Understanding MWI: A Newbie's Guide to Quantum Physics and the Multiverse

[Fra]

i'm probalby doing a mistake by adding this but...

you do not get out the same results than when you keep the electron wavefunction as a wavefunction and not a statistical mixture of positions.

We attack this differently but I think I see Vanesch point here. I am not quite prepared to present my view consistently yet (I'm working on it) but my information theoretic ideas to explain the connection between statistical (impure) mixtures and pure mixtures depends on how we define "addition of information".

I technically see different measurements, belonging to different probability spaces. And the exact relation between the spaces is needed to define addition of information.

Since normally conditional probabilities is defined like

$$P(A|B) := \frac{P(A \cap B)}{P(B)}$$

The question is what P(A|B) supposedly means unless they belong to the same event space?

This is IMO a _part_ key to explain the reason for different ways to "add information". And when we are dealing with systems of related "probability spaces" in between which we make transformations, then it does matter in which space we make the addition. This is something that isn't analysed to satisfaction normally. Normally the defining relation between momentum and position are postulated, one way or the other. This postulates away something I personally think there is a better explanation to.

I think there is a way to grow a new space based on the uncertainty in the existing one. New relations are defined in terms of patterns in the deviations of existing relations, and relations are "selected" as per a mechanics similar to natural selection. The selected relations implements an efficient encoding and retains a maxium amount of information about the environment in the observer.

The limit is determined by the observers information capacity, and beyond that the only further "progress" that can be made is for the observer/system to try to increase it's mass. Here I expect a deep connection to gravity.

Constraining the information capacity is also I think the key to staying away from infinities and generally divergent calculations. There will be a "natural cutoff" due to the observers limited information capacity.

I'm sorry if this makes no sense, but I hope that given more time I'll be able to put the pieces together, and it's designed to be all in the spirit of a minimum speculation information reasoning that I personally consider to be the most natural extension to the minimalist interpretation.

/Fredrik

 

[Hurkyl]

I technically see different measurements, belonging to different probability spaces.

I'm skeptical. Example, please?

 

[Fra]

Hmmm... you should be skeptical of course :) And I haven't yet finished this enough to present this properly, this is why I should perhaps be quiet, but consider different "measurements" (operators if you like), then each such measurements can loosely speaking be though of to span a probability space of it's own (the event space defined by the span of the possible events) - but how does different measurements relate to each other?

Then usually the relation between these operators are postulated as commutator relations etc, but if you instead consider the measuremetns or "operators" are dynamical objects then new possibilities open up. But how has the world evolve? does nature postulate relations? I think there is a more natural way view the relations.

However I can't give a complete explanation to this atm, that's why I perhaps should be quiet. I was curious if Reilly recognizes any of this or not. But from his last comments I fear not.

/Fredrik

 

[Fra]

comment on immature ideas

An example would be measuring position and measuring momentum.

We could postulate existence of two different observables and then postulate their relation, or do we postulate one observable and try to see if there is a principle that induces naturally complementing observables in a context of the first one an as needed basis?

If we take the set of distinguishable configurations (x) to make up configuration space, then all we can do is to see what the relative frequency is of the retained history. But what if we are to try to find a transformation, and use part of our memory (state of the observers microstructure in the general case) to store a transformed pattern which defines a new event space, could that give us lower uncertainty?

This transformation then relates two event spaces (probability spaces).

We can use a Fourier transform to define momentum (p) and try to find momentum patterns, but why Fourier transform? Is there a principle which renders this transform unique?

Given a limited information capacity, it seems the possible transformations must also be limited and transformation/patterns will be induced spontaneously. I have no answer on this, but my but feeling says there is one and I'm looking for it.

But there is intercommunication between the spaces, they can shrink or inflate at the expense of the other, as well as grow new dimensions, and there is supposedly a particular configuration that is "most favourable" in a given environment, where the observers microsctructure is selected for maximum fitness given the limited information capacity.

What I am trying but haven't yet succeeded with is to build structures starting from a minimalist starting point of a boolean observable which can be seen as an axiom claiming there is a concept of distinguishability defined to each observer. From that complexity and new structures are built as the observer is interacting with the environment. The relations formed are selected by the interaction with the environment.

So higher dimensions are built as extensions and aggregates from a basic notion of distinguishability. The dimensions and structures are emergent in the learning/evolution process. Structures selected represents an efficient encoding of information. Inefficient structures are destabilised and will loose information capacity because they consistently fail to keep up with the environmental perturbations.

Anyway the idea is that then all observables are connected by a natural connection, and we can "add information" from any part of the structure but the actual "additions" would have to be made by respecting the selected connections. It may give rise to apparently twisted statistics, but when seen from the unified view, with all the information transformed into the original event space, I think it will take on a simple view.

This when understood isn't just "an interpretation" that makes no difference, it could be realized as a complete self correcting strategy, which is why I personally like it.

My point in mentioning admittedly incomplete and in progress ideas is only to provoce the questions and possible ways in the spirit of information ideas and talking about going back to probabilit theory. I still think we choose to focus on different things.

I have no idea how long time it takes to mature this. I wouldn't be surprised if it takes several years considering that this is my hobby time, but I know there are other who share the ideas, and I look forward to seeing more work along these lines so I like to encourage any ideas relating to this, because I realize it's a minority idea.

/Fredrik

 

[Fra]

small note

P(X) = P(X|A) P(A) + P(X|B) P(B) + P(X|C) P(C)

and that *doesn't work* in quantum theory if we don't keep any information (by entanglement for instance) about A, B or C. So when we have the wavefunction expanded over A, B and C, we CANNOT see it as giving rise to a probability distribution.

I guess this obvious but a reason for this can still be seen within normal probability is because it's using yet another hidden implicit condition, namely the microstructure/background(M) that A B and C refers to. Making this dependence explicit we get

P(X|M) = P(X|A) P(A|M) + P(X|B) P(B|M) + P(X|C) P(C|M)

but
$$P(X) = \sum_{M} P(X|M)P(M)$$

If this is a sum over all possible background.

and thus
$$P(X) \neq P(X|A) P(A|M) + P(X|B) P(B|M) + P(X|C) P(C|M)$$

unless A B and C make a complete partitioning, which is exactly related to the selection of a particular background M.

I ultimately envision that this background makes up the identity of the observer, and contains a selection of retained history.

The only real problem here is how does one in a sensible way resolve the set of all possible backgrounds? This is what I've been given some thought and this problem is what gives rise to dynamics. It is not possible to resolve all possible backgrounds just like that, since it somehow must involve information storage and computing. This is why i think that summation must be intrisically related to change and ultimately time. Given any set of microstructures, one can always raise the same complaint. Which suggests that a sensible solution a non-unitary evolution.

The ultimate reason for this is that we do not easily "measure our probability space" without getting into silly things like infinite experiments and infinite data storage.

/Fredrik

 

[reilly]

vanesch:

I'm looking at the mathematical properties of wave equation solutions. In particular, that wave function norms are preserved by wave equations guarantees that the absolute square of any normalized wave function can be viewed as a probability measure, we'll call it QP for quantum probability, energy density for others, intensity, or what have you. I agree that, particularly in classical physics, the probability density associated with a wave equation has only a formal meaning, and is usually differently interpreted.

Thus, it seems to me that the QP is indeed always a probability density, given it's mathematical properties.. Why not?

