Why MWI cannot explain the Born rule

[Fra]

A discrete path integral over graphs.
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When you ask for the probability that the kth node, link or plaquette has the value Qo you obtain Z(Qk=Qo)/Z (Z is a partition function since we're using a Euclidean path integral), which is the discrete counterpart to QFT. I
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We have a paper under review at Foundations of Physics. If that gets accepted and we find our classical solution, then maybe I'll start a thread We can't discuss this anymore here, it's out of context.

Thanks for the hints, just a couple of quick questions :)

I got the impression from other threads you are seeking a reconstruction of the continuum in terms of a discrete model that is more "fundamental"? I symphatise because I also find the continuum starting point inherently unphysical uncountable redundance that isn't helping at all.

Does this mean you also reconstruct a "discrete" probability theory, where the measure rather spans a discrete rational subset of [0,1], constrained by complexity?

Somehow this latter thing, is I think relevant to the discussion of defining probability since it allows to avoid the issue of "infinite measurements" and frequency limits. Instead finite information might imply that the probability measure itself is discretized and not covering a continuum. This would suggest that one could actually "count" the truncated contiuum and also define measures on stuff like the space of possibilities (which is used for feynmann summation).

/Fredrik

 

[RUTA]

Thanks for the hints, just a couple of quick questions :)

I got the impression from other threads you are seeking a reconstruction of the continuum in terms of a discrete model that is more "fundamental"? I symphatise because I also find the continuum starting point inherently unphysical uncountable redundance that isn't helping at all.

Does this mean you also reconstruct a "discrete" probability theory, where the measure rather spans a discrete rational subset of [0,1], constrained by complexity?

Somehow this latter thing, is I think relevant to the discussion of defining probability since it allows to avoid the issue of "infinite measurements" and frequency limits. Instead finite information might imply that the probability measure itself is discretized and not covering a continuum. This would suggest that one could actually "count" the truncated contiuum and also define measures on stuff like the space of possibilities (which is used for feynmann summation).

Yes, the discrete structure is fundamental to the continuum structure, not a mere approximation thereto.

Our partition function (transition amplitude, Z) is defined over a countable number of graphical elements, but each element can have an uncountable number of possible field values.

 

[Fredrik]

I don't think that decomposition into subsystems is sufficient (maybe not even necessary) to produce decoherence. Consider two sets of interacting two-level systems. The Hilbert space of this composite system we describe as a tensor product of the individual Hilbert spaces. If we are only interested in the properties (observables) of one of the subsystems we may trace out the other one to produce a reduced density matrix. The reduced density matrix will generally be a mixed state but the coherence factors (off diagonal elements of the density matrix) need not vanish irreversibly. Most likely you would observe an oscillatory behaviour at a certain frequency. To observe irreversible behaviour you need to also assume that the system you are tracing out contains a large (infinite) number of degrees of freedom so that different frequencies add up to produce a decay on average.

Sounds like you know a few things about decoherence that I don't. But you specifically mention reduced density matrices, and those can't even be defined without a tensor product decomposition. I have started reading a pdf version of Schlosshauer (I'm still buying the real one) and in the intro, he describes decoherence as the system getting more and more entangled with the environment. Everything I have seen indicates that you need to consider at least two component subsystems: "the system" and "the environment".

 

[Fredrik]

What about http://arxiv.org/abs/0903.5082 which tries to derive the Born rule from...

Thanks. I intend to check it out, but I'll probably wait until I've read some more in Schlosshauer's book. I expect that I will have objections about the use of the tensor product and the use of density matrices. The former has some connection with probabilities that I don't fully understand yet (see #139), and the latter seems impossible to justify without the Born rule (see #108).

 

[jensa]

But you specifically mention reduced density matrices, and those can't even be defined without a tensor product decomposition.

Yes of course you are right, but my point with the passage you quoted was that even if we can decompose a system into subsystems (with tensor product) and look at the reduced density matrix of the particular subsystem we are interested in, you still do not necessarily get irreversible loss of coherence! Sure, the reduced density matrix will generally be of the mixed kind but will it irreversibly go towards a total loss of coherence?

