A In what sense does MWI fail to predict the Born Rule?

[Derek P]

I would say that decoherence plus tracing out environmental degrees of freedom transforms a superposition into a mixed state, not decoherence alone.

Exactly! Until this thread I had always assumed that when people talk about decoherence they mean entanglement with those environmental degrees of freedom. I don't see how it can be avoided unless they think of decoherence as a sort of phase noise introduced by a decidedly non-unitary operator - which then leaves the model with no "handle" on probabilities except a dogmatic assertion of the Born Rule a priori. Counting states replaces the requirement for the Born Rule with a much weaker requirement for the Principle of Indifference to be applicable to those environmental degrees of freedom. Would you agree?

I didn't answer your post with the ∏ operator as I wanted time to digest it but this thread is racing too fast to keep up! It seems to have branched more than an Everettian universe and, appropriately enough, different subthreads seem to be unaware of each other I'll get back to ∏ when I can.

 

[Derek P]

The physical process of decoherence is fully unitary, I think we all can agree. After it's occurred, you can pick subsystems that will be evolving mostly independently. That doesn't seem to require the Born rule to observe in a simulation/calculation, does it? But I guess it requires some kind of measure which doesn't amplify the shared terms that have vanished.

See post #150 and my comment following it.

 

[DarMM]

The word "decoherence" is used to denote the physical process of entanglement with a large system. You are using it to denote diagonalisation. Both are legitimate uses of the word and the latter follows from the former. So deriving the trace heuristic from the dynamics is not circular.

Okay, entanglement with the measuring device and environment does not require the Born rule. If that is what you mean by Decoherence, then yes it is not circulur, it can be seen to be a dynamic property.

However, it is not enough for Many-Worlds, where the branch structure only emerges within subsystems. For that you do need the trace.

So more accurately the emergence of the branch structure requires tracing.

 

[A. Neumaier]

I have always called the trace formula the Born rule - what would you call it?

It is just the definition of an expectation value, without any commitment to its interpretation in terms of measurements. This is done routinely in the (purely mathematical) theory of $C^*$-algebras, and hence part of the theoretical (shut-up-and-calculate) fraction of quantum mechanics. See my post #140.

The interpretation only starts when one is postulating that ''Upon measuring $A$ an ensemble of systems independently prepared in a fixed state, the expected mean of the measurement results equals the formal expectation value $\langle A \rangle$ corresponding to this state.'' This is van Fraasen's postulate (p.9 of the article mentioned in post #141).

 

[A. Neumaier]

I have always called the trace formula the Born rule - what would you call it?

The trace formula says less than the standard form of Born's rule, as it is silent about the possible values of a measurement. One can prove from the trace rule only that a value from the spectrum occurs with probability 1, but this allows finitely many measurements with other results, as probabilities are insensitive to a finite number of results.

On a purist note, we can only take finitely many measurements on a system. But the expected mean is also insensitive to a finite number of results. Thus, strictly speaking, the trace rule says nothing at all about measurement.

 

[stevendaryl]

Okay, entanglement with the measuring device and environment does not require the Born rule. If that is what you mean by Decoherence, then yes it is not circulur, it can be seen to be a dynamic property.

However, it is not enough for Many-Worlds, where the branch structure only emerges within subsystems. For that you do need the trace.

So more accurately the emergence of the branch structure requires tracing.

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates. These coarse-grained states are subspaces of the Hilbert space that can be specified by projection operators. The evolution of the projection operators (in the Heisenberg representation) follows from ordinary quantum mechanics, and the probability for being in one of the subspaces is just the Born rule applied to the projection operators (which are Hermitian observables, after all). With this approach, it's not clear where tracing or decoherence comes into play.

My feeling is that the decoherence approach is approximately equivalent, in some sense, but it seems very different.

 

[DarMM]

Okay so here are Wallace's axioms on the set of acts $\mathcal{U}_{E}$ available at an event $E$ and his axioms on the preference order on those acts $>^{\psi}$
Once again events $E$ are either classical "worlds" or superpositions there of. The set of worlds, excluding their superpositions, is denoted $\mathcal{M}$, elements $M$.

There is also the set of awards $\mathcal{R}$ which is a coarsening of worlds, as you might have the same reward in separate worlds. Elements are $r$.

If I perform an act $U$ at event $E$, $\mathcal{O}_{U}$ is used to denote the smallest event resulting from that act, e.g. $E$ might be me ready to perform a spin measurement, $U$ the act of performing the measurement and $\mathcal{O}_{U}$ the branched world afterward. I'll just call $\mathcal{O}_{U}$ the outcome of $U$. For worlds the notation $U(M)$ is common for their outcome.

I will list the axioms informally, they can be found more accurately in Mandolesi's first paper on p.11-17

Axioms on $\mathcal{U}_{E}$:

1. Acts can be restricted to subevents, e.g. $F \subset E$ means for any $U \in \mathcal{U}_{E}$ there is $U |_{F} \in \mathcal{U}_{F}$. So an example might be if I perform a spin measurement and split the world in three. A position measurement being available upon the superposition of worlds, would mean it is available in any branch.
   
