Many worlds interpretation with unequal probabilities

[Sherwood Botsford]

TL;DR Summary

How does the many worlds interpretation deal with events where there are two possible outcomes with unequal probability. Or worse, when the probabiliy ratio is an irrational number?

Let's pay a visit to one of Schrodinger's cats.

In the classical statement of the case, we have to decide if the cat is alive or dead when the probability of the radio-active decay mechanism has a 50/50 chance of releasing the cyanide, most often posed as 60 minutes.

If I understand the MW interpretation, at the point we make the observation, the universe splits into two, one with a dead cat, and one with a live cat, and a less unhappy chapter of PETA.

But suppose I make the observation at 23 minutes. This would require a review for me of exponential functions to calculate, but I think it's about 25%. In MWI are there 4 universes with 3 live cats and one dead one?

 

[DaveC426913]

But suppose I make the observation at 23 minutes. This would require a review for me of exponential functions to calculate, but I think it's about 25%. In MWI are there 4 universes with 3 live cats and one dead one?

We can rule out that conclusion pretty quickly I think. It takes math to get from 25% to 1-in-4. And math was invented by humans. 25% just as easily converts to 2-in-8, or 5-in-20, or 25 in 100, so why would the universe not instead produce 75 live cats and 25 dead ones?

 

[Sherwood Botsford]

We can rule out that conclusion pretty quickly I think. It takes math to get from 25% to 1-in-4. And math was invented by humans. 25% just as easily converts to 2-in-8, or 5-in-20, or 25 in 100, so why would the universe not instead produce 75 live cats and 25 dead ones?

Minimum effort? Automatic merging of identical universes? Some conservation principle?

But suppose the ratios are .5 * sqrt(2) and 1- (.5 * sqrt(2))

 

[Halc]

If I understand the MW interpretation, at the point we make the observation, the universe splits into two, one with a dead cat, and one with a live cat

You seem to be referencing a common misconception. Given what you say, the cat would be in countless states, each corresponding to a death after X much time and the non-stationary cat dying in a different position, plus the more common outcome where it lives.

Very few physicists have actually read Everett’s book [ref], which has lead to a common misconception that it contains a second postulate along the following lines:
What Everett does NOT postulate:
At certain magic instances, the the world undergoes some sort of metaphysical “split” into two branches that subsequently never interact.

https://arxiv.org/pdf/quant-ph/9709032.pdf

Read the shortish paper. It's quite informative and addresses several misconceptions and criticisms.

 

[hutchphd]

For you there is only the present universe. You are either feeding your kitty or digging a hole out back. If you had 100 similar kitties then you would most likely be digging 25 holes and filling 75 dishes. The shorter the time period in the box , the more Kibble you will need.

 

[Vanadium 50]

In many worlds, there aren't many worlds. There is only one world. As such, irrational probabilities are no problem.

 

[gentzen]

https://arxiv.org/pdf/quant-ph/9709032.pdf

Read the shortish paper. It's quite informative and addresses several misconceptions and criticisms.

This paper says more about its author (and his ideas about mathematics) than about the MWI. And what it actually says about MWI (IV.B) is worse than merely being wrong.
Maybe mathematicians should have discussed his ideas more generously, and ignored the many errors in details (in his main papers and the book about his ideas). Even so logic might best be interpreted as a language, and math is closely related to logic, it seems to have substance beyond merely being a language. And also substance beyond mere consistency of large cardinals and ordinals.
But all that has very little to do with MWI.

 

[PeterDonis]

what it actually says about MWI (IV.B)

That's not all it says about the MWI. The discussion in section II looks fine to me.

 

[gentzen]

That's not all it says about the MWI. The discussion in section II looks fine to me.

Perhaps it was fine, but not very informative (for me). So I decided that papers about MWI are better evaluated with respect to the important ideas and goal of their authors (I am thinking of people like D. Deutsch, D. Wallace, S. Carroll, H. Everett, D. Zeh, …) than with respect to the details of their interpretational content. This made me notice that M. Tegmark‘s ideas and goals have more to do with philosophy of mathematics than with physics.