This depends on what the other subsystem is. Mostly when people talk about "environment" I believe it is implied that it consists of an infinite amount of degrees of freedom (i.e. it is macroscopic). In other words, the macroscopic nature is at least as important to the concept of decoherence as the decomposition into subsystems. My general point is that you can actually remove the decomposition as a necessity.

Let me try one last time to convince you that you can get a loss of coherence even without the decomposition feature by returning to my original example of Schrödingers cat.

Let us write the general microscopic state of the macroscopic system (cat) as:

$$|\psi\rangle = \sum_i c_i(t)|i\rangle$$

where $i$ here denotes a set of labels completely characterizing the microscopic state of all particles the cat consists of (clearly a huge amount). Let us assume that we have chosen a basis in such a way that we can clearly distinguish for which microscopic state $|i\rangle$ the cat is either dead or alive. We can formally define a set of projection operators:

$$\hat{P}_\text{alive}=\sum_{i\in alive}|i\rangle\langle i|, \quad \hat{P}_\text{dead}=\sum_{i\in dead}|i\rangle\langle i|$$

We can then associate the state $\hat{P}_\text{alive}|\psi\rangle=\sum_{i\in \text{alive}}c_i(t)|i\rangle$ with a macroscopic state of the cat being alive and vice versa with the macroscopic "dead state". If we, for a moment, trust the conventional probability rule we have:

$$\text{Prob. alive}=\langle \psi|\hat{P}_\text{alive}|\psi\rangle=\sum_{i\in \text{alive}}|c_i(t)|^2, \quad \text{Prob. dead}=\langle \psi|\hat{P}_\text{dead}|\psi\rangle=\sum_{i\in \text{dead}}|c_i(t)|^2$$

So far so good, but is it possible to define a "relative phase" between the two macroscopic states "dead" and "alive"? In principle it should be clear already here that such a feat is difficult and designing an interference experiment without knowing the microscopic configuration of the cat is pretty much impossible. However, we could analyze the operator

$$\hat{P}_\text{a-d}=\sum_{i\in \text{alive}}\sum_{j\in \text{dead}}|i\rangle\langle j|$$

which connects the dead and alive subspaces and thus the object

$$\langle \psi|\hat{P}_\text{a-d}|\psi\rangle=\sum_{i\in \text{alive}}\sum_{j\in \text{dead}}c_i^*(t)c_j(t)$$

is directly related to the "coherence". Now there are two issues here: 1) The $c_i$'s are determined by the exact microscopic state (determined by initial conditions and exact many-particle hamiltonian) of which we are clearly ignorant. 2) the time scale of variation of the $c_i(t)$ is very short compared to macroscopic time scales. We thus expect this object to fluctuate wildly (both statistically and temporally) i.e. it is essentially a chaotic variable. Performing an average (coarse graining) over this object will average it out. The probabilities on the other hand must of course add up to unity and, while it may fluctuate depending on initial conditions and as a consequence of time dependence of $c_i(t)$, because it is always a positive quantity between 0 and 1 it will average to some constant. What this is supposed to illustrate is that the off-diagonal elements will vanish upon statistical and temporal averaging, and it is effectively impossible to create an interference experiment that may be used to observe the relative phase between the macroscopic states "dead" and "alive".

I have started reading a pdf version of Schlosshauer (I'm still buying the real one) and in the intro, he describes decoherence as the system getting more and more entangled with the environment. Everything I have seen indicates that you need to consider at least two component subsystems: "the system" and "the environment".

Yes he seems to credit the non-locality of quantum mechanics, which of course is related to system+environment. In fact most textbooks seem to use this notion to describe decoherence. However, I feel like this is only one specific type of decoherence (environmentally induced decoherence) and that decoherence in general can be described without the use of an external environment.EDIT: Sorry everyone for the long post..hope I didn't derail the discussion too much.