2. Branches do not later interfere, this is equivalent to existence of an identity act.
3. Composed acts are available. By composed acts I mean if I perform $U \in \mathcal{U}_{E}$, then if there is an act $V \in \mathcal{O}_{U}$, then $VU$ is a possible act.
4. There is an act available that results in a given award. If you are in a world $M$, then there is an act with outcome $U(M)$ such that $U(M) \subset r$ with $r$ some reward subspace.
5. There are acts that result in branching, but don't change the reward. i.e. acts that cause a world to evolve into a superposition of worlds, but remain in the same $r$
6. There are erasure acts. Essentially if two superpositions of worlds (including the special case of them being each a single world) are in the same award space, there exists an act on each that evolves them into the same state, i.e. an act where we "throw away" any differences between them.
7. Act continuity. If $U$ is available, acts "nearby" in the operator norm topology are available.

Axioms on $>^{\psi}$, these are given for $\mathcal{U}_{\mathcal{M}}$, that is they are the axioms on the preference order within one world:

1. $>^{\psi}$ is a total order on $\mathcal{U}_{\mathcal{M}}$
2. Acts that result purely in branching, without changing the reward space, are ignored, that is preferred as the identity act. If $M \subset r$ and $U(M) \subset r$ then $U \sim^{\psi} \mathbb{I}_{M}$. Nobody cares about pure branching with no reward change
3. Preferences depend only on final states, not acts or initial states (this is called State Supervenience in Wallace). There is a fully accurate form in Mandolesi, but let us just say if $U\psi = V\psi$ then $U \sim^{\psi} V$. This axiom basically encodes a weaker form of noncontexuality, or something related to noncontextuality (I'm not sure which yet). However it does not mean Wallace assumes a form of noncontextuality as strong as Gleason's theorem, hence criticism that it is a basically Gleason's theorem are unfounded I think. Of course he later uses the axioms to prove something like Gleason's noncontextuality, from which point the proof is like Gleason's.
4. The preference order is continuous. i.e. if $U >^{\psi} V$ then nearby $U', V'$ have $U' >^{\psi} V'$. Small changes don't alter act preferences.
5. Preferences only depend on branches where acts differ. If an act is preferred in all branches, then it is preferred overall. If two acts are only considered non-equivalent in only one branch, they are considered non-equivalent overall.
6. The situation under consideration is not degenerate, i.e. there are at least two acts with $U >^{\psi} V$.

So that's 13 axioms, with which Mandolesi finds 17 problems. I'll discuss this in the next post.

 

[DarMM]

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates. These coarse-grained states are subspaces of the Hilbert space that can be specified by projection operators. The evolution of the projection operators (in the Heisenberg representation) follows from ordinary quantum mechanics, and the probability for being in one of the subspaces is just the Born rule applied to the projection operators (which are Hermitian observables, after all). With this approach, it's not clear where tracing or decoherence comes into play.

My feeling is that the decoherence approach is approximately equivalent, in some sense, but it seems very different.

Wallace uses this form of coarse-graining as well. Tracing is used in the usual decoherence based approach to the emergence of worlds. I'll be discussing such coarse grainings as they are used by Wallace in his proof later.

You would in essence view the Born rule as arising from the "volume" of a coarse-graining, an idea like branch counting, but of course not exactly the same. Wallace says similar in his talks.

 

[akvadrako]

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates.

But in this scheme, wouldn't any course-graining be equally permissible? So you could have "possible worlds" that don't behave approximately classically. With decoherence and darwinian evolution, you end up with something recognisable as branching classical worlds.

 

[Derek P]

Okay, entanglement with the measuring device and environment does not require the Born rule. If that is what you mean by Decoherence, then yes it is not circulur, it can be seen to be a dynamic property.

Yay! Agreement!

However, it is not enough for Many-Worlds, where the branch structure only emerges within subsystems. For that you do need the trace.So more accurately the emergence of the branch structure requires tracing.

Oh dear! Why did you have to spoil it? Yes of course you have to perform a trace. But you do not need the Born Rule to calculate the matrix or to justify tracing.

 

[Derek P]

I don't see that as a necessity. As I sketched in another thread, it seems to me that you can just define a "possible world" as a coarse-graining of the state to define macrostates. These coarse-grained states are subspaces of the Hilbert space that can be specified by projection operators. The evolution of the projection operators (in the Heisenberg representation) follows from ordinary quantum mechanics, and the probability for being in one of the subspaces is just the Born rule applied to the projection operators (which are Hermitian observables, after all). With this approach, it's not clear where tracing or decoherence comes into play.

It comes from mapping the fine grain states to the coarse grained ones. Phenomenally the fine grain states add linearly. In the entanglement expression they add as vectors. Ergo Born's Rule. End of subject? No, probably not!