 

[gentzen]

That's not all it says about the MWI. The discussion in section II looks fine to me.

It is true that I accused S. Hossenfelder of committing a similar mistake with respect to MWI as described in section II. In this sense, I agree that section II is fine.

 

[kered rettop]

TL;DR Summary: How does the many worlds interpretation deal with events where there are two possible outcomes with unequal probability. Or worse, when the probabiliy ratio is an irrational number?

Let's pay a visit to one of Schrodinger's cats.

In the classical statement of the case, we have to decide if the cat is alive or dead when the probability of the radio-active decay mechanism has a 50/50 chance of releasing the cyanide, most often posed as 60 minutes.

If I understand the MW interpretation, at the point we make the observation, the universe splits into two, one with a dead cat, and one with a live cat, and a less unhappy chapter of PETA.

But suppose I make the observation at 23 minutes. This would require a review for me of exponential functions to calculate, but I think it's about 25%. In MWI are there 4 universes with 3 live cats and one dead one?

I think you mean worlds, not universes. If MWI talked about universes it would be MUI, not MWI.

The term world is still ambiguous but the simplest definition would be a macroscopic quantum state that contains information about a specific outcome of a measurement. So that means there are still only two worlds, a dead-cat world and a living-cat world. States have magnitudes, and the magnitudes will be different (and quite likely irrational).

 

[PeroK]

I think you mean worlds, not universes. If MWI talked about universes it would be MUI, not MWI.

The term world is still ambiguous but the simplest definition would be a macroscopic quantum state that contains information about a specific outcome of a measurement. So that means there are still only two worlds, a dead-cat world and a living-cat world. States have magnitudes, and the magnitudes will be different (and quite likely irrational).

That can't be correct. The cat could have died at any time during the hour. The precise time of death could be established by a close examination of the cat. There is, in any case, no uniquely identifiable state that corresponds to a dead cat. Nor to a live cat. To suppose there are only two states seems to invoke the "magic split" mentioned above.

If we take radioactive decay to have a continuous set of outcomes based on time, then there are potentially an uncountably infinite number of dead cat states.

 

[joshlevin10]

We can rule out that conclusion pretty quickly I think. It takes math to get from 25% to 1-in-4. And math was invented by humans. 25% just as easily converts to 2-in-8, or 5-in-20, or 25 in 100, so why would the universe not instead produce 75 live cats and 25 dead ones

I think that math being “invented” is an opinion of yours but not necessarily a factual statement. It’s possible that math is a discovered phenomena rather than an invented one to describe the universe. The human counting system is based on the number 10 because that is how many fingers we have so the number 100 representing 100 percent is also a factor of 10 because we have 10 fingers. A wave function is not based on the number 10 but based on infinity. That mathematically should mean that every probability is realized in a wave function and that each probability is equal to every other probability…. Meaning that something that was statistically more significant than something else would still have an equal probability of occurrence in MWI.

 

[kered rettop]

That can't be correct.

Oh yes it can.

The cat could have died at any time during the hour. The precise time of death could be established by a close examination of the cat. There is, in any case, no uniquely identifiable state that corresponds to a dead cat. Nor to a live cat. To suppose there are only two states seems to invoke the "magic split" mentioned above.

There doesn't need to be a uniquely identifiable state. There just needs to be an operator which resolves the cat-state into |dead> and |alive> That such a thing exists is obvious since we can, in fact, tell a dead cat from a living one.

If we take radioactive decay to have a continuous set of outcomes based on time, then there are potentially an uncountably infinite number of dead cat states.

Again, depending on the fine-graining. Applying the dead/alive operator to the wave function gives us two states. But if you switch to a microstate basis then it's misleading to talk about the number of dead-cat states - there are only microstates. To bridge between the two you could say "there are are an infinite number of microstates in the dead cat-state".