The difference between a classical wave superposition and a quantum interference is the following: in the classical superposition of a wave from slit 1 and a wave from slit 2, we had not a PROBABILITY 50% - 50% that the "wave" went through slit 1 or slit 2, it went physically through BOTH. In classical physics, intensity has nothing to do with probability, but just the amount of energy that is in a particular slot. HALF of the energy went through slit 1 and HALF of the energy went through slit 2 in the classical wave experiment.

But we cannot say that "half of the particle" went through slit 1 and "half of the particle" went through slit 2, and then equate this with the *probability that the entire particle* went through slit 1 is 50% and the probability that it went through slit 2 is also 50%, because if that were true, we wouldn't have a LUMPED impact on the screen, but rather 20% of a particle here, 10% of a particle there, etc...

Why would I want to say, "half the particle went through slit 1,...?" Such a statement makes no sense, except, perhaps, as highly figurative language.
In the symmetrical case, the initial conditions for slits state that the probability to find A in the left slit is equal to the probability to find A in the right slit. It's either there or here, but not in both places. The way we describe such a case is with probability in a specified portion of configuration space. As I noted above, I've been talking mainly about the mathematical parallels amongst wave equations. The interpretation of the equations and solutions can be all over the map.

[/QUOTE]

Now, it is so that the application of the Born rule to a quantum optics problem gives you in many cases just the classical intensity as a probability, which instores somehow the confusion between "classical power density" and "quantum probability" but these are entirely different concepts. The classical power density has nothing of a probability density in a classical setting, and a probability has nothing of a power density in a quantum setting.

Right, I agree. As I noted above, I've been talking mainly about the mathematical parallels amongst wave equations. The interpretation of the equations and solutions can be all over the map.

The classical superposition of waves has as much to do with probabilities as the superposition of, say, forces in Newtonian mechanics. Consider the forces on an apple lying on the table: there's the force of gravity, downward, and there's the force of reaction of the table, upward. Now, does that mean that we have 50% chance for the apple to undergo a downward fall, and 50% chance for it to be lifted upward ? This is exactly the same kind of reasoning that is applied when saying that two classical fields interfere is equivalent to two quantum states interfering. The classical fields were BOTH physically there (just as the two forces are). Their superposition gives rise to a certain overall field (just as there is a resultant force 0 on the apple). At no point, a probability is invoked.

But in the quantum setting, the wave function is made up of two parts. If this "made up of two parts" is interpreted as a probability at a certain point, you'd be able to track the system on the condition that case 1 was true, and to track the system on the condition that case 2 was true. But that's not what the result is when you compute the interference of the two parts. So when the wave function was made up of 2 parts, you CANNOT consider it as a probability distribution. But when the interference pattern is there, suddenly it DID become a probability distribution. THAT'S where a sleight of hand took place.

What are the two parts?

Below, I beg to differ. That is, I said "effectively" and "sorta equivalent" to two beams -- we are talking a metaphor, helps some of us in understanding the two slit situation. That is, the two slit expt.. is much like a scattering one -- the mathematics describing the two slit QM experiment is very similar to that for scattering..-- look at the Lippman-Schwinger eq. for example. Again, it's as if there were two beams emanating from the slits.

Absolutely. But classically, that would mean that BOTH beams are physically present, and NOT that they represent a 50/50 probability!

This is ONLY true when ACTUAL measurements took place! You cannot (such as at the slits in the screen) "pretend to have done a measurement" and then sum over all cases, which you can normally do with a genuine evolving probability distribution.
If the wave function decomposes, at a time t1, into disjoint states A, B and C, then, if it were possible to interpret the wave function as a probability density ALL THE TIME, it would mean that we can do:

P(X) = P(X|A) P(A) + P(X|B) P(B) + P(X|C) P(C)

and that *doesn't work* in quantum theory if we don't keep any information (by entanglement for instance) about A, B or C. So when we have the wave function expanded over A, B and C, we CANNOT see it as giving rise to a probability distribution.

Now, of course, for an actual setup, quantum theory generates a consistent set of probabilities for observation in the given setup. But the way the wave function behaves *in between* is NOT interpretable as an evolving probability distribution.
So the wave function is sometimes giving rise to a probability distribution (namely at the point of "measurements"), but sometimes not.
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
Now, you can take the attitude that the wave function is just "a trick relating preparations and observations". Fine. Or you can try to give a "physical picture" of the wave function machinery. Then you're looking into interpretations.

Conditional probabilities are a bit tricky at the wave function level, but not so much at the density matrix level. In your example,

P(X) = | WA(X)+WB(X) +WC(X)| ^^2, where WA is the wave function for state A.

Or, P(X) = WA^^2+ 2*WA*WB + …..

(I've dropped the X and the complex conjugate signs)

Now this is a quadratic form that can be diagonalized, which means three disjoint states with probability densities once they are properly normalized. These states are linear combinations of all states, A,B,C, These disjoint states are nicely described by the standard Bayes Thrm , which you site above, and contain interference terms between A,B,C.

Regards,
Reilly

 

[vanesch]

Thus, it seems to me that the QP is indeed always a probability density, given it's mathematical properties.. Why not?

Well, for the very reason I repeat again. If we take the wavefunction of the particle, and we let it evolve unitarily, then at the slit, the wavefunction takes on the form:
|psi1> = |slit1> + |slit2>
which are essentially orthogonal states at this point (in position representation, |slit1> has a bump at slit 1 and nothing at slit 2, and vice versa).

Now, if this is to have a *probability interpretation*, then we have to say that at this point, our particle has 50% chance to be at slit 1 and 50% chance to be at slit 2, right ?

A bit later, we evolve |psi1> unitarily into |psi2> and this time, we have an interference pattern. We write psi2 in the position representation, as:
|psi2> = sum over x of f(x) |x> with f(x) the wavefunction.

This time, we interpret |f|^2 as a probability density to be at point x.

Now, if at the first instance, we had 50% chance for the particle to be at slit 1, 50% chance to be at slit 2, then it is clear that |f|^2 = 0.5 P(x|slit1) + 0.5 P(x|slit2), because this is a theorem in probability theory:

P(X) = P(X|A) P(A) + P(X|B) P(B)

if events A and B are mutually exclusive and complete, which is the case for "slit 1" and "slit 2".

But we know very well that |f|^2 = 0.5 P(x|slit1) + 0.5 P(x|slit2) is NOT true for an interference pattern!

So in no way, we can see |psi1> as a probability density to have 50% chance to go through slit 1 and 50% chance to go through slit 2.

Conditional probabilities are a bit tricky at the wave function level, but not so much at the density matrix level. In your example,

P(X) = | WA(X)+WB(X) +WC(X)| ^^2, where WA is the wave function for state A.

Or, P(X) = WA^^2+ 2*WA*WB + …..

(I've dropped the X and the complex conjugate signs)

Now this is a quadratic form that can be diagonalized, which means three disjoint states with probability densities once they are properly normalized. These states are linear combinations of all states, A,B,C, These disjoint states are nicely described by the standard Bayes Thrm , which you site above, and contain interference terms between A,B,C.

Regards,
Reilly

The point is that a pure state, converted into a density matrix, after diagonalisation, always results in a TRIVIAL density matrix: zero everywhere, and a single ONE somewhere on the diagonal, corresponding to the pure state (which is part of the basis in which the matrix is diagonal).

As such, your density matrix will simply tell you that you have 100% probability to be in the state...

If you don't believe me, for a pure state, we have that rho^2 = rho. The only diagonal elements that can satisfy this are 0 and 1. We also have that Tr(rho) = 1, hence we can only have one single 1.

