How Many Worlds are in a Universe?

[S.Daedalus]

So, this is a simple question that's been bugging me for a while. Let's consider a particularly simple universe (and its wavefunction): a single qubit. This might be in a superposed state, wrt the computational basis ($\lbrace |0\rangle, |1\rangle\rbrace$), such as $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$. Now, measurement in this basis yields with 50% probability either 0 or 1. Naively, in a many-worlds picture, one might thus consider this as 'two worlds', one in which the qubit is 0, and the other in which it is 1.

However, measuring the same qubit in the $\lbrace |+\rangle, |-\rangle \rbrace$ basis, we get '+' with certainty (since $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, so we'd be inclined to consider this as 'one world'. More generally, any wavefunction in an eigenstate wrt some observable can be written as a superposition of eigenstates of some other observable.

Now, ordinarily, perhaps the environment, or some scientist (who's after all part of the environment) decides the basis to 'measure' in. Or, perhaps the canonical answer in the many-worlds picture is that world-splitting happens only upon decoherence, when some thermodynamically irreversible interaction occurs.

This is fine for everyday systems, but it runs into problems when the whole universe is considered: there's neither an environment to account for decoherence (the universe being, you know, all there is), nor is there an experimenter to do measurements (unless, I suppose, you count God, but let's just not go there). So, is the wave function of the universe in a superposition, or not? In what sense are there then 'many worlds' as opposed to just one, defined by whatever observable the universal wavefunction happens to be an eigenstate of? Or is my whole thinking just muddled?

 

[bcrowell]

So, is the wave function of the universe in a superposition, or not? In what sense are there then 'many worlds' as opposed to just one, defined by whatever observable the universal wavefunction happens to be an eigenstate of? Or is my whole thinking just muddled?

Most physicists (including me) do not consider interpretations of quantum mechanics such as the many-worlds interpretation (MWI) and Copenhagen interpretation (CI) to be testable physical theories, in the sense that although we have done experiments that had the logical capability of falsifying MWI or CI, they could have done this only by falsifying quantum mechanics itself. This would have falsified *both* MWI and CI, which means that they had no capability of establishing that one interpretation was to be preferred over the other.

IMO, interpretations of quantum mechanics are psychological descriptions of what it's like to make measurements. They succeed at the psychological level, not the physical level.

Both MWI and CI get silly if you aren't careful to distinguish the psychological and physical levels of explanation.

CI treats measurement and observation as distinct from other processes, but they're not distinct at the physical level. There's clearly a continuum involved. A human can observe the result of a radioactive decay, but the decay could also act as a stimulus to a cat, a flatworm, a bacterium, or a virus.

MWI can't answer questions about the rate at which universes proliferate, but that's OK because they don't proliferate at the physical level. The vector space describing Schrodinger's cat doesn't change discontinuously at any point; it always keeps the same dimension and never gains or loses information.

 

[Jazzdude]

So, is the wave function of the universe in a superposition, or not? In what sense are there then 'many worlds' as opposed to just one, defined by whatever observable the universal wavefunction happens to be an eigenstate of? Or is my whole thinking just muddled?

Whether a state is in a superposition depends on the basis, obviously. Being in a superposition is not a objective physical property. So, yes, the universe is in a superposition with some basis and not with with other. If you're asking if the universe is in a superposition of classical states then the answer is definitely yes, even though it's hard to even think of a basis of classical states.

Also, the worlds in MWI are relative to an observer, it's called the relative state interpretation for that reason. The observer becomes entangled with the outcome of a measurement so that you get a state of the sorts |ObserverSeesA>|A> + |ObserverSeesB>|B>. Now it is argued that the observer's reality splits up into two separate world branches determined by this decomposition. There are several problems with this argument, including the non-unique decomposition into an observation basis. But more importantly, the statistics of measurement are a problem with this approach. There is no good reason why the probability to find yourself in one world or another should depend on the amplitude of the corresponding branch. MWI followers argue that with clever counting you can get back the Born rule, but in order to do that clever counting they introduce some additional postulate that allows to break the linearity of the state space in some way. That in principle is not a great problem, however MWI argues that it's the natural approach that doesn't need any additional structure and is just the logical consequence of applying unitary quantum theory on everything. So these additional postulates break the most attractive feature of MWI and make the notion of infinitely many parallel worlds hard to accept.