 

[DarMM]

Yay! Agreement!

Oh dear! Why did you have to spoil it? Yes of course you have to perform a trace. But you do not need the Born Rule to calculate the matrix or to justify tracing.

What's the justification for tracing if not to preserve Born weights?

 

[Derek P]

But in this scheme, wouldn't any course-graining be equally permissible?

Fine-grain states are tagged by the original interaction - a qubit or whatever - and also by the observer experience - which you can quantify by an operator ∏ as Steven does. Does that address what you're getting at?

 

[Derek P]

What's the justification for tracing if not to preserve Born weights?

? That doesn't make sense.

 

[Derek P]

But in this scheme, wouldn't any course-graining be equally permissible? So you could have "possible worlds" that don't behave approximately classically.

"that haven't behaved approximately classically to date". Non-classicality doesn't become baked into that branch for ever more. We are talking about a statistical ensemble based on frequencies in a history. There are plenty of rogue worlds.

 

[DarMM]

? That doesn't make sense.

How not, that's the justification for tracing given in most textbooks. If you have a justification of tracing that ignores Born Weights, what is it?

Also you still haven't said what proof of Born's rule you consider to close the issue.

 

[akvadrako]

Fine-grain states are tagged by the original interaction - a qubit or whatever - and also by the observer experience - which you can quantify by an operator ∏ as Steven does. Does that address what you're getting at?

I admit I didn't read his post carefully the first time. Now it's clear what you mean by micro-states and I imagine each one corresponds to the idea of a fixed volume of state space from Wallace's explanation. Maybe that works (I can't tell) and it would seem to be an alternative to Gleason's theorem — in the end what you have is a metric for Hilbert space. Is that accurate?

Oh and by non-classical, I didn't mean maverick worlds, I meant something like like |alive> + |dead>. But I guess you are assuming Alice's mind states are classical.

 

[bhobba]

What's the justification for tracing if not to preserve Born weights?

It will help I think to see exactly what partial tracing is. I generally do not like to give links to stuff by Lubos but to save me time will post the details from him on exactly what it is:
https://physics.stackexchange.com/q...ake-the-partial-trace-to-describe-a-subsystem

The the expectation value of a system composed of a number of likely entangled parts uses the Generalized Born Rule (it is what I will call it in view of the previous discussion - for simplicity now just called the Born Rule). Chug through the math and the trace breaks into two parts - the trace about what's being observed and the trace of the rest. Since we are only interested in what's being observed you can do the trace and give the formula as the trace of something else. This something else is of the form of the Born Rule on a mixed state. It is said the other stuff we are not interested in has been traced out and leads to, as all the stuff is likely entangled, interpreting the mixed state from this math as the state of the system. Mathematically the superposition has been converted to a mixed state by this manipulation which is the essence of decoherence. We say what we are not interested in has been traced out. The Born rule is central to it. Even interpreting the resulting mixed state needs the Born Rule. As I have I think said before given a general mixed state ∑pi |xi><xi| you need the Born Rule to be able to say pi is the probability of getting a yes if you observe it to see if it is in the state |xi><xi| (assuming the other states in the mixture are orthogonal).

The whole thing basically revolves around the Born Rule.

Thanks
Bill

 

[DarMM]

I'll have a separate post for each problem. This is not Mandolesi's list, as if the same problem affects two axioms, he lists them separately. I'm just stating the root problem. I'll call axioms on $\mathcal{U}_{E}$ U1, etc and those on $>^{\psi}$ O1, etc

Okay, so one of the major problems with Wallace's proof as far as Madolesi and others are concerned is related to posts we have seen here, the validity of assuming the existence, or rather the dominance, of robust quasi-classical branches. That is, a working solution of the preferred basis problem.

Wallace's axioms make use of this assumption both in the plain existence of a set $M$ of macroworlds and in axioms U2, U4, U5 and O3.

How so?

Axioms on $\mathcal{U}_{E}$:

U2. You need robust quasi-classical branches to have no interference
U4. To have an act, which is mathematically a unitary operator, result in a given fixed reward, there has to be no "leakage" under the Schrodinger evolution into separate reward spaces.
U5. Again to ensure branching does not cross reward subspace boundaries there can be no leakage. Physically I'd have to be guaranteed the existence of some experiment whose branches could never result in interference or overlap with Macroworlds with separate rewards.

Axioms on $>^{\psi}$:
O3. This is not as easy to see. If 2. is violated, it might still be the case that the proof can be recovered provided this axioms holds approximately. There might be a small amount of interference, but if this axiom is extended from:
$U\psi = V\psi \rightarrow U \sim^{\psi} V$
to
$U\psi = V\psi + \phi \rightarrow U \sim^{\psi} V$
Where $\phi$ is as "small" as interference terms between quasi-classical branches are. This would allow this axiom to generate an effective version of U2. There might be a small amount of interference, but it doesn't effect our ordering.