 

[Fra]

the wavefunction takes on the form:
|psi1> = |slit1> + |slit2>
which are essentially orthogonal states at this point (in position representation, |slit1> has a bump at slit 1 and nothing at slit 2, and vice versa).

I don't see the deduction that they are orthogonal? I consider this an assumption originating from the use of "classical intuition", not something we know.

because this is a theorem in probability theory:

P(X) = P(X|A) P(A) + P(X|B) P(B)

if events A and B are mutually exclusive and complete, which is the case for "slit 1" and "slit 2".

I agree completely that it's difficult to find the proper interpretation, but before giving up I have two objections.

(1) Assumption of mutual exclusive options.

In general we have
$$P(x|b \vee c) = \left[P(x|b)P(b) + P(x|c)P(c) - P(x|b \wedge c)P(b \wedge c) \right]\frac{1}{P(b \vee c)}$$

(2) Ambigousness in mixing information from different probability spaces.

Even accepting the forumla above, it seems non-trivial to interpret this in terms of

$$\psi\psi^* = \frac{|b|^2|\psi_b|^2 + |c|^2|\psi_c|^2 + 2Re(bc^* \psi_b \psi_c^*) }{\sum_x |b\psi_b(x) + c\psi_c(x)|^2}$$

IMO, x and b does not belong to the same event space, considered as "simple sets", therefor the definition of P(x|b) needs to be analysed.

P(x|b) is defined only if the conjunction of x anb b i defined.

So there is a problem here, I agree with Vanesch. But I think that the solution is to define the relation between the two spaces where x and b belongs. This relation will provide a consistent definition of P(x|b). Right not P(x|b) is intuitive, but the formal definition is not - it is dependent on a background defining the relation between the two observables. This is where I am personally currently looking for the connection that will restore a consistent "probabilistic" reasoning. I think we both need a revision of priobability theory itself, in the sense that he probability space NEED to be allowed to by dynamical, anything else just seems hopeless AND we need to understand how the relations between observables are formed. Postulating relations are guesses, I suggest this guessing is made more systematical.

/Fredrik

 

[vanesch]

I don't see the deduction that they are orthogonal? I consider this an assumption originating from the use of "classical intuition", not something we know.

No, it is pretty simple and straightforward. Consider the slits on the x-axis: slit one is from x = 0.3 to x = 0.4 and slit 2 is from x = 1.3 to 1.4.
The wavefunction at the slit, psi, will be a function that is zero outside of the above intervals. So we can write psi(x) as psi1(x) + psi2(x), where psi1(x) will have non-zero values between 0.3 and 0.4, and be 0 everywhere else, while psi2(x) will be non-zero only between 1.3 and 1.4, and zero everywhere else.

As such, psi1(x) and psi2(x) are orthogonal, because integral psi1(x) psi2(x) dx is 0.

I agree completely that it's difficult to find the proper interpretation, but before giving up I have two objections.

(1) Assumption of mutual exclusive options.

In general we have
$$P(x|b \vee c) = \left[P(x|b)P(b) + P(x|c)P(c) - P(x|b \wedge c)P(b \wedge c) \right]\frac{1}{P(b \vee c)}$$

Yes, but if the wavefunction was to be interpreted as a probability density, then there was 50% chance for psi1 and 50% chance for psi2 and nothing else, because, exactly, psi1 and psi2 are orthogonal. In other words, we can see them as elements of an orthogonal system in which we apply the Born rule (that's what it means, to give a probability density interpretation to a wavefunction: pick an orthogonal basis, and apply the Born rule).
The events corresponding to two orthogonal states are mutually exclusive.

(2) Ambigousness in mixing information from different probability spaces.

Even accepting the forumla above, it seems non-trivial to interpret this in terms of

$$\psi\psi^* = \frac{|b|^2|\psi_b|^2 + |c|^2|\psi_c|^2 + 2Re(bc^* \psi_b \psi_c^*) }{\sum_x |b\psi_b(x) + c\psi_c(x)|^2}$$

IMO, x and b does not belong to the same event space, considered as "simple sets", therefor the definition of P(x|b) needs to be analysed.

I'm fighting the claim that the wavefunction can ALWAYS be interpreted as a probability density. I'm claiming that this statement is not true and that at moments, you CANNOT see the wavefunction as just a probability density. You can in fact ONLY see it as a probability density when "measurements" are performed, but NOT in between, where it has ANOTHER (physical!) meaning, in the sense of things like the classical electromagnetic field. As such, one cannot simply wave away interpretational issues such as wavefunction collapse as just "changes of probability of stuff because we've learned something".

I'm trying to show here that the wavefunction is, most of the time, NOT to be seen as a probability density, but "jumps into that suit" when we say that we do measurements.

We take as basic example the 2-slit experiment, and I'm trying to show that when the wavefunction is at the two slits, it cannot be seen as just saying us that there is 50% chance that the particle went through slit 1 and 50% chance that the particle went through slit 2, but rather that there was something PHYSICAL that went through both, in the same way as a classical EM wave goes through both and doesn't surprise us when it generates interference patterns. The classical EM wave that goes through both slits is NOT to be seen as 50% chance that the lightbeam goes through the first hole, and 50% chance that the lightbeam goes through the second hole, and in the same way, the wavefunction at that point can also not be seen like that.

Now, saying that "it still might be a probability density, but not in the probability space in which we are looking at our results" is an equivalent statement to "it wasn't, after all, a probability distribution", because after all, things are a probability distribution only if they are part of the one and only "global" probability space of events.

ALL quantum weirdness reduces to the fact that quantum-mechanical superposition is NOT always statistical mixture. This is shown here in the double-slit experiment, but it is also the case in more sophisticated situations, including EPR experiments.

It is, for me, the basic motivation to look upon quantum theory through MWI goggles in fact: there, it is obvious that superposition is not statistical mixture ; as such, this view avoids one to fall into the most serious trap, IMO, in looking at quantum theory: namely to interpret too lightly a superposition as a statistical mixture. It is also the reason why MWI is fundamentally different from the "universe of possibilities" that might be postulated in a random but classical universe, and many people don't seem to realize this.

 

[Fra]

> it is obvious that superposition is not statistical mixture

Right. In the way you refer to this, this is clearly correct and I see what you are saying. It would be foolish to argue on this. I completely understand the different between superposition of wavefunction vs incoherent mixtures of classical probability distributions.

But we at least seem to disagree on how to resolve this and what conclusions to draw, or part of the problem is I think that _maybe_ we need terminology and maybe we misunderstand each other, because were both trying to extend the current description.

If we are going right by the book I would say that the Kolmogorov probabilitiy theory and it's axioms are an imperfect choice of axioms to describe nature. The most severe objection is howto attach the probability space to observation. This space is (in standard QM) not measured. Instead there are a lot of silly ideas of infinite measurements. This is one problem, but there are more of them.

Maybe I caused the confusion, but IMO there difference lies not in probability theory mathematically, but in the application of the axioms to reality. It depends on how you see it. As you know there has been philosophical debate of the meaning of probability, having nothing to do with quantum mechanics. I'm relating to this as well.

Now, saying that "it still might be a probability density, but not in the probability space in which we are looking at our results" is an equivalent statement to "it wasn't, after all, a probability distribution", because after all, things are a probability distribution only if they are part of the one and only "global" probability space of events.

You can put it that way. We are getting closer.

