worlds exist, and so observers in "most" of the worlds will not observe the born rule. The problem is that "most" is not well-defined by itself, because the space is continuous. So to specify what you mean by "most", you need to introduce a measure on the space.

Remember, the two components A and B are not actually independent. They are only approximately independent. It's not as if the "blobs" of amplitude compsing A and B are sharply defined, so trying to count A and B as "two" worlds (or more, if A and B are individually also fragmented into separate amplitude blobs, but let's assume not), is misguided. Indeed, during the transition where A and B separate out from one interacting blob of amplitude into two, at what point do you transition from counting "one" to counting "two"? Do you count "1.5" at some point? What about cases where a component splits off that has so little amplitude relative to other things that it's barely distinguishable from any "background" amplitude? What about when components temporarily separate but then later rejoin and interact, as with interference? It's all a bunch of smooth interacting ripples and blobs of amplitude, which indicates that it's is terribly arbitrary and unnatural to attempt to "count" in any discrete fashion here.

So who cares if "most" observers don't see the correct statistics under some arbitrary unnatural measure based on discrete counting? There is a much more natural measure to use here. There are continuum-many worlds in the space, each with continuously varying amplitude, so the natural thing to do is to integrate over the space in a continuous fashion. And since the evolution is unitary, the correct measure is obvious, it's the one that corresponds to the conserved quantity. Under this measure, the problem goes away.

Thank you for putting this into these words.

@SU3SU2U1: Arguing with literature only is problematic, especially with old literature. I don't say the papers you refer to are useless, but if you stick to them only you can never go beyond them.

>> Yes, they [the "blobs"] are sharply defined. If they aren't, many worlds can't make any predictions.
It can, as shown by lightvector. They are as good as Copenhagens, Borns, ... in the sense that you get a measure. Whether you want to interpret this as probability or not is your choice. I didn't see a good answer to my question "what does it mean to have a future event with 80% probability?" yet. It is easy to interpret this in terms of experiments done in the past, but not so easy for the future. With Copenhagen, only one of this will happen. How is it possible to talk about probabilities then?

>> See Zurek's work for timescales of decoherence.
Timescales, not points in time. So where is the single point in time where you count "2" instead of "1"?


>> There are many measures on the space.
Non-trivial, unitarity conserving measures?

[SU3SU2U1]

Arguing with literature only is problematic, especially with old literature. I don't say the papers you refer to are useless, but if you stick to them only you can never go beyond them.

The majority of the papers I referenced are modern (Deutsch and Wallace, Hsu/Zee,Weismann,etc), the exception being Everett which is there for historical reasons. Also, the goal of this thread isn't to present original research, its to explain the state of the literature.

I didn't see a good answer to my question "what does it mean to have a future event with 80% probability?" yet. It is easy to interpret this in terms of experiments done in the past, but not so easy for the future

Your just asking "what is a probability?" The standard definition for frequentists is in terms of ensembles, as I've already stated. 'What is a probability?' is the opening section of any book on probability.

Non-trivial, unitarity conserving measures?

The simplest is related to the symplectic volume form on the phase space manifold. The measure would be density which is preserved under evolution by Louville's theorem. Remember that phase space has lots of built-in structure. Its related strictly to a probability, which is why it is conserved, but its different than amplitude squared counting for a given basis. In particular, the density will have a universal set of 'probability' while the measurement amplitudes we are discussing are relative probabilities. Basically, density in this sense takes into account all the worlds, so if we take expectations of operators, you'll get very weird things. This is why you need a good prescription for distinguishing worlds.

i.e. if we use density as our measure, then there is only an x% probability that I'm currently 'me' and not some 'other me'. Gleason's prescription is (according to copenhagen) the obvious better choice (probability defined on a linear sub-subspace of the full hilbert space), but why?

Also, have you ever sat down and proved Gleason's theorem? Does the countability of 'outcomes' matter for the proof?

[mfb]

The standard definition for frequentists is in terms of ensembles, as I've already stated. 'What is a probability?' is the opening section of any book on probability.

How do you get an ensemble with Copenhagen and only one universe?
Using ensembles, why is the probability that "you" are in a universe the same for all universes* (of which 80% have state 1 and 20% state 2)? Oh, I like this .

Non-trivial, unitarity conserving measures?

The simplest is related to the symplectic volume form on the phase space manifold. The measure would be density which is preserved under evolution by Louville's theorem.

Density of what? Defined on a global deterministic wave function?

Also, have you ever sat down and proved Gleason's theorem?