The big problem? How do you ensure the non-quasi classical histories are small or negligible without the Born rule?

As Schrodinger evolution does not preserve compact support of wavefunctions, the state will always "leak" into having a small overlap with another macroscopic world. Now such tails have small Born Weight, but how do you ignore them without the Born rule. Purely by counting they will be "most" of your branches.

Wallace claims one can use the Hilbert Space metric, small under the metric means little physical effect. However as has been pointed out by others, states can be very similar physically and be "far" under the metric, or very different physically and be "close", to quote Mandolesi:
"Note that, no matter how far apart two wave packets get, their states remain at an almost constant distance in Hilbert space. Which is smaller than the distance between the physically equivalent states $\psi$ and $-\psi$. If this metric is such a lousy measure of how different two states are, why, without a Born-like rule, should we expect it to be a good measure of similarity?"

It has also been claimed that one can use the volume of the macrostates in Hilbert space, ones with small volumes being negligible, but it is hard to see how this affects probabilities.

Say the state has become as follows after a measurement:
$\psi = \alpha_{M}\psi_{M} + \Sigma_{i} \alpha_{A_i}\psi_{A_i}$

where $\psi_{M}$ is a microstate which at the macro level looks like the classical world $M$ and the $\psi_{A_i}$ are correspond to non-classical bizarro macroworlds $A_i$. Yes the world volume of states with appearance $M$ might be larger, but the $A_i$ are still part of your state, simply states like them are rarer. In terms of resulting branches from your act, bizarro worlds are more common, they just have lower "world entropy" in the sense of being less generic.

(Also note the above even assumes you can carve out these world volumes consistently in Hilbert space, for which there is no clear demonstration).

I don't know of any reason that these tail evolutions can be discarded without the Born rule. They are equally real parts of the wavefunction, the only difference being they have small coefficients. Since no meaning has been given to the coefficients that would allow them to be discarded, they can't be, especially seeing as how they dominate world count following a measurement.

Mandolesi has an idea, that I will return to at the end. It's not a bad one in my opinion, but it has a common feature with many attempts to save the many worlds, the transformation of the theory into a many-minds theory. Zurek's envariance proof has also gone to a Many-Minds form to prevent certain problems, which is interesting as it follows a completely separate line of reasoning.

EDIT: I think i should add that this is not to say one cannot define negligible worlds or prove the existence of a branching structure without Born's Rule, just that this has currently not been done and is more a hope rather than a rigorous result of some research program.

I think many will have expected this problem, so I will move onto more subtle ones.

 

[A. Neumaier]

Which is smaller than the distance between the physically equivalent states ψ and −ψ

Shouldn't one use the standard metric on the projective space?

volume of the macrostates in Hilbert space

The Hilbert spaces are infinite-dimensional. So volume is not really defined!

 

[DarMM]

Shouldn't one use the standard metric on the projective space?

Yes, probably. However would this show the other terms can be discarded?

The Hilbert spaces are infinite-dimensional. So volume is not really defined!

Yes, of course. I think the hope is that the quasi-classical macroworld subspaces are larger in some appropriate manner than odd macroworld subspaces, for which volume is a vague shorthand. However since the actual definition of these subspaces is quite loose, just assumed to exist, at least in Wallace, I'm not sure how this is realized or justified.

For the purpose of honesty let me just say my preferred interpretation of QM is "I'm really confused and the more I learn about QM the less I understand", that is the "I wish I could shut up and calculate" interpretation.

 

[A. Neumaier]

my preferred interpretation of QM is "I'm really confused and the more I learn about QM the less I understand", that is the "I wish I could shut up and calculate" interpretation.

Maybe you'd like my thermal interpretation! It is the result of my gradual coping with the confusion I found when I studied the available options many years ago.

 

[Derek P]

I admit I didn't read his post carefully the first time. Now it's clear what you mean by micro-states and I imagine each one corresponds to the idea of a fixed volume of state space from Wallace's explanation. Maybe that works (I can't tell) and it would seem to be an alternative to Gleason's theorem — in the end what you have is a metric for Hilbert space. Is that accurate?

I don't know Wallace's explanation so I can't really say. And I'm not sure what "a metric for Hilbert space" means. It sounds right. You should ask @stevendaryl. "My" version would be something more simplistic like the one I posted by Price, which was instantly torn to shreds. <shrug> I'm not arguing one way or the other, I'm just asking why the more-or-less obvious approach is said to fail as well as all others. I didn't make the claim. If you read my OP you'll see I'm rusty as hell after nearly half a century.

Oh and by non-classical, I didn't mean maverick worlds, I meant something like like |alive> + |dead>. But I guess you are assuming Alice's mind states are classical.

Yes it's quite possible that there is an assumption of a preferred basis.

 

[stevendaryl]

Oh and by non-classical, I didn't mean maverick worlds, I meant something like like |alive> + |dead>. But I guess you are assuming Alice's mind states are classical.