What I am suggesting is that there never exists a certain probability space, because the probability space is grown out of the retained history. But one can actually still define a conditional probability, that is defined, not as the predicted relative frequency of an event - because this can never be known, only guessed - rather this conditional probability is technically and expectation of a probability, based on an expected probability space. This is becauase the set of events and their relations are generally uncertain and changing.

I guess what I am saying is that in the way I try to see things, I see no such thing as a definite probability, for the reason that neither the probability itself, nor the probability space can be measured with infinite precision. One way to appreciate this is if you think that the observers ability to retain information is limited.

My standpoint has originated from trying to understand the process of inducing a probabiltiy space from data. This is related to datacompression, but also to responsetimes to change the estimates time. In essence the probability space is under constant "drift", as the observers memory is remodeled. The dynamics of the remodelling is like a movie beeing played, that is projected from the environement. The observer can never ever determined a exact probability of an external event, it can only deduce subjective expectations of effective proabilitties - which is better see as a bet distribution in a gaming perspective.

To not mix up language, we can eithre forget about probabilities and invent new lables for it. But the thing is that it will still end up closely related to a probability theory, just a relational one.

I magine that even space is structures and are spontaneosuly formed in observers memory (and since I think Zurek's idea that "What the observer knows is inseparable from what the observer is" is dead on), this is also responsible for "physical structure formation". And in this context, the particle zoo and the spacetime itself should be treated in a in principle equal basis.

Sometimes I wonder if the different universes the MWI people talk about are either the set of all possible "images of universes projected on observers" considering all possible "observers"? If so, we may be closer than what it seems. If that is so, then what I am saying is the the different universes do interact. but calling the universes is a very strange terminology IMO - I would call _that_ the projection of the unknown(some may call this the universe) to the observer. Wether this match is correct or not, I am not entirely sure.

I guess when all of these ideas are worked on, testable predictions will come and then it's probabably also easier to compare them? What I envision, is not just an interpretation, when it's finished I expect a lot of things to pop out. But it seems not so many people are actually spending time on these things, otherwise it's a mystery why not more has been accomplished.

/Fredrik

 

[Fra]

Vanesh, in your view... do you ever attempt to explain where the complex amplitude formalism comes from, or is it just taken to be a fact, and you are trying to interpret it?

What I suggest, is that there may ultimately be an explanation to exactly what a superposition means. And I suggest that it can be seen as a statistical mixture, but not of probability distributions in the one and same space, but rather in a related space. The easiest example is the momentum space with a phase. These two spaces are related by a Fourier transform - but why Fourier transform? Is there an answer to this? I think so.

What is a superposition in one space, is just a simple mixture in another space. so what I tried to suggest above is that in P(x|b or c)

The formula
$$P(x|b \vee c) = \left[P(x|b)P(b) + P(x|c)P(c) - P(x|b \wedge c)P(b \wedge c) \right]\frac{1}{P(b \vee c)}$$

Does NOT apply if the condition doesn't live in the same space as x does. So probability theory here does not really fail, it's possibly just IMO misapplied.

What I'm suggesting is thus that the operation of making a union of conditions - defined in a related space, needs to be transformed to make sense. This I see analogous to the fact that you need to parallelltransport vectors from different tangentspaces before you can add them. Adding vectors from different tangentsspaces by coordinate is nonsense.

I am working on this and have not completeed the formalism yet, but maybe you see that there is at least a possibility to accomplish this? If I make it or not will be evident in time. But I am very optimistic.

/Fredrik

 

[Fra]

So I see two issues.

1) the superposition vs mixture issue can *I think* be solve by transforming the events to the same event space before making the union of the condition. However to understand the consistency of reasoning of tihs approach (ie to motivate it) I think one needs to consider the uncertainty of the probabiltiy space in a larger context which actually connects to the second point

2) The uncertainty in establishing your probability space, and how this is constrained by the memory capacity, and the implication that incoming information must be balanced by releasing information, unless the "mass" is increasing... which can happen, and it can also shirnk... which is sort of the third issue... but there may be a way to wrap this up.

At minimum and early level this should unless it's a total failure

1) explain the mystery of the "complex wavefunction" and how it can be introduced from first principles as something unique

2) explain the relation between mass and information capacity and how it relates to inertia

3) Explain the arrow of time as a sort of simple diffusion over a network of related probabiltiy spaces.

I think this can unite standard formalism from statistical mechanics with the QM stuff, and probably get gravity for free.

Either it works, or it doesn't.

/Fredrik

 

[vanesch]

Vanesh, in your view... do you ever attempt to explain where the complex amplitude formalism comes from, or is it just taken to be a fact, and you are trying to interpret it?

Yes. I take it as given, and I try to make sense of it. That's why I call it "interpretation of quantum theory", and not "develloping quantum theory further".

One might try to modify/improve/change/... quantum theory into something vaster, but my bet is that without any experimental hint what so ever, this is a risky (and IMO futile) undertaking. For most practical purposes, quantum theory as it stands is "good enough", and so the only thing that is really needed is a STORY around it, that makes its principles stand out more clearly (even though the story might sound crazy), and as thus, help you understand the *machinery*.
By no means, I consider it a "final story", and by no means I think that quantum theory is "the ultimate theory", but I have no alternatives, and it is good enough.

I think it has something mentally healthy to it to have a theory that is "good enough" but of which you suspect that it is not "the final word" (without secretly having an idea of how that final word should look like, but to be honestly agnostic about it) and to have a totally crazy story about the workings of this theory that illustrates perfectly how the theory works (and which might, or might not, be true). It helps one to put things - especially physics - in perspective.

It avoids one to enter into a kind of delirium of "knowing the ultimate truth" and grandiose (and childish) ideas about knowing the "meaning of life, the universe and everything"... and it makes for amazingly good stories around the camp fire On top of that, it works very well in practice! And, like with all good ghost stories... you never know !

 

[Fra]

I think I understand your perspective a little better now.

To speak for myself, If I didn't have the (not matter how naive! :) idea that i could improve the current theory and a hint of how to do it, I would be perfectly satisfied with the "shut up and calculate" philosophy.

Without such an ambition of improvement or extension, the best "understanding" of the current models is to me to simply understand the historical development of the theory. The history of science and physics is interesting and it's not that hard to imagine how ideas has developed.

/Fredrik

 

[ripcurl1016]

If we do not know what something is doing, then it is allowed to do everything possible simultaneously. In the case of the double slit experiment, we do not know whether the electron passed through the left slit or the right slit, so we assume that it passed through both slits simultaneously. Each possibility being a "state", and because the electron fullfills both possibilities it is said to be in a "superposition of states."


So MWI claims that the electron has two definite choices-either it passes through the left slit or the right slit-at which point the universe divides into two universes, and in one universe the electron goes through the left slit, and in the other universe the electron goes through the right slit. These two universes somehow interfere with each other causes the intereference pattern. So whenever an object has the potential to enter one of several possible states, the universe splits into many universes, so that each potential is fulfilled in a different universe.

 

[vanesch]

If we do not know what something is doing, then it is allowed to do everything possible simultaneously. In the case of the double slit experiment, we do not know whether the electron passed through the left slit or the right slit, so we assume that it passed through both slits simultaneously. Each possibility being a "state", and because the electron fullfills both possibilities it is said to be in a "superposition of states."

Well, I don't know if this is what you are saying, but it illustrates a point in the discussion with reilly.

There is a big difference between: "not knowing in what state a system is" (but assuming it is in one or another), and knowing (or not) that the system is in a *superposition* of states.