/Jens

 

[Fredrik]

The argument by Hartle allows you to replace the Born rule by the weaker rule that says that measuring an observable of a system if the system is in an eigenstate of that observable, will yield the corresponding eigenvalue with certainty.

I've been doing some more thinking about this. I still think that Hartle's argument is useless, and proves nothing, but I've realized that Gleason's theorem says something very similar to the above. It says that if $\mu$ is a probability measure on the set of subspaces on a separable Hilbert space (real or complex, and at least 3-dimensional), there exists a density operator $\rho$ such that

$$\mu(M)=\mbox{Tr}(\rho P_M)$$

where $P_M$ is the projection operator associated with the closed subspace M.

Consider the simplest possible case, i.e. when $\rho$ is a pure state $|\psi\rangle\langle\psi|$, and M is a 1-dimensional eigenspace corresponding to the eigenvalue b of an observable B. The theorem says that the only possible probability measure assigns probabilty

$$\mbox{Tr}(|\psi\rangle\langle\psi|b\rangle\langle b|)=\sum_{b'}\langle b'|\psi\rangle\langle\psi|b\rangle\langle b|b'\rangle=|\langle b|\psi\rangle|^2$$

to that eigenspace. So it certainly looks like Gleason has derived the Born rule. There are however several subtle points worth noting here.

1. Why are we looking for probability measures on the set of closed subspaces of a separable Hilbert space? A partial answer is that the closed subspaces can be thought of as representing "properties" of physical systems in QM*. This is an axiom in the quantum logic approach to QM**, but in the traditional Hilbert space approach to QM, we would have to use the Born rule to prove that this is true.

2. Where did we "put probabilities into get probabilities out"? This is explained by item 1 and the footnotes. The probability measure (which is uniquely determined by the state) assigns non-trivial probabilities to mathematical objects (closed subspaces) that are already associated with assignments of probability 1 to possible events in the real word. (This is where it's very similar to what Count Iblis claimed above).

3. This is clearly not a derivation of the sort originally envisioned by Everett. It isn't a derivation from the assumption that the state of the universe can be described by a state vector satisfying a Schrödinger equation. Gleason didn't assume that. Instead he started with the assumption that the set of "properties" is represented by the simplest possible mathematical structure that's consistent with a set of axioms that we expect all theories to satisfy. Of course, before QM was discovered, we would have guessed that all theories must satisfy a much stronger set of axioms, so this set of axioms was chosen specifically to ensure that QM (with its Born rule) qualifies as a theory.

*) What I call "properties" goes by many names in the literature, including "propositions", "elements of reality" and "experimentally verifiable statements". The last one is probably the most appropriate, since these phrases all refer to the possible results of experiments that are assigned probability 1 by QM.

**) Technically they use another set of axioms in order to associate a mathematical structure with the set of properties, but then they define a "standard" structure as one that's isomorphic to the lattice of closed subspaces of a complex separable Hilbert space

 

[jensa]

Just to conclude my somewhat tangential series of posts: What I have been describing falls under a category which Joos calls "Fake Decoherence"*, while he prefers to restrict the word decoherence to the system+environment stuff. Personally I find the term "fake decoherence" quite misleading; the effect of practical inability to observe coherence is quite real. I would prefer to call the effect itself decoherence and then use "environmentally induced decoherence" to refer to the system+environment stuff. Of course it is quite possible that for the purposes of MWI only environmentally induced decoherence is important, although I cannot see immediately why this would be so.

*) See the book "Decoherence and the appearance of a classical world in quantum theory" by Joos, Zeh, Kiefer, Giulini, Kupsch and Stamatescu in the section entitled "True, False and Fake decoherence". I personally prefer this book over Schlosshauer's.

 

[Demystifier]

*) See the book "Decoherence and the appearance of a classical world in quantum theory" by Joos, Zeh, Kiefer, Giulini, Kupsch and Stamatescu in the section entitled "True, False and Fake decoherence". I personally prefer this book over Schlosshauer's.

This is an excellent book too. However, I cannot find the section you mention above. Can you help me (section number, page number, contributor name, or something like that)?