I'm not at all suggesting a principled reason for choosing one coarse-graining over another. But I was postulating a very specific coarse-graining, which is that the coarse-grained state determines, for every volume of space down to some minimal volume, the particle content, total energy, total momentum, average electric field, average magnetic field, etc. I'm assuming that a single coarse-grained state would not be compatible with a cat being both alive and dead. As far as what's going on in Alice's mind, I don't actually know about that. That could require very fine-grained information, or maybe not.

The idea of the coarse-grained state is that if you care about gross quantities, then such things are approximately commuting---they can have simultaneous values.

 

[DarMM]

Next points.

Contradiction between U2 and U6:
The axiom U2 supposes that branches can never re-interfere once branching has occurred. In essence there is thermodynamic irreversibility post measurement. However the erasure axiom, U6, requires there to be a way to undo differences between acts if they don't affect rewards, including branchings that don't affect rewards, hence some branching acts need to be reversible.

The overt nonphysical power of U6:
Is the capacity to reverse the branchings, as is required by erasure in some cases, really physically possible?
Consider a superposition of two microstates resulting in separate macroworlds (which are orthogonal), i.e. $\psi_{1} \in M_{1}$ and $\psi_{2} \in M_{2}$.

Now assume both Macroworlds are in the same reward subspace. Hence we can erase differences between them by some acts $U$ and $V$:

$$U\psi_{1} = V\psi_{2}$$

However in the superposition

$$\psi = \psi_{1} + \psi_{2}$$

One could perform both acts of erasure, one in each branch and hence $U$ and $V$ are just restrictions of some $W$ and so:

$$W\psi_{1} = V\psi_{2}$$

in violation of unitarity

Branching indifference not justified:
O2 assumes the agent does not care about branching, if the branching remains within the same reward subspace, that is the agent does not care about branching alone. Many (Kent, Maudlin, Mallah) have criticised this for basically assuming the agent reasons in a one world manner. If $\psi_{1} \in M_{1}$ has me receive a reward and $\psi_{2} \in M_{2}$ has me receive the same reward, yes I would be indifferent between them, but why would I also consider them as valuable as:

$$\psi = \psi_{1} + \psi_{2}$$

as there are now two of me enjoying the reward?

This even links back into the definition of the reward sets as vector subspaces. Is a superposition of two worlds with the same reward, again itself the same reward? If not, the view of the reward space as a vector subspace fails, affecting the entire proof.

This one has quite a bit of back and forth within the literature, see Mandolesi's second paper, p.23

Branch indifference not justified:
Axiom O5 declares that act preferences only depends on branches where they differ. I won't go into Mandolesi's specific example, but this axiom is equivaklenbt to saying your decisions shouldn't take other branches into account, only your own and its descendants.

Again it can be argued that this sneaks one-world style reasoning into your notion of rationality. Other worlds don't exist from you perspective and hence don't matter.

I'm not sure how I feel about this one.

Do preferences only depend on final states:
This is claimed by axiom O3. However this is ultimately the very strong claim that I have no reason to care about the path taking me to the final state, i.e. the intermediate states I pass through. Wallace acknowledges this as a weakness of his taking fixed unitary operators (i.e. ones that simply map intial state to final state, without the path), but defends it by saying one can consider the unitary operators to act on time scales to small to notice.

 

[Mentz114]

[..]
For the purpose of honesty let me just say my preferred interpretation of QM is "I'm really confused and the more I learn about QM the less I understand", that is the "I wish I could shut up and calculate" interpretation.

I want to thank you for your work in explaining the Wallace opus to those of us who don't have the inclination/time/ability required to read it. Your efforts have not been a waste of time.

A good antidote to your confusion (and mine) is A.Neumaiers Coherent foundations article
which has a prospectus for the thermal interpretation which promises

The measurement problem turns from a philosophical riddle into a scientific problem in the domain of quantum statistical mechanics, namely how the quantum dynamics correlates macroscopic readings from an instrument with properties of the state of a measured microscopic system.

This puts the current difficulties into perspective.

I've downloaded the first Allahverdyan et al. paper (147 pages !) and I'll drool over it for while before atempting to read it.

 

[DarMM]

Summing up, in case my previous posts are confusing (and because summaries stick better in one's head I find), Wallace's argument has four classes of contentions:

1. It assumes one can obtain a stable robust quasi-classical branching structure without using the usual trace based derivations of decoherence. This is acknowledged by Wallace, but he sketches reasons why he thinks this can be done. In my opinion, this flat out needs to be shown by a research program, not assumed as justified by handwaving arguments.
   Wallace's argument in places assumes an idealised branching structure, not an approximate one. Since it is likely the above research program, even if successful, will only have the Macrostates as approximate structures, the proof will need to be modified to reflect this.
2. Some of the preference order axioms assume that the unique way of being rational is similar to one held by somebody who believes in one world. Of these I think the objections to "Branching indifference" are the most valid. This is currently undergoing discussion in the literature, but I do agree with the main point that it is odd that a superposition of identical rewards must always be held as being as valuable as simply evolving into one world with that reward.
3. Contradictions arising from axiom U6, erasure. This seems overly powerful, non-physical and in direct contradiction to the irreversibility assumed by the other axioms in places.
4. Even if we can avoid 1,2,3 all we have shown is that using the Born Weights to make decisions is rational. How does this connect to actually seeing Born frequencies in experimental results? This is the topic of a forthcoming paper by Mandolesi, so I will leave it.