The first one (not knowing in which state...) is a statement about my knowledge, and enters in a probability description (with a Bayesian view). The second one is about an objective phsysical property.

In the first case, we can "treat all possible cases": we can "loop over" all possible states the system could be in, and reason under each hypothesis, to come to a final conclusion with uncertainties (because we have uncertainties about which statement was correct).

This was, in our example, the "uncertainty" about whether the particle went through the left or the right slit. But it means that we can say:
A) let's assume it was the left slit, blah blah...
B) let's assume it was the right slit, bluh bluh...

and so the endresult is 50% chance to have blahblah and 50% chance to have bluh bluh.

But in the second case (objective physical state), the *superposition of the two states* is a different physical state than "left state" or "right state" and lack of knowledge. It is as different as any other state from these two. So the superposition is then NOT a "lack of knowledge" on my part, but a physically different situation, which can best be seen that the particle goes through both slits at the same time (which is of course physically different from "the particle goes through slit 1" and from "the particle goes through slit 2").

The wavefunction (in collapse interpretations) dances between both. Indeed, at the moment of observation, the wavefunction (written in the "right" basis!) is clearly generating a "probability distribution", that is: it is saying something about a lack of knowledge on our part: namely WHICH outcome we're going to obtain. The wavefunction at the moment of detection doesn't seem to tell us that the particle is AT THE SAME TIME at x1, at x2, at x3, at x4 ... but rather that it will be measured at x1 OR at x2 OR at x3... and that we don't KNOW this.

But when it was at the slits, it wasn't telling us that the particle went through slit 1 OR through slit 2, but rather that it did something different (most reasonably go through both slits at once).

In classical settings, a lack of knowledge is usually NOT assumed to correspond to a DIFFERENT state than the "possible ones". If Joe has thrown a dice, but he didn't tell me the outcome, I assume that the result was 1, OR 2 OR 3 OR... OR 6, but not some state different from these 6 possible ones. I can deduce everything by assuming first that he threw 1, then assuming that he threw 2 ... and in the end, weight all possible results with a probability 1/6 (fair dice).
For instance, if I would assume that he had ALL RESULTS AT ONCE, I would probably have a totally different outcome (like Joe writing to the local newspaper of having seen something quite amazing!).

 

[reilly]

You are right to point out the inadequacy of my density matrix example.

After a good bit of thought, I still conclude that your convincing argument about conditional probabilities is, in fact incorrect.

It all boils down to the requirement that if

P(X) = P(X|A) P(A) + P(X|B) P(B)

then events A and B must be disjoint and independent. That is, they cannot occur simultaneously, so there is no interference involved in the above probability scheme.

In the quantum situation, the situation is very different. For all practical purposes there are three independent( well almost) distinct events for the two slit expt., and they are the initial experimental conditions, slit A open, or slit B open, or slits A and B open. Properly normalized and orthogonalized, the QM measurement pattern will be reproduced as the sum of conditional QM probs based on the three slit states.

Regards,
Reilly

Well, for the very reason I repeat again. If we take the wavefunction of the particle, and we let it evolve unitarily, then at the slit, the wavefunction takes on the form:
|psi1> = |slit1> + |slit2>
which are essentially orthogonal states at this point (in position representation, |slit1> has a bump at slit 1 and nothing at slit 2, and vice versa).

Now, if this is to have a *probability interpretation*, then we have to say that at this point, our particle has 50% chance to be at slit 1 and 50% chance to be at slit 2, right ?

A bit later, we evolve |psi1> unitarily into |psi2> and this time, we have an interference pattern. We write psi2 in the position representation, as:
|psi2> = sum over x of f(x) |x> with f(x) the wavefunction.

This time, we interpret |f|^2 as a probability density to be at point x.

Now, if at the first instance, we had 50% chance for the particle to be at slit 1, 50% chance to be at slit 2, then it is clear that |f|^2 = 0.5 P(x|slit1) + 0.5 P(x|slit2), because this is a theorem in probability theory:

P(X) = P(X|A) P(A) + P(X|B) P(B)

if events A and B are mutually exclusive and complete, which is the case for "slit 1" and "slit 2".

But we know very well that |f|^2 = 0.5 P(x|slit1) + 0.5 P(x|slit2) is NOT true for an interference pattern!

So in no way, we can see |psi1> as a probability density to have 50% chance to go through slit 1 and 50% chance to go through slit 2.



The point is that a pure state, converted into a density matrix, after diagonalisation, always results in a TRIVIAL density matrix: zero everywhere, and a single ONE somewhere on the diagonal, corresponding to the pure state (which is part of the basis in which the matrix is diagonal).

As such, your density matrix will simply tell you that you have 100% probability to be in the state...

If you don't believe me, for a pure state, we have that rho^2 = rho. The only diagonal elements that can satisfy this are 0 and 1. We also have that Tr(rho) = 1, hence we can only have one single 1.

 

[reilly]

You are right to point out the inadequacy of my density matrix example.

After a good bit of thought, I still conclude that your convincing argument about conditional probabilities is, in fact incorrect.

It all boils down to the requirement that if

P(X) = P(X|A) P(A) + P(X|B) P(B)

then events A and B must be disjoint and independent. That is, they cannot occur simultaneously, so there is no interference involved in the above probability scheme.

In the quantum situation, the situation is very different. For all practical purposes there are three independent( well almost) distinct events for the two slit expt., and they are the initial experimental conditions, slit A open, or slit B open, or slits A and B open. Properly normalized and orthogonalized, the QM measurement pattern will be reproduced as the sum of conditional QM probs based on the three slit states.

Regards,
Reilly

Well, for the very reason I repeat again. If we take the wavefunction of the particle, and we let it evolve unitarily, then at the slit, the wavefunction takes on the form:
|psi1> = |slit1> + |slit2>
which are essentially orthogonal states at this point (in position representation, |slit1> has a bump at slit 1 and nothing at slit 2, and vice versa).

Now, if this is to have a *probability interpretation*, then we have to say that at this point, our particle has 50% chance to be at slit 1 and 50% chance to be at slit 2, right ?

A bit later, we evolve |psi1> unitarily into |psi2> and this time, we have an interference pattern. We write psi2 in the position representation, as:
|psi2> = sum over x of f(x) |x> with f(x) the wavefunction.

This time, we interpret |f|^2 as a probability density to be at point x.

Now, if at the first instance, we had 50% chance for the particle to be at slit 1, 50% chance to be at slit 2, then it is clear that |f|^2 = 0.5 P(x|slit1) + 0.5 P(x|slit2), because this is a theorem in probability theory:

P(X) = P(X|A) P(A) + P(X|B) P(B)

if events A and B are mutually exclusive and complete, which is the case for "slit 1" and "slit 2".

But we know very well that |f|^2 = 0.5 P(x|slit1) + 0.5 P(x|slit2) is NOT true for an interference pattern!

So in no way, we can see |psi1> as a probability density to have 50% chance to go through slit 1 and 50% chance to go through slit 2.



The point is that a pure state, converted into a density matrix, after diagonalisation, always results in a TRIVIAL density matrix: zero everywhere, and a single ONE somewhere on the diagonal, corresponding to the pure state (which is part of the basis in which the matrix is diagonal).

As such, your density matrix will simply tell you that you have 100% probability to be in the state...

If you don't believe me, for a pure state, we have that rho^2 = rho. The only diagonal elements that can satisfy this are 0 and 1. We also have that Tr(rho) = 1, hence we can only have one single 1.