 

[jensa]

This is an excellent book too. However, I cannot find the section you mention above. Can you help me (section number, page number, contributor name, or something like that)?

I have the second edition where it is in Chapter 3 "Decoherence Through interaction with the Environment" by Joos, section 3.4.3. Perhaps it is absent in the first edition?

Edit: You can find the second edition by searching for it in Google Books.

 

[Demystifier]

I have the second edition where it is in Chapter 3 "Decoherence Through interaction with the Environment" by Joos, section 3.4.3. Perhaps it is absent in the first edition?

Edit: You can find the second edition by searching for it in Google Books.

Yes, I have the first edition. It does not even contain Sec. 3.4.3.

The preview by Google Books gives first 101 pages, which does not cover Sec. 3.4.3 either.

 

[jensa]

That is odd. I can view it just fine. What happens if you use this link:

http://books.google.com/books?id=6eTHcxeNxdUC&lpg=PP1&pg=PA175#v=onepage&q=&f=false

 

[Demystifier]

That is odd. I can view it just fine. What happens if you use this link:

http://books.google.com/books?id=6eTHcxeNxdUC&lpg=PP1&pg=PA175#v=onepage&q=&f=false

It gives me no more than 101 pages plus the back cover. How many pages can you see?

 

[jensa]

It gives me no more than 101 pages plus the back cover. How many pages can you see?

Oddly I can see pages 134-180 + the first few pages with table of contents etc + backcover. I am a little confused as to how the preview feature works...perhaps its either 101 first pages or part of a chapter of your choosing.

You could try amazon as well, at least I am able to look at that particular section (searching for page 175).

https://www.amazon.com/dp/3540003908/?tag=pfamazon01-20

 

[Demystifier]

Oddly I can see pages 134-180 + the first few pages with table of contents etc + backcover. I am a little confused as to how the preview feature works...perhaps its either 101 first pages or part of a chapter of your choosing.

Perhaps different pages are allowed in different countries?
It is the fact that some web pages have different appearance in different countries. For example, on the top of page of this Forum I often see an advertisement for an IQ test in CROATIAN language (I live in Croatia).

 

[Fredrik]

Google books uses cookies for something. (I'm not sure what). So I would at least try clearing out the cookies from the browser, and also make sure that it allows cookies from that site.

 

[Demystifier]

*) See the book "Decoherence and the appearance of a classical world in quantum theory" by Joos, Zeh, Kiefer, Giulini, Kupsch and Stamatescu in the section entitled "True, False and Fake decoherence". I personally prefer this book over Schlosshauer's.

Now I have noticed that Schlosshauer also explains the notions of true and fake decoherence, but does not mention false decoherence. Could you be so kind to explain to me (by your own words) what false decoherence is and what is the difference between false decoherence and fake decoherence?

 

[jensa]

Now I have noticed that Schlosshauer also explains the notions of true and fake decoherence, but does not mention false decoherence. Could you be so kind to explain to me (by your own words) what false decoherence is and what is the difference between false decoherence and fake decoherence?

Well, I hope I didn't give the impression that I am some kind of expert in decoherence, I'm certainly not. But I can try to explain what I understand from the the book. Joos describes false decoherence as when "Coherence is trivially lost if one of the required components [of the wavefunction] disappears". I'm not sure really how to interpret this sentence, but he gives two examples of such cases 1) Relaxation 2) When we are considering only a subset of all available states.

Let's start with what I think is the easiest one no. 2). It is sometimes convenient to use a truncated model to describe a system. For example, many qubit implementations are in fact multilevel systems although we choose only to consider the two lowest energy levels as an operational subspace. It is clear that there is a certain probability of "leakage" out of this subspace, this process also produces suppression of the off-diagonal elements of our density matrix, however it also does not conserve probability (within the subspace). Coherence is not really lost here, it's just a consequence of our truncated model.

