"The wavefunction never collapses"

[sevensages]

TL;DR

How does a wavefunction collapse?

I recently read Max Tegmark's book Our Mathematical Universe. In Our Mathematical Universe, Tegmark asserts that Niels Bohr and other scientists supported the Copenhagen Interpretation that the wavefunction collapses. Tegmark asserts that Hugh Everett originated the competing thesis that the wavefunction never collapses. I don't know what the word collapse means in this context. It's my understanding that the wave function is purely a mathematical object, not something physical. So how does a wave function collapse? What does a wave function collapse even mean?

 

[Baluncore]

I don't know what the word collapse means in this context.

Maybe it is quantum mechanics that collapses.

 

[PeterDonis]

Different QM interpretations say different things about wave function collapse. Roughly speaking, at a high level, there are three different general things that different interpretations say:

Some interpretations say that "wave function collapse" is something we do in the math to make correct predictions once we know the results of experiments, and doesn't reflect anything physical actually happening to an individual quantum system when it's measured. "Copenhagen" interpretations more or less take this approach (it's hard to be definite because "Copenhagen" is a very vague and general label for interpretations); so do statistical/ensemble interpretations such as the one used by Ballentine in his textbook. Intepretations in this category say that measurements do have single outcomes (unlike those in the third category below), but they offer no explanation of how that happens.

Some interpretations say that "wave function collapse" is an actual physical thing that happens to an individual quantum system when it's measured--in other words, during a measurement the actual dynamics of the wave function are not the same as the unitary dynamics it has between measurements. This is how such interpretations account for measurements having single outcomes. The main issue with this category of interpretations is that all attempts to construct an underlying model of what the dynamics would be during a measurement have failed.

Finally, interpretations like the Many Worlds Interpretation say that "wave function collapse" never actually happens, because measurements don't have single outcomes--all possible outcomes happen, each one in its own branch of the wave function. This is indeed what you get if you just apply unitary dynamics to the wave function all the time, including during a measurement. In interpretations like this, we ourselves, observing the results of measurements, have "branches" of our own wave functions, and when we think we've observed a measurement to have a single outcome, that's because "we" are just one branch of the wave function. In that particular branch, you can mathematically apply the "wave function collapse" rule to make predictions about what you, in that particular branch, will observe in the future, because the other branches will never interfere with yours.

So there is no single answer to your question; all we have are different QM interpretations that say different, mutually incompatible things.

 

[sevensages]

Different QM interpretations say different things about wave function collapse. Roughly speaking, at a high level, there are three different general things that different interpretations say:

Some interpretations say that "wave function collapse" is something we do in the math to make correct predictions once we know the results of experiments, and doesn't reflect anything physical actually happening to an individual quantum system when it's measured. "Copenhagen" interpretations more or less take this approach (it's hard to be definite because "Copenhagen" is a very vague and general label for interpretations); so do statistical/ensemble interpretations such as the one used by Ballentine in his textbook. Intepretations in this category say that measurements do have single outcomes (unlike those in the third category below), but they offer no explanation of how that happens.

Some interpretations say that "wave function collapse" is an actual physical thing that happens to an individual quantum system when it's measured--in other words, during a measurement the actual dynamics of the wave function are not the same as the unitary dynamics it has between measurements. This is how such interpretations account for measurements having single outcomes. The main issue with this category of interpretations is that all attempts to construct an underlying model of what the dynamics would be during a measurement have failed.

Finally, interpretations like the Many Worlds Interpretation say that "wave function collapse" never actually happens, because measurements don't have single outcomes--all possible outcomes happen, each one in its own branch of the wave function. This is indeed what you get if you just apply unitary dynamics to the wave function all the time, including during a measurement. In interpretations like this, we ourselves, observing the results of measurements, have "branches" of our own wave functions, and when we think we've observed a measurement to have a single outcome, that's because "we" are just one branch of the wave function. In that particular branch, you can mathematically apply the "wave function collapse" rule to make predictions about what you, in that particular branch, will observe in the future, because the other branches will never interfere with yours.

So there is no single answer to your question; all we have are different QM interpretations that say different, mutually incompatible things.

So is Everett's Interpretation the same as the Many Worlds Interpretation?

 

[PeterDonis]

So is Everett's Interpretation the same as the Many Worlds Interpretation?