 

[Fra]

Reilly, even if you use the proper formula I suggested above that accounts for the fact that in general A and B are not generally disjoint, it is not straightforward to interpret the terms in terms of the wavefunctions. Can you do it?

My suggestion is that the resolution is that the expression fails because x and A really doesn't refer to the same space. To explain this explicity it gets complicated though and the dynamics of the probability space need to be accounted for for it to make complete sense.

When I have finished this explicitly I'll post it. Due to limited time I suspect it will take me to next year. This will restors a consistent probability interpretation, and the "trick" is to consider dynamical and relational probability spaces.

/Fredrik

 

[Fra]

it is not straightforward to interpret the terms in terms of the wavefunctions

This is however only one problem, and most certainly the easiest part.

The other thing is to show that not only does the wavefunction make sense, I'd also like to show where it comes from. I think this can be done too, and this is where it gets more complicated. I think of it as a spontaneous decomposition of the microstructure into dual structures that are related. And I'm trying to understand/explain why this is spontaneous and what the mechanism is. In essence I think it's repartitioning of the observers data record, and the particular repartitioning selected, is simply more probably given the current incompletness. I haven't done it yet though, and maybe I'm wrong. But at least I'm convinced enough to take the risk of beeing wrong.

/Fredrik

 

[Hurkyl]

It all boils down to the requirement that if

P(X) = P(X|A) P(A) + P(X|B) P(B)

then events A and B must be disjoint and independent.

(Speaking purely about probability...)

That's not correct. Independent means that

P(A and B) = P(A) * P(B)

Since disjoint means P(A and B) = 0, we see that two events can be disjoint and independent if and only if the probability of one of them happening is zero.

Furthermore, disjointness doesn't follow from that requirement: the best we can say is that X, A, and B do not happen simultaneously with nonzero probability... it is possible for A and B to happen simultaneously without X happening.

The implications of that equation are:

P(X and not (A or B)) = 0
P(X and A and B) = 0

 

[Hurkyl]

Vanesh, in your view... do you ever attempt to explain where the complex amplitude formalism comes from, or is it just taken to be a fact, and you are trying to interpret it?

For the record...

If, given a quantum state and a measurement operator, you have some means of extracting the "expected value" of the operator... then the Gelfand-Naimark-Segal construction says that you can take the 'square root' of the state, giving you a bra and a ket that represent your quantum state. Such objects live in Hilbert spaces.

This applies to any state -- including statistical mixtures. In the case of a statistical mixtrue, the Hilbert spaces¹ produced by the GNS construction has a special form; it is reducible. You can split the Hilbert space into irreducible state spaces (e.g. you can split "particle with unit charge" into "particle with charge +1" and "particle with charge -1"). The state corresponding to a statistical mixture can always be decomposed into its individual parts.

The same cannot be said for pure states; the GNS construction provides you with an irreducible Hilbert space. Thus we see that pure states cannot be reinterpreted as statistical mixtures. (At least, not in any direct way)


1: more precisely, it's a unitary representation of the measurement algebra.

 

[vanesch]

You are right to point out the inadequacy of my density matrix example.

After a good bit of thought, I still conclude that your convincing argument about conditional probabilities is, in fact incorrect.

It all boils down to the requirement that if

P(X) = P(X|A) P(A) + P(X|B) P(B)

then events A and B must be disjoint and independent. That is, they cannot occur simultaneously, so there is no interference involved in the above probability scheme.

I think you misunderstand me. To address Fra's justified remark, I think that if one is going to say that something has "a probability interpretation", it has to be in one and the same probability space (one and the same event universe with Kolmogorov measure over it). It simply doesn't make sense, IMO, to talk about probabilities in different spaces when talking about something physical (eg. an actual, objective, physical event), because probability just means a description of our ignorance, and grossly justified by a frequentist repetition of the situation.

As such, when I say that the probability that you went through the left door of the building yesterday is 30%, then that means two things: it 1) describes my state of knowledge about that event and 2) it must correspond to a kind of "generic situation" compatible with all I know, that in 30% of the cases you go to the left door (and let's assume, 70% of the time you take the right door).
Of course, after asking you through which door you went, this probability changes, say, from 30% to 100% when you say "I went through the left door" (assuming you're not lying). But that is because my state of knowledge changes (namely, the additional phrase "I went through the left door" is added), so the set of generic events must now be that set where you also utter that phrase, and in THAT set, you ALWAYS go through the left door.

But all these events are part of one and the same probability space. The only thing that changes is the "generic subspace of events compatible with my knowledge", which is expressed in probability language with conditional probabilities.

If you have different probability spaces, you can always combine them in one bigger probability space, by adding a generic tag to each one. If Omega1 is a probability space, with probability measure p1 over it, and Omega2 is another probability space with probabiliity measure p2 over it, then it is possible to construct a new probability space,

Omega = Omega1 x {"1"} union Omega2x{"2"}
(I added the tags 1 and 2).

The probability measure p over Omega can be of different kinds, but we can, for instance, define p(x) = a p1 if x in omega1 and b p2 if x in omega2 ; when a + b = 1, then p will be again a probability measure. P(omega1) = a, and P(omega2) = b.

As such, we can recover the old measures as: p1(x) = P(x | x in omega1) and p2(x) = P(x | x in omega2).

It is in this sense that I understand that we work in ONE SINGLE probability space, and that it is in THIS unique global space that we cannot interpret the wavefunction, when it is at the height of the slits, as giving a probability density.

Indeed, let us assume omega1 as the universe of events of measured impacts on the screen. p1 is then nothing else but the wavefunction squared of the interference pattern. Right. But now we want to interpret the wavefunction at the height of the slits ALSO as a probability generating function. This would be then our Omega2 which consists of 2 events, namely "through slit 1" and "through slit 2", and p2 would then be 50 - 50.

It is the building of an overall probability space as I showed above, that doesn't work in this case. THIS is what I mean that you cannot see the wavefunction with a probability interpretation when it is "in between measurements".

In the quantum situation, the situation is very different. For all practical purposes there are three independent( well almost) distinct events for the two slit expt., and they are the initial experimental conditions, slit A open, or slit B open, or slits A and B open. Properly normalized and orthogonalized, the QM measurement pattern will be reproduced as the sum of conditional QM probs based on the three slit states.

This only works when considering only as events the "preparation" and the "measurement" events, and NOT when considering the "in between" states as EVENTS (necessary to give them a probability interpretation).

I agree of course with you that P(measured at x | slit situation 1) will give you the right answers. But we only consider "omega" here as consisting of "preparation" events and "measurement" events. This is fully in line with the Copenhagen view that you *cannot talk about what happens in between". And that "cannot talk" is not only about IGNORANCE, but simply about non-talkability! If it were *ignorance* then it could be seen as events in a probability space and that's what is impossible.

If the wavefunction at the height of the slit is to have an *ignorance* or probability interpretation - instead of NOT having an interpretation such as in Copenhagen - then it is to be seen as giving us a probability for "going through slit 1" and "going through slit 2".

And once we do that, we have completely "forgotten" the conditional probabilities of the experiment setup, simply because the wavefunction at the slits contains all "information" for going further.

Let us consider the following events: A = slit 1 open, B = slit 2 open, C = slit 1 & 2 open.
X = particle through slit 1, Y = particle through slit 2
U(x) impact at position x on screen

P(U(x) | A ) gives us a blob behind 1, P(U(x) | B) gives us a blob behind 2, and P(U(x) | C) gives us an interference pattern. That's what you get out of quantum mechanics, and out of Copenhagen. X and Y don't exist in the event space of Copenhagen.