 

[mfb]

You cannot perform measurements in a universe with just one qubit - measurements are always interactions in the universe. If that universe is so small that you do not get decoherence, I would consider it as "one world" in MWI, and as universe without collapse in CI.

The many worlds in MWI are not something fundamental (a full description of physics in MWI does not require to talk about "worlds") - they are just a convenient description how the universe looks like with unitary evolution.

 

[S.Daedalus]

Thanks Jazzdude, that was roughly the answer I was looking for. Of course, if we define worlds only relatively to observers, the problem doesn't arise, since they single out some basis for the decomposition. Conditioned on our existence, the splitting into worlds is unique, but there seems little fundamental reason for this conditioning. But basically, this does not seem much better than just assuming an a priori classical realm for the measuring apparatus.

 

[mfb]

Well, decoherence gives you a natural way to talk about "worlds". Unitary evolution leads to several areas in phase space which have no significant connection and evolve (nearly) independently. If you maintain a superposition in a lab experiment, the region of the wave function stays connected - until you let it interact with your environment in such a way that you get decoherence.

 

[bcrowell]

But more importantly, the statistics of measurement are a problem with this approach. There is no good reason why the probability to find yourself in one world or another should depend on the amplitude of the corresponding branch. MWI followers argue that with clever counting you can get back the Born rule, but in order to do that clever counting they introduce some additional postulate that allows to break the linearity of the state space in some way.

Interesting. Googling certainly shows that many people seem to believe that there's a problem with MWI because it can't derive the Born rule. IMO this is just another example of inappropriately mixing psychological and physical levels of explanation. A lot of physicists seem to expect that MWI should be stated as a set of postulates forming a theory that predicts the Born rule. But interpretations of quantum mechanics aren't physics, they're psychological descriptions of the experience of measurement. Since when do we expect psychology to be an axiomatic system in which results can be proved formally and can survive all empirical tests?

There are obvious reasons why any usable psychological interpretation of quantum mechanics must incorporate something like the Born rule. If I observe a diffraction pattern as a series of photon impacts, the average over a long time must agree with the classical theory, in which energy density is proportional to the square of the field. No, I haven't stated a set of axioms and derived this as a theorem from them, but that isn't the kind of thing you can do in psychology.

I'm obviously not the first person to say, "shut up and calculate!" Maybe the problem is that people see the mindset embodied by this slogan as an abdication of philosophical responsibility, when actually it's a statement of philosophical responsibility. There's a responsibility for carpenters not to pretend they can fix your plumbing, and for physicists not to pretend they're psychologists.

 

[Jazzdude]

Since when do we expect psychology to be an axiomatic system in which results can be proved formally and can survive all empirical tests?

The quantum measurement is not a problem of psychology. The outcome of experiments is the very subject of a theory, and MWI was created to explain that. So if it doesn't without additional postulates then it fails. I do agree that interpretations of quantum theory are not physics, that's why we need a theory that explains quantum measurement and the emergence of the Born rule. But just calling it psychology does not solve anything.

 

[S.Daedalus]

Well, decoherence gives you a natural way to talk about "worlds". Unitary evolution leads to several areas in phase space which have no significant connection and evolve (nearly) independently. If you maintain a superposition in a lab experiment, the region of the wave function stays connected - until you let it interact with your environment in such a way that you get decoherence.

Mainly, decoherence allows you to consider some approximate 'collapse' or 'branching', relative to a given interaction within an environment. However, this does not solve the problem, IMHO at least, but rather, just shunts it away from the currently considered context. Consider an experimenter doing experiments within some (really well closed-off) laboratory. Depending on his outcomes, relative to him, you would consider there to be 'many worlds'; but relative to an outside observer, until the lab's door is opened, there's only one world. Point being, since for the universe, there is no environment with respect to which to decohere, for any description using 'many worlds', I can find an equivalent one in which there's just one world. But then, what's the ultimate justification of talking about 'worlds' in an ontic sense at all?

There are obvious reasons why any usable psychological interpretation of quantum mechanics must incorporate something like the Born rule. If I observe a diffraction pattern as a series of photon impacts, the average over a long time must agree with the classical theory, in which energy density is proportional to the square of the field. No, I haven't stated a set of axioms and derived this as a theorem from them, but that isn't the kind of thing you can do in psychology.