Some would say yes. Others would say that Everett himself never made the "many worlds" claim (the term he used was "relative state"), and that others added that later. (And some would say that "many worlds" is not a good description of the interpretation that goes by that name. Unfortunately this kind of thing is typical in the QM interpretation literature.)

 

[PeterDonis]

@sevensages please do not quote entire posts. It just clutters up the thread. If you are responding to a specific statement, quote just that statement.

 

[Dale]

all we have are different QM interpretations that say different, mutually incompatible things.

I have wondered about this. Are they really mutually incompatible? They all share the same math and the same experimental predictions. So nature doesn’t seem to view them as incompatible. They are clearly all isomorphic to each other.

So what more besides an isomorphism is required for things to be “compatible”? Can things only be “compatible” if they are identical? Clearly different interpretations are not identical, but is “compatible” intended to be a synonym for “identical”? And if not shouldn’t we want non-identical interpretations to qualify?

 

[PeterDonis]

Are they really mutually incompatible?

As far as their claims about "what is really happening", yes. Of course they all make the same predictions for what we can test by experiment, but they don't limit themselves to that.

For example, the MWI says measurements have all possible outcomes, but other interpretations say measurements have single outcomes. Those are incompatible statements; they can't both be true. They both make the same predictions for what we can test by experiment, but their claims go beyond what we can test by experiment, and they do so in incompatible ways.

They are clearly all isomorphic to each other.

The math is, but QM interpretations don't limit themselves to the math and what it supports. Or anyway, none of them except the "shut up and calculate" interpretation do, and opinions differ on whether that's even an interpretation at all. The parts that go beyond the math are not isomorphic to each other.

 

[Roberto Pavani]

A naive concern about MWI: it implies the existence of branches where 1000 photons all pass through a 45° polarizer. In that branch, physicists would conclude the Born rule doesn't hold.

MWI's response is that such branches have small "measure", but that measure is the Born rule, which is what MWI should derive, not assume. And given that infinitely many quantum events have occurred since the beginning of the universe, why do we always find ourselves in the branch where statistics match the Born rule?

This circularity bothers me, and Occam's razor doesn't help me sleep at night. Am I missing something?

 

[PeroK]

A naive concern about MWI: it implies the existence of branches where 1000 photons all pass through a 45° polarizer. In that branch, physicists would conclude the Born rule doesn't hold.

MWI's response is that such branches have small "measure", but that measure is the Born rule, which is what MWI should derive, not assume. And given that infinitely many quantum events have occurred since the beginning of the universe, why do we always find ourselves in the branch where statistics match the Born rule?

This circularity bothers me, and Occam's razor doesn't help me sleep at night. Am I missing something?

MWI has no satisfactory answer for the Born rule, except where the branching is equally likely.

That's why almost all presentations of MWI assume a simple 50-50 branch.

That said, I've seen attempts at constructing a statistical basis for the more general cases. But, to me, they always seem unconvincing. There are some threads on here about that.

I think it's fair to say that the main problem with MWI is explaining the Born rule.

 

[pines-demon]

A naive concern about MWI: it implies the existence of branches where 1000 photons all pass through a 45° polarizer. In that branch, physicists would conclude the Born rule doesn't hold.

MWI's response is that such branches have small "measure", but that measure is the Born rule, which is what MWI should derive, not assume. And given that infinitely many quantum events have occurred since the beginning of the universe, why do we always find ourselves in the branch where statistics match the Born rule?

This circularity bothers me, and Occam's razor doesn't help me sleep at night. Am I missing something?

Indeed that is known as the probability definition problem of MWI.

 

[martinbn]

A naive concern about MWI: it implies the existence of branches where 1000 photons all pass through a 45° polarizer. In that branch, physicists would conclude the Born rule doesn't hold.

If many people toss 1000 coins each, one could find a 1000 heads. So? Nothing to do with QM or MWI.

MWI's response is that such branches have small "measure", but that measure is the Born rule, which is what MWI should derive, not assume.

Why should MWI derive it? Why shouldn't every interpretation?

And given that infinitely many quantum events have occurred since the beginning of the universe, why do we always find ourselves in the branch where statistics match the Born rule?

How could statistics not march Born's rule?!

This circularity bothers me, and Occam's razor doesn't help me sleep at night. Am I missing something?

 

[Roberto Pavani]

If there are all possibile outcome universes then there are some where born rule doesn't hold. The question is then: how come in our universe the born rule hold? Are we "special"?