But if we insist on having X and Y in there (probability interpretation of the wavefunction all the time), then the situation is the following:

P(U(x) |A and X) = P(U(x) | X). Indeed, if we KNOW that slit A was open, AND we know that the particle went through slit 1, then we can limit ourselves to just knowing that the particle went through slit 1.
(A and Y) = 0 so no conditional prob can be defined here
(B and X) = 0 idem
P(U(x) | B and Y) = P(U(x) | Y)

We know this also from QM: if we evolve the state "at slit 1" onto the screen, we will find the blob behind 1.

And now comes the crux:
P(U(x) |(C and X)) = P(U(x) |X). Indeed, IF WE KNOW that the particle went through slit 1, even if we opened slits 1 and 2, (for instance by measuring it at the slit), then we find simply the blob behind 1. In other words, knowing that the particle went through slit 1 is sufficient, and the "extra information" that both the slits are open doesn't change anything on the conditional probability.

And now we're home:

P(U(x) | (C and X) ) = P(U(x) | X ) = P(U(x) | (A and X))

In other words, when we count X and Y as events, and we take them conditionally, we don't care what were the preparations.

If that's true, thenP(X|C) * P(U(x) | (C and X)) + P(Y|C) P(U(x) | (C and Y) ) should be P(U(x) | C) and it isn't, because P(U(x) | (C and X) ) = P(U(x) | X ) = P(U(x) | A) and in the same way P(U(x) | (C and Y) ) = P(U(x) | B). P(X|C) = P(Y|C) = 0.5 (under the probability interpretation of the wavefunction)

so we should have: 0.5 P(U(x) | A ) + 0.5 P(U(x) | B) = P(U(x) |C), which is not true.

 

[Fra]

I agree to a good extent with vanesch here regarding the problem, but perhaps not on the cure.

I think that if one is going to say that something has "a probability interpretation", it has to be in one and the same probability space (one and the same event universe with Kolmogorov measure over it). It simply doesn't make sense, IMO, to talk about probabilities in different spaces when talking about something physical

Yes. This is why I mumble about that they are not statistically independent spaces, they are related, and this relation also has a physical interpretation. And information in one spaces induces, via relations, information in the connected spaces, until there is equilibrium. Also, new substructures can spontaneously form.

This thinking has similarities to decoherence selection by the environment.

If you have different probability spaces, you can always combine them in one bigger probability space, by adding a generic tag to each one. If Omega1 is a probability space, with probability measure p1 over it, and Omega2 is another probability space with probabiliity measure p2 over it, then it is possible to construct a new probability space,

I sort of agree, so in a sense what I was talking about is that we decompose the "probability spaces", or the microstructure (sometimes a better word), into substructures, that effectively work as standalone spaces, but they are connected to each other, so the decomposition of substructures serves a purpose, like a kind o self organisation, to optimise storage. Clearly the concept of dimension is strongly related to this. Dimensionality is in my thinking, related to splitting of the microstructure. In effect the structure selecetion is "driven" by interaction by the environment.

I think this tangents to what Hurky relates to. That measurements are selected, this is excellent. But there are still some details missing, isn't there? Not to mention gravity and intertia.

I see the foundational issues of QM so closely touching inertia concepts that I think this could explain also gravity. Or as I think, rather just "identify gravity", like we identify other properties.

If you consider information, and the related microstructures one can intuitively imagine intertia in subspaces, so there are a certain stability. The remodelling are unlikely to be chaotic. From my view at least, gravity/spacetime and these foundational issues of QM touch each other. I can't picture a consistent solution to one without the other in the information view.

/Fredrik

 

[Fra]

Yes. This is why I mumble about that they are not statistically independent spaces, they are related, and this relation also has a physical interpretation. And information in one spaces induces, via relations, information in the connected spaces, until there is equilibrium. Also, new substructures can spontaneously form.

It's clear that people use different words depending on their approach.

I would say that what I mean to say here, is closely related to the various concepts of symmetry breaking. How come a uniform ignorance spontaneously split and start to form structures? And how does the relations between substructures form? and are certain relation more likely to be formed than others? is there even an answer to this? and what about stability? why doesn't everything just happened all over the place yielding just chaos?

Why is spontanouse structure formation more likely than the opposite?

/Fredrik

 

[Fra]

grossly justified by a frequentist repetition of the situation.

If this is to be realistic, the information capacity of hte observer seemingly limits the confidence level of the probability estimate, or? what's your comment on tihs?

This is something that I try to take very seriously, and resolve.

A consistent frequentists interpretation seems to contradict with limited encoding capacity? If you make use of external memory, then aren't you modifying the condition C?

P(x|C)

Part of the implicit (not always written out) condition is information capacity in my thinking. I am trying to give a physical interpretation of "probabilities", but since they aren'y really unitary in time, the word microstates and microstructures is better.

/Fredrik

 

[reilly]

Hurkyl -- You are quite right, except I meant independent in time (Markovian). What happens now is independent of what happened then -- I refer to repetitions of the experiment.
Regards,
Reilly

(Speaking purely about probability...)

That's not correct. Independent means that

P(A and B) = P(A) * P(B)

Since disjoint means P(A and B) = 0, we see that two events can be disjoint and independent if and only if the probability of one of them happening is zero.

Furthermore, disjointness doesn't follow from that requirement: the best we can say is that X, A, and B do not happen simultaneously with nonzero probability... it is possible for A and B to happen simultaneously without X happening.

The implications of that equation are:

P(X and not (A or B)) = 0
P(X and A and B) = 0

 

[reilly]

A quick rejoinder, and a question.

RA
In the quantum situation, the situation is very different. For all practical purposes there are three independent( well almost) distinct events for the two slit expt., and they are the initial experimental conditions, slit A open, or slit B open, or slits A and B open. Properly normalized and orthogonalized, the QM measurement pattern will be reproduced as the sum of conditional QM probs based on the three slit states.

vanesch
This only works when considering only as events the "preparation" and the "measurement" events, and NOT when considering the "in between" states as EVENTS (necessary to give them a probability interpretation).

RA
I don't get it. Events are often defined as points in a probability space, as outcomes of some measurement procedure. So, |W(x)|^^2 represents a probability density in configuration space, defined by the eigenvalues and eigenstates of the position operator, x. So, in between the absolute square of the wave function will describe measurements showing interference effects related to those on other planes. And, of course experiment will confirm the validity of theory. What am I missing?

You and Fra talk about different spaces. What are they.
Regards,
Reilly

 

[vanesch]

I don't get it. Events are often defined as points in a probability space, as outcomes of some measurement procedure. So, |W(x)|^^2 represents a probability density in configuration space, defined by the eigenvalues and eigenstates of the position operator, x. So, in between the absolute square of the wave function will describe measurements showing interference effects related to those on other planes. And, of course experiment will confirm the validity of theory. What am I missing?

By definition, events related to observation can be seen as events in a probability space, and quantum mechanics describes that probability space perfectly. There's no problem with that. However, the wavefunction also "exists" (in whatever sense) *in between* observations. And it is there that one cannot give a probability interpretation to that wavefunction. Copenhagen has no problems with that: the wavefunction is just a tool to calculate the probabilities of measurements, so its meaning *in between* measurements is left undefined - it's mostly seen as a kind of calculational tool, that's all.
If you insist on giving the wavefunction a probabilistic interpretation *always* (so not only at observations, but also in between) you run into the kinds of problems I tried to explain. That's all.