As for Relaxation, it should originate from interaction with an environment (since energy is not conserved). So I suppose the point he is trying to make is that interaction with environment can produce "true decoherence" which is a pure quantum effect, as well as an exchange of energy which also occurs for classical systems. Relaxation, although it also suppresses the off-diagonal elements of the density matrix, should be distinguished from "true decoherence".



If you would like, I could scan the pages and send them to you. Would be nice to see what somebody else makes of it.

 

[Demystifier]

Thanks jensa, it was a quite clear explanation.

 

[Dmitry67]

Here is a radical view. What is Born rule does not need to be derived from MWI? What if Born rule is not objective but subjective?

Usually we look it from the inside (frogs perspective): I am looking at the world and I observe more frequent event than events with lower probability. How is it explained by MWI, while all possible outcomes exist in omnium? I think that this question is incorrect. We should ask an inversed question instead: We know that all branches exist, why there is “more” conscious observers with the higher “intensity of existence?”

I can give you another example. Say, we have a very powerful computer and we know TOE. So we were able to calculate the “wavefunction of the Universe” from moment 0 (initial conditions at BB) to, say, 1 second. Not just one “branch”, but omnium at whole, the global solution. There are no conscious observers, however. But do we need a Born rule as soon as we know the global solution?

In fact, once you look at the Universe from ‘bird’s view’, having the solution of the evolution of the omnium, there is no Born rule. We have a solution, that’s it. Born rule appear when we try to jump inside the Universe, converting from a single birds view to some arbitrary frog’s view. At the moment of the choice you already implicitly use the Born rule, choosing “more intensive” branch. In such case the Born rule – which is so obvious when we look around – is nothing more then an illusion, created by our consciousness, similar to so obvious flow of time

 

[Demystifier]

Dmitry, I don't see that this view explains the Born rule in any quantitative sense.

 

[Dmitry67]

Well, my idea was that it Born rule is just an illusion created by our consiousness.
It is a part of theory of consciousness, not a part of physics.

 

[Dmitry67]

Suppose, there are 2 outcomes of some experiment: (F)requent (90%) and (R)are 10%
We repeat this experiment 3 times.
Obviously, there are 8 branches, begiining from F-F-F to R-R-R
Why do we see F events more often then R events?

Check the picture.
There is no difference between F-F-F and R-R-R at the birds view.
In the frog's view (asking "what observer will see?") you must define the basis (a point on my diagram). So we chose (for some mysterious reason, I agree) the point showed by red cross and ask: why in the history of that observer there are more Fs then Rs?

So Born rule is not about physics, it is about how consicous observers are chosen.

 

[pellman]

So we chose (for some mysterious reason, I agree) the point showed by red cross and ask: why in the history of that observer there are more Fs then Rs?

The "mysterious reason" is the crux of the whole question.

Although the question "why in the history of that observer there are more Fs then Rs?" is somewhat troublesome. There are 8 potential observers. One of them sees F-F-F. And we chose to label that one with the X.

But what is special about that observer?

 

[Count Iblis]

In the bird's view there is one timeless multiverse. What we experience as time evolution is just a unitary mapping from one sector of the multiverse to another. At the level of the whole multiverse, time does not exist.

In this case you have some sector containing the initial observer |i>, which maps under the time evolution operator to a |F> + b |R>. It is this time evolution operator which is subjective to the observer. In the bird's view time does not exist and you could have considered any other unitary operator.

For the observer things are different. An observer is essentially an algorithm that processes information. To define the algorithm as it is in any particular computational state, you have to list the outputs for each possible input. This defines an operator which is essentially a sort of coarse grained Hamiltonian which generates the time evolution experienced by the observer.

You cannot just define an observer by looking at a sequence of the states it runs through, because you can then map that to the states of a clock which does not execute a nontrivial algorithm. You have include the information about counterfactual inputs and the corresponding counterfactual outputs, which will not be mapped correctly from the observer to the clock.

Note that the information about the correct Hamiltonian is not present in the bird's view, because if H is the "correct" Hamiltonian, the whole multiverse satisfies an equation like:

H|psi> = 0

There is then no way you can construct H back from|psi>. You can replace H by any other Hamiltonian that except for |psi> has different eigenvectors.