 

[Matterwave]

Some interpretations say that "wave function collapse" is an actual physical thing that happens to an individual quantum system when it's measured--in other words, during a measurement the actual dynamics of the wave function are not the same as the unitary dynamics it has between measurements.

There are also objective collapse theories (GRW, Penrose gravity-collapse) where the wave function collapse happens at dynamical timescales not related to measurement.

But getting back at the OPs original question. I think Peter's post is a good tornado-speed summary of the interpretational view of collapse.

I will expand a bit on the mathematical view, which was originally formalized by Von Neuman. Von Neumann's postulates regarding collapse basically says that collapse is a non-unitary evolution of the wave function into an eigenstate of the observable being measured. The probability of getting a particular value out as the measurement outcome is given by Born's rule $P(a_n)=|\langle n| \Psi\rangle|^2$ where $a_n$ is the eigenvalue corresponding to the eigenstate $| n\rangle$

More explicitly, the wave function gets projected into the eigenstate by the projector operator $P_n=|n\rangle\langle n|$:

$$|\psi_{\text{after}}\rangle = \frac{|n\rangle \langle n | \psi \rangle}{|\langle n | \psi \rangle|}$$

Note again that *which n should we use here* is not given by quantum mechanics. Only the probabilities are given by the Born rule.

 

[PeroK]

If there are all possibile outcome universes then there are some where born rule doesn't hold. The question is then: how come in our universe the born rule hold? Are we "special"?

If you toss a coin 1000 times, then you are almost certain to get approximately 500 heads and 500 tails. It's almost impossible to get.1000 heads.

However, each sequence of heads and tails is equally likely.

There is nothing special about our experience where the law of large numbers appears to hold as a statistical law.

 

[Roberto Pavani]

In classical coin tossing, only one sequence is realized. In MWI, all sequences are realized. The question isn't 'why is 1000 heads unlikely?', it's 'why are we in the branch with ~500 and not in the equally real branch with 1000 (or any other "unusual" distribution) ?

 

[martinbn]

In classical coin tossing, only one sequence is realized. In MWI, all sequences are realized. The question isn't 'why is 1000 heads unlikely?', it's 'why are we in the branch with ~500 and not in the equally real branch with 1000 (or any other "unusual" distribution) ?

My point was that if you repeat the toss of 1000 coins experiment we will eventually see the 1000 heads outcome.

 

[martinbn]

If there are all possibile outcome universes then there are some where born rule doesn't hold. The question is then: how come in our universe the born rule hold? Are we "special"?

In some universes the multiverse theory is wrong, so how come... This is how I see your logic. What makes you think that the Born rule is wrong in some branches?

 

[Roberto Pavani]

In a single universe, yes, once every ~2^1000 repetitions. But in MWI, that branch exists right now, at every branching. You don't need to wait, all outcomes are already real, simultaneously.

 

[PeroK]

In a single universe, yes, once every ~2^1000 repetitions. But in MWI, that branch exists right now, at every branching. You don't need to wait, all outcomes are already real, simultaneously.

Yes, in MWI there are possibly branches where people are throwing away their physics books.

However: there is no way to distinguish between MWI where there are a large finite number of branches and MWI where there are an infinite number of branches.

It could be that statistically every branch in MWI obeys the law of large numbers and looks normal.

This doesn't work with the simple analogy of coin tossing, but perhaps with more complex branching. Although, the Born rule is still a problem.

For example, I could write two computer programs. The first generates one of a countable infinity of outcomes, each with probability $1/2^n$. Every outcome is possible.

The other approximates this, but only allows the first 1000 outcomes.

It would take a long time to distinguish the output!

The fundamental problem is that neither MWI nor its extreme conclusions are testable. There may be infinite branches. Or, there may be a finite number of branches, where statistically something unusual at a macroscopic scale rarely if ever happens. How could we tell the difference?

All the hypothetical strange branches could be figments of the physicist's imagination.

 

[PeroK]

PS it's also not clear how our conscious experience arises from MWI. In a collapse interpretation, QM is a model for a single universe where we exist as conscious beings.

In MWI we ourselves are continually branching into however many descendants, each with a slightly different experience.

And, of course, in MWI the probability that you and I even exist in a given branch is close to zero. Insofar as probability makes sense in that context.

 

[martinbn]

In a single universe, yes, once every ~2^1000 repetitions. But in MWI, that branch exists right now, at every branching. You don't need to wait, all outcomes are already real, simultaneously.

And?