The basic problem is that the naive account of probability in many worlds actually ends up making the wrong predictions, i.e. disagreeing with the ordinary quantum statistics. So you have to introduce some 'existential measure' or some equivalent in order to recover the original meaning of the Born rule. And even then, the account of probability in the MWI is controversial: usually, if we say that something has a certain probability of occurring, then we mean by that 'occurring to the exclusion of certain alternatives'; but the co-occurrence of all alternatives is in a sense the point of many worlds. There's been some interesting work by Deutsch and Wallace to recover the notion of probability from decision theory, i.e. they argue that the most rational behaviour for an observer in a many-worlds universe is to act as if the Born rule held; also, Saunders has an interesting account of 'relational probability', which I however can't claim to be familiar with. But neither is, to my knowledge, uncontroversial.

 

[mfb]

I doubt that you can isolate and prepare a system with a human inside sufficiently precise to allow any superposition afterwards - and if you cannot do that, the outside world can talk about many worlds, too.
But the human does not really matter, you can consider it as a quantum system anyway. Humans are nothing special, they are just collections of protons, neutrons and electrons.

The basic problem is that the naive account of probability in many worlds actually ends up making the wrong predictions, i.e. disagreeing with the ordinary quantum statistics.

It is as naive as saying "in collapse interpretations, all possible measurements have the same probability just because they are possible". Naive, and wrong.

You do not need probabilities at all to do physics. Something I posted in a previous thread:

Assume that the whole universe can be described as the evolution of a wave function. Develop some theory about the world which can describe the evolution of the wave function. Note that, given appropriate initial conditions, the wave function can split into a number of different parts, which are (in really good approximation) independent of each other. Each part is called a world.
Define a measure for worlds, which is just the integral over the amplitude squared over their region in phase space.
The evolution has to conserve the measure for all worlds (PI equivalent: probabilities always add up to 1). The theory now allows to predict how the wave function can split into several parts by things usually called "observation".

Define a test as a series of measurements in some way. Sort the possible results in two groups: One with a large measure (and preferably, but not necessarily, a small set of different measurement results) and one with all other results with a small sum of amplitude squares. Publish that you will perform this test (that is good scientific practice!) and that the test is a success in all worlds which are part of the group with large measure. Perform the test.
Now, some worlds will see a success and some will not. But here is the trick: If the theory was right, most of the measure will see a success. If not (in a significant way), a large part of the measure will see a failure.
With more and more tests, every theory which is wrong gets discarded in worlds containing a measure of ~1*. A correct theory will see a lot of confirmations and kept in worlds containing a measure of ~1*. A lot of worlds will come to wrong scientific conclusions, but their total measure goes to 0.

Comparison to PIs: There are many ways how experiments can go wrong, but the probability of all experiments going wrong tends to 0.

*strictly speaking, human scientists do not exist in all worlds, but this does not matter. It can be scaled to the fraction of the world with humans inside.

 

[Jazzdude]

It is as naive as saying "in collapse interpretations, all possible measurements have the same probability just because they are possible". Naive, and wrong.

You do not need probabilities at all to do physics. Something I posted in a previous thread:

mfb, you assume that someone who disagree with what you wrote above does not understand the argument. But in fact, I understand the argument very well and still disagree entirely.

Probability is something that emerges from counting events, it's not something you can simply postulate as a measure. Events are defined by the dynamics of a system and must emerge from a dynamical description. Saying we simply assign a probability measure to each branch so that is is mathematically consistent does not solve anything, especially not in an interpretation that tries to derive things from first principles. It doesn't matter what kind of arguments you bring in later, as this point is already problematic (at least) and cannot be justified a posteriori.

So while MWI possibly explains the state collapse it does not explain the frequency of events. If you think differently then that's ok, but you won't be able to convince me otherwise because I've already seen all the arguments. So the relative state interpretation is in no better shape than any other interpretation I'm afraid.

I only accept a solution to the measurement problem that derives both, the perceived (or possibly even objective) collapse and the correct event count. If there's no way to derive that from linear unitary quantum theory (as diverse no-go theorems suggest) then one might have to consider nonlinear modifications of quantum theory.

 

[mfb]

mfb, you assume that someone who disagree with what you wrote above does not understand the argument. But in fact, I understand the argument very well and still disagree entirely.