No


*Bonus question: Assuming that the Born probabilities are perfect, would you need an infinite number of universes? If that is the case, counting does not work - how do you measure that?

[SU3SU2U1]

How do you get an ensemble with Copenhagen and only one universe?

You do the same experiment many times.

This is ranging too far afield- lets refocus. I've said my piece, and linked to the literature. If anyone has questions about the literature or is looking for more information, let me know. Otherwise, I'm going to drop this line of argument. People who argue the number of universes is uncountable should ask themselves what it would mean that results of experiments are uncountable. If they have a satisfactory answer (and one that leads to Born probabilities) they should publish it. Repeating the same arguments over and over is not helpful or instructive. Also, if you believe in many worlds, but have never proved Gleason's theorem, I recommend you try. Its fun and instructive. I further recommend trying to lay down an axiom for your preferred method to get Born into many worlds and see where it leads. Also fun and instructive.


So lets look at a philosophical aspect, and Occam's razor. We'll ignore the technical issues with many worlds for the time being. Quantum mechanics is a response to two stylized facts (to borrow a term from the economists)-

1. there are classes of experiment where we can't predict the outcome
2. doing many of those experiments we can always predict the distribution

Rephrased in terms of observables-
1. certain observables do not evolve 'nicely'
2. the observable's expectation values do.

If we send one photon through two slits, no idea where on the screen it hits. Send a million, and we know what the distribution looks like.

The minimal interpretation, then, is to say that the wavefunction ultimately represents the ensemble of experiments, not a single experiment. What it actually predicts are distributions- after all thats what behaves deterministically! The wavefunction never 'collapses', it just never makes predictions about single systems. When we do a single experiment, we take one sample from the predicted distribution.

Now, what the many worlds advocates say is- no the wavefunction doesn't represent a distribution of experiments! It represents each single experiment! It LOOKS like a distribution, but thats because each experiment creates a distribution of WORLDS- but all of them exist. It sure looks like they are introducing a whole lot extraneous 'worlds' in order to deny empirical fact 1. They want to say that the ability to predict distributions somehow implies the ability to predict single experiments, as long as you can add as many extra worlds as you need.

This looks to me like a pretty severe violation of Occam, but to each their own.

[mfb]

Well, we have a different understanding of "complexity" with respect to Occam's razor.
All interpretations use the wave function in some way, but most of them add something more to it - collapses, particles or other things. MWI does not: "Shut up and let the wave function evolve" .

>> Send a million, and we know what the distribution looks like.
You know what the most probable distribution / the one with the highest amplitude looks like. You can look at your 1 million photons as a single experiment, with a lot of different outcomes. Most of them (made finite through detector resolution or similar ways) look like the common double-slit pattern.
Repeating experiments just moves probabilities/amplitudes around to give a larger space of measurements/worlds. There is nothing special about them. If you can explain 1 million photons, you can explain 1, and vice versa.

[SU3SU2U1]

All interpretations use the wave function in some way, but most of them add something more to it - collapses, particles or other things.

The interpretation I described above adds nothing, has no collapse, and doesn't involve multiple universes.

If you can explain 1 million photons, you can explain 1, and vice versa.

So when I send a single photon through a two slit experiment, you can tell me where it will hit the screen?

[mfb]

If you can explain 1 million photons, you can explain 1, and vice versa.

So when I send a single photon through a two slit experiment, you can tell me where it will hit the screen?

Can you explain it for 1 million photons? For each detector position, can you give me the number of photons detected? No. You can give me a probability that 10000 +-1% photons will be detected at position x (for each x), you can give me confidence levels and so on. But the same thing is possible for 1 photon. And multiplying probabilities/amplitudes is a trivial task.

Actually, MWI can, taking your question literally: Everywhere where the amplitude is not zero.

[Xanthir]

Hey, SU3SU2U1, can I ask for a quick clarification on your position? I just want to make sure I understand what you're arguing so I'm not thinking against a strawman.

Here's a simplified situation: you have an event with 11 possible outcomes. One of these outcomes (call it A) has a 90% probability, the other ten (call them 0-9) have 1% probability each.

By trivial probability, if you repeat the event 6 times, you expect a slight majority (~53%) of the time to get six As. However, if you simply count the total number of possibilities, a slight majority of them (~56%) get *no* As at all, only digits.

Is this what you mean by "most observers don't see the Born probabilities"?