So, OR we have a probabilistic description of events at preparation/measurement and NOTHING in between (except for calculational aids), OR we have something physical there, which is *sometimes* to be seen as probability, and *sometimes* not.

But I agree with you that this is NOT observable, because to be observable, means, to be a measurement, and then we know that we can use probability theory. I'm talking about the wavefunction of a thing *in between* measurements - like we can talk about the EM wave in between emission and detection. I'm trying to say that - in as much as we want to have a physical picture - we have no difficulties taking that there IS an E/B field between the emitter and the receiver, and applying a similar logic, we should accept that there is something physical like a wavefunction in between preparation and measurement, and that we can't see that simply as a kind of probability description.
Copenhagen denies that. It is a position.

 

[ripcurl1016]

So in the absence of observation is when the wave function exists.

 

[Fra]

It's clear that we disagree on somethings and agree on some parts. I'm not in the position to currently make a full explanation of what I mean, beause it's in progress but here are some more comments.

By definition, events related to observation can be seen as events in a probability space, and quantum mechanics describes that probability space perfectly.

I can't accept that as a definition. It's usually a postulate of QM - I call it an assumption. And the reason I point it out is of course because I disagree with the assumption, except in special cases.

What I suggest is that generall, the decomposition of just observed "raw data" if I may use that word, into a specific probability space seems to be ambigous. I haven't seen a rigorous reasoning that deduces a uniqe structure from the raw data.

Any my point is not that there exists such a deduction, buy what I think is that there exists an induction, that ultimately selects this structure/space. But this is foundamentally "dynamical", and actually relational to the history of data.

It's exactly such an induction I look for. And I think even such an induction itself is not static, it's evolving too. Everything is "sliding" and then how can we get stability and predictability? Here where I think the rescue is intertia and time seen as a parametrisation of this remodelling.

There's no problem with that. However, the wavefunction also "exists" (in whatever sense) *in between* observations. And it is there that one cannot give a probability interpretation to that wavefunction. Copenhagen has no problems with that: the wavefunction is just a tool to calculate the probabilities of measurements, so its meaning *in between* measurements is left undefined - it's mostly seen as a kind of calculational tool, that's all.

Since I argue for a reconstruction of quantum foundations, I am not yet sure how big modifications of the wavefunction terminology that is to be seen, but if we ignore that details for a second and with wavefunction included what it is today and any possible improvements of it, taking it's effective place, then the way I see it the wavefunction exists and is defined by the microstructure and microstate of an observer. This means that the wavefunction is not fully objective, it's subjective and part of hte observers identity, meaning it's observable to the observer himeself, but not fully to others.

The "wavefunction" can change even without new observations, and that's due to the observers internal "dynamics" that ideally is in harmony with the environment. It's the "expected" projection of the environment. But this expectation is updated on each observation or release of information.

The way I picture the probability-like interpretation in between measurements is by connection the state of the microstructure of the observers to a probability, and thus the evolution of the wavefunction are simply the only consistent "self-evolution" of the observes information. This takes place because typically the information actually contains also information about the dynamics, and uncertainty. Thus, even left unperturbed, this information is bound to be subject of self-evolution to respect itself. This I imagine is also the basis for something like an "internal clock".

I am unable to make this reasoning without connecting to intertia and time. I have been fascinated but this fact, that a consistency analysis of QM alone, seems to point towards a direction that smells strongly of relativity.

/Fredrik

 

[Fra]

You and Fra talk about different spaces. What are they.

If you like, perhaps considering them my personal pet ideas is the easiest rescue here. But I've tried to elaborate the ideas.

But the different spaces can be seen as "substructures" of the one full structure.

Lets consider the data perspective, and let's not talk about time.

For example suppose you have 9 samples? What determines wether you have 9 samples from a one dimensional structure, or 3 triples from a 3 dimensional space? The raw data is the same? Or 4 pairs and a mismatched sample?

That's as silly example, but what I mean to say is that the raw data can be sub structured in various ways, but not just this simple way, a more sophisticated thing starts when you have a limited mmeory record. Suppose you can only store 9 samples. Then when you get the 10'th sample you need to make a decisions. How do you incorporate/remodel your current record, to update your information about the new sample, and and the same time get rid of one sample? You obviously need to throw away the least important sample - you need to make a decision. You want to maximise your learning given some constraints.

More advanced options starts when you consider performation a transformation on the whole, or parts of your record, and hten instead store the transformed record? MAYBE that is a way to retain more information with same storate capacity? This is in effect data compression. Think - Fourier transform for an example. Then, what transformation is more efficient? This may, many complicated things can be done and since we continously train and remodel the record, we are no longer talking about equivalent information.

I'm not sure if this makes sense. If not I point out again that most of this is my personal thinking, no need to give anyone headache at least until i am in the position to present an explicit formalism.

But to connect to reilly's original post I connected to, subjective bayesian probability reasoning and inductive reasoning are key tools for me in this.

/Fredrik

 

[Fra]

If the idea occurred to anyone I don't think this will generally come out as clean as a markov process. But possibly "almost" like a markov process in some limiting cases.

Similarly I expect _in general_ violation of unitarity, but in many cases an effective unitarity will be emergent, so this will be entirely consistent with the effective unitarity we have experience with from QM.

/Fredrik

 

[reilly]

vanesch -- Finally I get it, or I get why we have never quite agreed. Now, I still go with Born, in that, ultimately in between is properly described in terms of quantum probabilities.

First, how do you know that B and E exist in between? Yes, there's enormous circumstantial evidence to support continuity in electromagnetism. But you really don't know until you measure, and the quantum world will cause a few problems of interpretation . Same is true for photons or electrons going through slits; it's the possible measurements, which define the events of probability spaces.

One way, in principle, to show the in-between would be to use muons, or pi naughts, or ..., in a two slit experiment. Then as muons go through the slits, the decay will show the in-between probabilities -- much like Faraday's iron filings in a magnetic field.(The clever experimenter will, of course, fix things so that the bulk of decays occur between the transmission and receptor screens. )

Regards,
Reilly

By definition, events related to observation can be seen as events in a probability space, and quantum mechanics describes that probability space perfectly. There's no problem with that. However, the wavefunction also "exists" (in whatever sense) *in between* observations. And it is there that one cannot give a probability interpretation to that wavefunction. Copenhagen has no problems with that: the wavefunction is just a tool to calculate the probabilities of measurements, so its meaning *in between* measurements is left undefined - it's mostly seen as a kind of calculational tool, that's all.
If you insist on giving the wavefunction a probabilistic interpretation *always* (so not only at observations, but also in between) you run into the kinds of problems I tried to explain. That's all.

So, OR we have a probabilistic description of events at preparation/measurement and NOTHING in between (except for calculational aids), OR we have something physical there, which is *sometimes* to be seen as probability, and *sometimes* not.

But I agree with you that this is NOT observable, because to be observable, means, to be a measurement, and then we know that we can use probability theory. I'm talking about the wavefunction of a thing *in between* measurements - like we can talk about the EM wave in between emission and detection. I'm trying to say that - in as much as we want to have a physical picture - we have no difficulties taking that there IS an E/B field between the emitter and the receiver, and applying a similar logic, we should accept that there is something physical like a wavefunction in between preparation and measurement, and that we can't see that simply as a kind of probability description.
Copenhagen denies that. It is a position.