I think (and hope) we all understand it :).

Probability is something that emerges from counting events, it's not something you can simply postulate as a measure.

I did not do this. The description of a MWI-universe does not need any probability or measure at all. You have laws of unitary evolution and an initial state of the universe (which might somehow follow from the laws, too). This is a full description of the universe.
Note that it is much simpler than collapse interpretations. There, you need initial state and laws of physics, too, but additionally you need a collapse mechanism and a long list of the result of >10¹⁰⁰ collapses.

However, we want to test theories about the universe. And this is the part where a measure is introduced - it is a very natural one (I would even call it the natural one), as it is conserved under time-evolution. There are no probabilities involved at all.

With MWI, we have a method to find laws of physics in a set with measure ~1. There is no way to find and confirm them in every world.
With collapse interpretations, we have a method to find laws of physics in a set (of possible futures) with probability ~1. There is no way to find and confirm them with all possible experimental results.

 

[bcrowell]

But just calling it psychology does not solve anything.

It solves the problem by saying that the problem was a non-problem to start with. AFAIK the majority of physicists deal with these issues in exactly this way. This is why "shut up and calculate" is so popular: it correctly identifies a non-problem as a non-problem.

The quantum measurement is not a problem of psychology. The outcome of experiments is the very subject of a theory, and MWI was created to explain that. So if it doesn't without additional postulates then it fails.

Let's break this down carefully. "The outcome of experiments is the very subject of a theory[...]" I agree. Shut-up-and-calculate quantum mechanics is a theory, and it does successfully predict the outcome of experiments. "[...]MWI was created to explain that." But you and I agree that MWI and CI are not physical theories. "So if it doesn't without additional postulates then it fails." Yes, all interpretations of quantum mechanics fail in some sense, because they're attempts to make axiomatic theories of a psychological process.

I do agree that interpretations of quantum theory are not physics, that's why we need a theory that explains quantum measurement and the emergence of the Born rule.

Hmm...if you agree that interpretations of QM are not physics, but you disagree that they're psychology, what are you saying that they are?

But just calling it psychology does not solve anything.

No, it doesn't solve anything. It shows why it can't be solved by constructing axiomatic systems -- because psychology doesn't work with axiomatic systems.

Just as a random example of the success of calling it psychology, I think a lot of people have worried about the fact that in the CI, wavefunction collapse seems to violate unitarity and time-reversal. Exteme CI enthusiasts like Penrose have even postulated that there's some actual physical process that occurs, and that it's non-unitary. But to me, the solution seems pretty obvious. Memory is a psychological process. We have kind of a rough understanding of the physics behind memory, from a variety of approaches, such as the second law of thermodynamics and physical descriptions of how neurons work. To the extent that we understand the origin of the second law, we seem to have some understanding for the fact that we can remember the past but not the future, despite the time-reversal symmetry of quantum mechanics. IMO this shows that the problem has been solved to the extent that we can expect physics to solve any problem in psychology. We can't derive it from an axiomatic system describing memory and the brain, but that's OK.

IMO the common-sense reason I gave in #7 for the Born rule falls into the same category. It solves the psychological problem to the extent that we can expect physics to solve any problem in psychology.

 

[Jazzdude]

It solves the problem by saying that the problem was a non-problem to start with. AFAIK the majority of physicists deal with these issues in exactly this way. This is why "shut up and calculate" is so popular: it correctly identifies a non-problem as a non-problem.

Clearly the majority of physicists is not deeply into the measurement problem and on a superficial level an explanation of this kind may seem sufficient. However, if you look closer you realize that it's not so simple.

Hmm...if you agree that interpretations of QM are not physics, but you disagree that they're psychology, what are you saying that they are?

Interpretations are psychological aids to help us deal with the fact that we have not understood everything related to quantum measurement yet. Each single interpretation has some kind of familiarity device that calms our mind, but fails terribly at actually explaining anything. The only difference is really the rug under which the problems are swept.

"Shut up and calculate" also only works in a specific domain where experiment and observer are sufficiently separated and the experiment itself mostly isolated from its environment. There are many situations where we don't know when to apply a projection evolution or unitary evolution including decoherence. We still don't know exactly what a measurement device is, even though decoherence followers usually argue we do.