[WarDaft]

It's not even that, if I understand it. It's that, by quantum weirdness, you see A 0.9*0.9/0.01/0.01 times as often as any given digit, IE 8100 times more often, not just 90 times. That is, there's a certain 'size' that could be said to associate with any of the outcomes in MWI, but the relative frequency of observing* these outcomes is proportional to the size squared... at least, in our experience.

But I am not even slightly qualified as a physicist.

*Being part of a computational branch for which possible interaction with the surrounding stimulus results in the high level 'opinion' that these outcomes are what has 'occurred.'

[Xanthir]

I explicitly said "probability" rather than "amplitude" to avoid dealing with the square. If it's easier to answer with amplitudes, assume that the amplitude of the A event is ~.95, and the amplitude of each of the digit events is .1.

[SU3SU2U1]

Hey, SU3SU2U1, can I ask for a quick clarification on your position? I just want to make sure I understand what you're arguing so I'm not thinking against a strawman.

Here's a simplified situation: you have an event with 11 possible outcomes. One of these outcomes (call it A) has a 90% probability, the other ten (call them 0-9) have 1% probability each.

By trivial probability, if you repeat the event 6 times, you expect a slight majority (~53%) of the time to get six As. However, if you simply count the total number of possibilities, a slight majority of them (~56%) get *no* As at all, only digits.

Is this what you mean by "most observers don't see the Born probabilities"?

Remember- literally all we have is a wavefunction. For many worlds to work, we first need some way to define observers or worlds within the wavefunction. If we can't do that, its impossible to talk about a measurement. I argue that Zurek solves this problem, though Yakk and mfb have expressed some doubts (which would be detrimental for many worlds).

If we take Zurek seriously, then the wavefunction after a single measurement of your type:

\left|\psi\right> = C(0.1*\left|O_0\right>\left|0\right>+...0.1*\left|O_9\right>\left|9\right>+0.95\left|O_A\right>\left|A\right>)+D

Here each O_x is an observer who measured x. C is some totally unknown amplitude, and D is an unknown function representing all the phase space where no measurement was made. When I say "most observers" I mean most of the Os in the wavefunction.

After one measurement, your wavefunction contains 10 observers, unambiguously. That means each observer EXISTS with probability 1. Each observer has an amplitude associated with them, but no way to detect that amplitude. The common argument is that we have no reason to introduce a probability measure on this space. Probability of what?

How can I make a prediction just from the wavefunction above? What do I need to do?

[Xanthir]

...so the shorter version of your answer is "Yes, your understanding is correct."? ^_^

[WarDaft]

After one measurement, your wavefunction contains 10 observers, unambiguously. That means each observer EXISTS with probability 1. Each observer has an amplitude associated with them, but no way to detect that amplitude. The common argument is that we have no reason to introduce a probability measure on this space. Probability of what?

Future computational history maybe?

I mean, if the universe didn't contain any operations that could combine to form computation at some scale, we wouldn't be here asking this question, so that has to be important somehow.

If you have an NDTM with observers in it, then two *different* observers can have memories of past observations that are in *literally* the same observations. Later on, both clearly exist separate from each other in the NDTM model. The model however, has clear distinction between branches, and so we are tempted to just draw the line there, but there can easily be analogue or continuous models that can encode an NDTM's operations (QM is perhaps even such a construct even at the smallest scales, we aren't sure yet) in which the "question what are we counting" becomes hazy. But the existence of the observer is still coming from the fact that it's modeling an NDTM, the fuzziness is just making things complicated (or perhaps, simpler) and... well... fuzzy. But even if no two observers ever become fully de-coupled in their possible interaction, there are still differing computational histories in the full description of the world (there are of course 'more' of them) and if there's a concept of history in a deterministic function, then surely there is also a concept of future. When an observer at T=n is multiple observers at T=n+k, then the observer at T=n cannot possibly be every observer at T=n+k, and must have some range of expectation for future past observations.

The only part that seems weird to me is explaining the actual probabilities we have been observing... why are they this, with undeniable repeatability, and not something else... or why this probability is coming from a term being multiplied by a complex number. I realize this is still one of the main objections to many worlds, I'm just saying I don't see why the existence of observers or having some probability associated with the wave function is strange. My gut reaction is actually that it shouldn't be using complex numbers at all, that they're just an abstraction over a deeper model that just works this way in the tests we've been able to bring to bear so far, which makes the probabilities observed and the complex factors reconcilable. We still don't have quantum gravity, and gravitons would be so much smaller than anything else that they might just reveal another layer of humdingery going on that the current model emerges from. That could (as in almost certainly is) just be my near complete ignorance of the field and general hopefulness talking however.