Make of these objections what you will. For me it is really 1. and 3. that prevent me from viewing this as a derivation of the Born weights.

 

[Derek P]

Summing up, in case my previous posts are confusing
[...]
Make of these objections what you will. For me it is really 1. and 3. that prevent me from viewing this as a derivation of the Born weights.

Thanks for all of that. I put my hand up to being one of @Mentz114's "those of us who don't have the inclination/time/ability required to read it". All three in my case, at least until adequately motivated to spend a year focussed on it . Which on current showing appears to be vanishingly unlikely. Or, as I am learning to say, "On the assumption that I am a rational agent, my optimisal strategy for maximising my chosen reward, that of understanding quantum mechanics better, is to act as if there is not a snowflake in hell's chance of getting anything from Wallace".

But I have a problem with 4.

1. Wallace's proof presumably (purports to) eliminate personal preferences from the argument, so that using the Born Weights is optimal whether you just want to get to work on time or have a penchant for quantum suicide.
2. So let's say it optimises your reward if you act as if there were a 50/50 chance of Schroedinger's cat being alive or dead.
3. You appear to be saying that this does not imply 50% frequency.
4. It seems to me that it does.
5. Because of 1, the decision to act as if the frequency were 50% must include cases where the reward is to estimate the frequencies correctly.
6. So the optimised stategy for estimating the frequencies correctly is to use the Born Weights.
7. If the Born Weights are well-defined and the frequencies are well-defined then there is only one correct estimate.
8. Therefore using the Born Weights for estimating the frequencies does, in fact, yield the correct value.

I've probably missed what you're saying or perhaps my argument is invalid. But if it seems valid to you we had better tell Mandolesi that he can stop what he's doing and ... ...

 

[Derek P]

I'm not at all suggesting a principled reason for choosing one coarse-graining over another. But I was postulating a very specific coarse-graining, which is that the coarse-grained state determines, for every volume of space down to some minimal volume, the particle content, total energy, total momentum, average electric field, average magnetic field, etc. I'm assuming that a single coarse-grained state would not be compatible with a cat being both alive and dead. As far as what's going on in Alice's mind, I don't actually know about that. That could require very fine-grained information, or maybe not.

Well your ∏ operator seemed to extract exactly one bit of information from a ket, namely whether Alice thinks "dead ket" or "alive ket". Her mind as such doesn't come into it. She could be an electron whose spin preparation is determined by the state of the detector. One qubit in, one classical bit out! I'll talk with you about coarse-graining soon.

 

[DarMM]

But I have a problem with 4

I get what you're saying. To some people it's because there isn't a clear mechanism given for the rationality as discussed in Wallace's paper. I think Wallace can't give a clear mechanism because it depends on exactly how the non-trace based derivation of decoherence will actually work when it is brought to fruition.

Mandolesi for example has another paper where he sketches a research program for deriving decoherence via causal stability. In it macrostates with large amplitudes are more resilient to interference and hence have more microstates "implementing" them where interference isn't so strong as to prevent the existence of rational agents (which require stable memory, an arrow of time, etc). That is they have a higher microworld count where they can give rise to beings like us. Hence you'd take the Born weight into account when assessing a macroworld, because there will more microworlds refining it with a you in them. I'm still digesting that paper, so I might be slightly off in that summary. I'll return with a correction if I'm significantly wrong. Though note the many-minds nature of the explanation.

So even accepting Wallace's proof, we don't know "why" the Born weights are the rational weights, just that they are. I think I'll await Mandolesi's third paper before commenting on this more, because he might have explicit examples where the Born weights are rational to use, but don't manifest in a world's experimental history.

 

[akvadrako]

I'm just asking why the more-or-less obvious approach is said to fail as well as all others.

There are quite a few approaches and even if you discover one is technically sound, you haven't dealt with the main issue of why they are all said to fail. Your focus on macro-states and course-graining is an attempt to do basically the same thing as Gleason's theorem, which says roughly if there is a probabilistic interpretation of WMI, it must be the Born rule. So it's not that there are viable alternatives to the Born rule, but you have to accept the premise.

That is why all the other approaches exist; an attempt to demonstrate this premise. Given a deterministic system where almost everything happens, how can we interpret it probabilistically from the perspective of something inside the system?

 

[DarMM]

I'm currently reading Zurek's papers and follow up articles concerning his envariance proof of the Born rule. This is the second method to derive the Born rule in Many-Worlds and does not use decision theory as Wallace does. I will again provide a summary here, so that we have both of the major derivations of the Born rule covered.