 

[Dmitry67]

The "mysterious reason" is the crux of the whole question.

Although the question "why in the history of that observer there are more Fs then Rs?" is somewhat troublesome. There are 8 potential observers. One of them sees F-F-F. And we chose to label that one with the X.

But what is special about that observer?

I can provide a part of an answer by giving another example:

the moment of "NOW"

This is also a big red cross, a moment very special to consciousness. But as we believe in block time, physically there is absolutely nothing special about NOW!

So if our consicouness can break symmetry on T axis, creating an illusion of a special moment 'NOW', then why it can't create another illusion on B-axis (Branch axis. Yes, I am aware that it is not an axis at all)

 

[Dmitry67]

You cannot just define an observer by looking at a sequence of the states it runs through, because you can then map that to the states of a clock which does not execute a nontrivial algorithm. You have include the information about counterfactual inputs and the corresponding counterfactual outputs, which will not be mapped correctly from the observer to the clock.

This sounds very interesting but I am lost at the part I quoted
Could you elaborate?

 

[Demystifier]

Well, my idea was that it Born rule is just an illusion created by our consiousness.
It is a part of theory of consciousness, not a part of physics.

Does this mean that such a theory does not need to have a quantitative form?

 

[Dmitry67]

Yes, but we can use Born rule for practical purposes. As a rule of thumb

For example, NOW we are 13.7 billion years from the BB, or 8.2*10^60 plank times from the Big Bang

Can you derive somehow this integer dimensionless number (8.2*10^60) from BM?

 

[Demystifier]

You can use the Born rule as a rule of thumb.
However, if it is the best that MWI can do about the Born rule, then BM is much more successfull than MWI because in BM the Born rule is much more than a rule of thumb.

 

[Demystifier]

Can you derive somehow this integer dimensionless number (8.2*10^60) from BM?

Of course I can't. If you give me a theory that can, I will accept that theory immediately. Moreover, if that theory turns out to be incompatible with BM, I will reject BM as well. But in the meantime, I will keep BM as the most attractive possibility currently known.

 

[Dmitry67]

You can use the Born rule as a rule of thumb.
However, if it is the best that MWI can do about the Born rule, then BM is much more successfull than MWI because in BM the Born rule is much more than a rule of thumb.

Yes, there is some advantage, but it is too weak.
Observer is point is some space, and history is a curve.
While BM limits the number of curves to 1, it does not limit the curve to a single point. So I think the claim 'MWI does not explain what is observed now' is to the full extent applicable to BM. It just slightly limits the number of degrees of freedom, where we can put that red cross.

 

[Dmitry67]

Of course I can't. If you give me a theory that can, I will accept that theory immediately. Moreover, if that theory turns out to be incompatible with BM, I will reject BM as well. But in the meantime, I will keep BM as the most attractive possibility currently known.

Hm. Do you think that symmetry breaking (NOW vs not-NOW) must be explained by physical theiry? It sounds very Smolin-like.

 

[Dmitry67]

Note that the information about the correct Hamiltonian is not present in the bird's view, because if H is the "correct" Hamiltonian, the whole multiverse satisfies an equation like:

So are you saying that some properties visible in the frog's view can't be derived mathematically, even in principle, from birds view? So even when we have an ultimate TOE equation, we can't explain everyhting we observe?

 

[Demystifier]

Hm. Do you think that symmetry breaking (NOW vs not-NOW) must be explained by physical theiry?

Maybe yes, maybe not.

 

[toho]

Regarding deriving the Born rule from the MWI, here is a conjecture:

The Born rule is the only probability measure, Q, consistent with the criteria that
i) P(A)=0 iff L2-norm of psi over A = 0, for every A in the limit as time -> infinity
ii) Q is a function of psi, for any psi consistent with QM

I.e. if you want a probabilistic interpretation of the MWI interpretation of QM, it has got to be the Born rule. (It think it would be possible to relax the t->inf critera, btw)