No, it doesn't solve anything. It shows why it can't be solved by constructing axiomatic systems -- because psychology doesn't work with axiomatic systems.

Just as a random example of the success of calling it psychology, I think a lot of people have worried about the fact that in the CI, wavefunction collapse seems to violate unitarity and time-reversal. Exteme CI enthusiasts like Penrose have even postulated that there's some actual physical process that occurs, and that it's non-unitary. But to me, the solution seems pretty obvious. Memory is a psychological process. We have kind of a rough understanding of the physics behind memory, from a variety of approaches, such as the second law of thermodynamics and physical descriptions of how neurons work. To the extent that we understand the origin of the second law, we seem to have some understanding for the fact that we can remember the past but not the future, despite the time-reversal symmetry of quantum mechanics. IMO this shows that the problem has been solved to the extent that we can expect physics to solve any problem in psychology. We can't derive it from an axiomatic system describing memory and the brain, but that's OK.

It's easy to blame the complexity of the human mind for weird effects, but actually not really considered to be a valid way out anymore. Even the human mind is subject to the linearity of the state space, and all decoherence, world splitting and the like cannot explain how a nonlinear switch between specific results only based on the amplitudes could possibly happen. In fact, it's quite easy to show that this is not possible, no matter how complex the machinery and how subjective the event.

The general position regarding the measurement problem is probably best described as resignation, not as a non-issue. The frequency of events must be explainable from a physical theory. Just saying it's all psychology is both wrong and dangerous, because it let's us lose focus of the problems that indeed exist. The truth is very simple: We don't understand it, and it very likely requires a completely new idea to approach the problem.

 

[S.Daedalus]

To the extent that we understand the origin of the second law

Which we don't, of course, without understanding the collapse dynamics (or whatever plays its role), since the unitary dynamics only takes pure states to pure states, and thus, doesn't increase entropy...

OK, that's maybe a little facetious an argument, but I think there are good reasons to think about measurement, and quantum foundations in general. One is that otherwise, much of what we think we know is kind of built on sand -- which there may not be a way around, but that's no excuse for not trying (though this kind of goes against my own conviction that effective theories can contain complete explanations, by virtue of their independence from the detailed microdynamics). Another is simple curiosity: I wouldn't be satisfied if somebody just gave me a black box and a recipe by which to twiddle its knobs and turn its dials in order to predict outcomes for experiments, which kinda is the state quantum theory is in right now; I'd want to know how the box works. I think an argument can be made that somebody in possession of the box does not actually possesses greater knowledge than somebody without it, despite his greater ability to predict experimental outcomes.

Lastly (I can think of more arguments, but honestly, this is not a discussion I'm very interested in having; this may be more a matter of taste than anything), and I think perhaps of greater appeal to you, there's a genuine possibility of there being new physics to discover in the answer to the measurement problem, and by extension, in the interpretation of quantum mechanics, and the 'shut up and calculate' approach is blind to this possibility. You mention Penrose, who is one of those who have brought up the possibility of actually modifying quantum theory in order to explain the collapse, by virtue of turning it into an actual, physical event. This would introduce a nonlinearity in quantum theory that can be experimentally tested (in fact, such tests have been proposed). (And perhaps as a side remark, wondering about the interpretation of the wave function is what brought us Bell's theorem, the ideas of nonlocality, contextuality etc., plus their marvelous experimental verification.)

Anyway, as I said, I'm not going to try and convert anyone to quantum interpretionism; certainly, most physicists (and practically everybody else) can manage without worrying about quantum foundations in their day to day lives. But I think there's some legitimate worrying to be done on that front.

 

[audioloop]

Mainly, decoherence you to consider some approximate 'collapse' or 'branching', relative to a given interaction within an environment. However, this does not solve the problem, IMHO at least, but rather, just shunts it away from the currently considered context. Consider an experimenter doing experiments within some (really well closed-off) laboratory. Depending on his outcomes, relative to him, you would consider there to be 'many '; but relative to an outside observer, until the lab's door is opened, there's only one world. Point being, since for the universe, there is no environment with respect to which to decohere, for any description using 'many ', I can find an equivalent one in which there's just one world. But then, what's the ultimate justification of talking about '' in an ontic sense at all?

can be solved with an objective collapse model i.e. a nonlinear quantum mechanics.
no need of enviroment.