I will say already it is much clearer than the decision theory derivations, as Zurek gives an explicit physical meaning to everything he says.

 

[A. Neumaier]

I'm not at all suggesting a principled reason for choosing one coarse-graining over another. But I was postulating a very specific coarse-graining, which is that the coarse-grained state determines, for every volume of space down to some minimal volume, the particle content, total energy, total momentum, average electric field, average magnetic field, etc.

This will give you precisely the hydromechanic field variables. There is a very principled way of doing this, described in the book on nonequilibrium quantum field theory by Calzetta and Hu, resulting (after some approximation) in classical hydromechanic equtions. Or, for the nonrelativistic case, see the statistical physics book by Linda Reichel, who starts from N-particle quantum mechanics.

The coarse-graned observables are precisely the expectation values of associated quantum fields. Thus with your vision you come very close to my thermal interpretation.

 

[Derek P]

I get what you're saying. To some people it's because there isn't a clear mechanism given for the rationality as discussed in Wallace's paper.

That's the sort of statement that has me baffled. I really can't attach any meaning to "mechanism for rationality". I'm guessing it means something like a physical model for a rational decision. Which is a very specific philosophical stance, essentially that "we of the xyz school do not accept that logic can be decoupled from human thinking, and therefore logical propositions are experimental hypotheses subject to Popperian falsification and empirical verification." I may be overstating it a little. But maybe you're talking about something completely different.

So even accepting Wallace's proof, we don't know "why" the Born weights are the rational weights, just that they are. I think I'll await Mandolesi's third paper before commenting on this more, because he might have explicit examples where the Born weights are rational to use, but don't manifest in a world's experimental history.

Well I bet he can't except in the trivial sense of inherently one-off events, if there are such things. We shall see, And I'll let someone else digest the proffered argument for me!

 

[stevendaryl]

I guess I've made this point before, so it's a little redundant, but it's a little ridiculous to me to ask about what's a rational agent's motivation for accepting the Born rule. I don't think there is any more to it than the fact that his experience of previous experiments show relative frequencies in agreement with the Born rule. That seems necessary and sufficient for the rational agent to accept the rule.

The problem with Many Worlds is that there are possible worlds where the Born rule gives the wrong relative frequencies. Well, in those worlds, people just aren't going to accept the Born rule. They are not going to be persuaded by a decision-theoretic argument, no matter how mathematically sound.

 

[DarMM]

That's the sort of statement that has me baffled. I really can't attach any meaning to "mechanism for rationality".

I think it just means "why is it the rational way to act?"

Is it because for worlds with high Born weight there are more copies of that world, or worlds with that outcome are more stable, or worlds with that outcome have more subworlds with rational agents, etc?

 

[DarMM]

I guess I've made this point before, so it's a little redundant, but it's a little ridiculous to me to ask about what's a rational agent's motivation for accepting the Born rule. I don't think there is any more to it than the fact that his experience of previous experiments show relative frequencies in agreement with the Born rule. That seems necessary and sufficient for the rational agent to accept the rule.

The problem with Many Worlds is that there are possible worlds where the Born rule gives the wrong relative frequencies. Well, in those worlds, people just aren't going to accept the Born rule. They are not going to be persuaded by a decision-theoretic argument, no matter how mathematically sound.

That's fine, but it really implies the coefficients have no real intrinsic meaning physically. There are just some worlds where the ratios between outcomes randomly happen to align with them, but such worlds aren't "common" in any sense, worlds with uniform ratios are just as common. Nor should you expect the Born rule to continue to work, as your history has just randomly happened to align with it thus far, but since they don't really mean anything why would you expect this to continue.

I think to trust the Born rule regarding the future, you have to have some reason why an agent would use or trust them.

 

[stevendaryl]

I guess I've made this point before, so it's a little redundant, but it's a little ridiculous to me to ask about what's a rational agent's motivation for accepting the Born rule. I don't think there is any more to it than the fact that his experience of previous experiments show relative frequencies in agreement with the Born rule. That seems necessary and sufficient for the rational agent to accept the rule.

I guess the issue is whether the rational agent would continue to believe the Born rule after you've explain "Many Worlds" to him.

 

[Derek P]

I guess the issue is whether the rational agent would continue to believe the Born rule after you've explain "Many Worlds" to him.

I wonder how Wallace deals with that.

 

[Derek P]

That's fine, but it really implies the coefficients have no real intrinsic meaning physically. There are just some worlds where the ratios between outcomes randomly happen to align with them, but such worlds aren't "common" in any sense, worlds with uniform ratios are just as common. Nor should you expect the Born rule to continue to work, as your history has just randomly happened to align with it thus far, but since they don't really mean anything why would you expect this to continue.

I think to trust the Born rule regarding the future, you have to have some reason why an agent would use or trust them.

It's called the Principle of Induction. It's a principle that has always worked in the past so it's bound to work in the future

 

[DarMM]

It's called the Principle of Induction. It's a principle that has always worked in the past so it's bound to work in the future

The point would be without some physical account of the Born weightings and given the truth of Many Worlds, the principle of induction would be invalid to apply to the Born rule. You would know that worlds where it continues to hold, even given it has held, are in an almost vanishing minority.

 

[Derek P]

I think it just means "why is it the rational way to act?"

Is it because for worlds with high Born weight there are more copies of that world, or worlds with that outcome are more stable, or worlds with that outcome have more subworlds with rational agents, etc?

Well obviously it's no use asking me what Wallace thinks, if that's still the context. I cannot imagine how he escapes using a statistical argument to define an expectation value. But if, as this would imply, his method freely uses the idea of probability when applied to propositions - the chances of P being true - then you don't need any tarradiddle about rational agents... But what would I know?

 

[DarMM]

But if, as this would imply, his method freely uses the idea of probability when applied to propositions - the chances of P being true

It doesn't. More clearly what he does is show that assuming a branching structure can be derived without use of tracing (something that has yet to be even half-rigorously shown), a certain level of control over the environment and assuming a certain form to rational reasoning (that is a definition of rationality in an Everettian world), then the Born weights can be shown to be involved in any rational decision.

This doesn't, as such, show they have anything to do with probabilities. It just shows that any rational decision (where "rational" is defined by his axioms) will involve them.

 

[Derek P]

The point would be without some physical account of the Born weightings and given the truth of Many Worlds, the principle of induction would be invalid to apply to the Born rule. You would know that worlds where it continues to hold, even given it has held, are in an almost vanishing minority.

The Principle of Induction does not depend on the physical account. That's kind of the point of it.

 

[DarMM]

The Principle of Induction does not depend on the physical account. That's kind of the point of it.

I think something is confused here.

Many Worlds, without some physical meaning behind the Born weights, would predict that they shouldn't continue to hold as ratios of experiments with almost certain probability.

Consider General Relativity and a being who has only experienced Newtonian gravitational wells. If you place him near a Neutron Star, by the principle of induction he would say Newtonian gravity should continue to hold, but General Relativity will say this is going to be wrong and it is in fact wrong.

Many-Worlds without a physical meaning for the Born Weights, which I'll just call Raw Many Worlds, would predict they will fail to hold as ratios of outcomes. You can use the principle of induction if you want, but Raw Many Worlds will say you will be wrong.

Hence the issue here is a prediction of Raw Many Worlds, I don't think the principle of induction matters to that.

 

[DarMM]

The Principle of Induction does not depend on the physical account. That's kind of the point of it.

Another way of coming at this, of course one can use the Principle of Induction, but why is that relevant when the theory is predicting that Born ratios are going to fail to hold.

 

[Derek P]

And another way of coming to it is to notice the smily!

 

[Derek P]

I think something is confused here.

Many Worlds, without some physical meaning behind the Born weights, would predict that they shouldn't continue to hold as ratios of experiments with almost certain probability.

Consider General Relativity and a being who has only experienced Newtonian gravitational wells. If you place him near a Neutron Star, by the principle of induction he would say Newtonian gravity should continue to hold, but General Relativity will say this is going to be wrong and it is in fact wrong.

Many-Worlds without a physical meaning for the Born Weights, which I'll just call Raw Many Worlds, would predict they will fail to hold as ratios of outcomes. You can use the principle of induction if you want, but Raw Many Worlds will say you will be wrong.

Hence the issue here is a prediction of Raw Many Worlds, I don't think the principle of induction matters to that.

You've lost me. There's always a physical meaning to the Born Weights. The principle of induction doesn't depend on it though and yes it will mislead you on occasion. I don't know what you're getting at at all.

 

[Derek P]

There are quite a few approaches and even if you discover one is technically sound, you haven't dealt with the main issue of why they are all said to fail. Your focus on macro-states and course-graining is an attempt to do basically the same thing as Gleason's theorem, which says roughly if there is a probabilistic interpretation of WMI, it must be the Born rule. So it's not that there are viable alternatives to the Born rule, but you have to accept the premise.

That is why all the other approaches exist; an attempt to demonstrate this premise. Given a deterministic system where almost everything happens, how can we interpret it probabilistically from the perspective of something inside the system?

There is no difficulty at all in having probabilities in a deterministic theory. (I have a horrible feeling I may have said the opposite at some point, but if so I was wrong.) Probabilities arise through someone inside the system not knowing which "part" they are in. That would not help if there was only one observer or observer-state. But if each "part" has its own version, the observer can say "the probabilty of my being in such-and-such a subset of all the states is ..."

 

[DarMM]

There's always a physical meaning to the Born Weights

I was talking about Wallace's proof where this physical meaning is not obvious or stated. Or Many-Worlds prior to the Born Rule where it is absent.

What is the physical meaning in the version of Many Worlds you are considering?