 

[DaveC426913]

I think that math being “invented” is an opinion of yours but not necessarily a factual statement. It’s possible that math is a discovered phenomena rather than an invented one to describe the universe.

Then you haven't addressed the question I posed to Sherman Botsford. He went from 25% to 1-in-4, I.e. four permutations. Why not 2 in 8? Or 19 in 76?

The human counting system is based on the number 10 because that is how many fingers

That's awfully specious and kinda Euro centric. Some human counting systems are based on 60. I use base 2 a lot.

"The Mayans used a vigesimal (base 20) number system, the Babylonians used a sexagesimal (base 60) number system, and the Egyptians used a duo-decimal (base 12) number system. It is believed that the Mayans used base 20 because they lived in a warm climate where they did not wear shoes, thus giving 20 fingers and toes."
https://www.math.drexel.edu/~jsteuber/Educ525/History/history.html

Base 10 was just an early one that caught on in a particularly virulent part of civilisation and spread.

A wave function is not based on the number 10 but based on infinity.

Is it? What does it mean to be "based on infinity"? Citation?None of this addresses the challenge I posed, which is the crux: Sherman thought it was pretty plain that 4 universes would be created, corresponding to his 25% dead/alive calculations. Why not 76? Why not 2¹⁰¹?

 

[PeroK]

Oh yes it can.

There doesn't need to be a uniquely identifiable state. There just needs to be an operator which resolves the cat-state into |dead> and |alive> That such a thing exists is obvious since we can, in fact, tell a dead cat from a living one.Again, depending on the fine-graining. Applying the dead/alive operator to the wave function gives us two states. But if you switch to a microstate basis then it's misleading to talk about the number of dead-cat states - there are only microstates. To bridge between the two you could say "there are are an infinite number of microstates in the dead cat-state".

The MWI must be about microstates, as macroscopic objects like cats don't behave like elementary quantum systems. When you observe a cat, in no sense is that observation what determines whether the cat is alive or dead. A cat is not like a radioactive isotope that is in a single stable state and then randomly decays.

Even if you say that a cat is alive and then suddenly it's dead, the macroscopic process involving ageing or illness or poisoning is a completely different process from radioactive decay.

Observing a cat does not force it into an eigenstate of the live/dead operator. Live and dead cannot be eigenstates of any macroscopic observable.

A macroscopic observation about a cat is a generic, not a specific, observation. You also observe what colour it is, whether it's asleep or scratching itself. This is fundamentally different from observing the spin on an electron. Likewise, Observing a cat is not what determines whether it is asleep or awake.

If a cat did behave like that then there would be no mystery to the experiment. The whole point of Schrodinger's cat is that observing the cat should not affect the state of the cat. In that sense it is not an elementary quantum object.

The other problem with your argument is that all cats are unique systems. Quantum Mechanics deals with elementary, indistinguishable systems, governed by quantum statistics. Even macroscopic objects like red snooker balls that appear identical obey classical statistics and classical probability theory. Red snooker balls are neither bosons nor fermions,nor do they obey Pauli's Exclusion Principle.

 

[kered rettop]

MWI is predicated on QM being universal. A great deal of quantum theory is devoted to showing that classical physics is emergent on QM rather than being fundamentally different. So I don't see the point in discussing cat-physics which isn't quantum.

 

[DaveC426913]

MWI is predicated on QM being universal. A great deal of quantum theory is devoted to showing that classical physics is emergent on QM rather than being fundamentally different. So I don't see the point in discussing cat-physics which isn't quantum.

And yet, it is very difficult to describe the parameters of a cat in QM terms, rather than classical terms.

And as always, where does one draw the line? Is a 30cm cat classical or quantum mechanical?
30 micrometers? 30 angstroms?

 

[kered rettop]

And yet, it is very difficult to describe the parameters of a cat in QM terms, rather than classical terms.

And as always, where does one draw the line? Is a 30cm cat classical or quantum mechanical?
30 micrometers? 30 angstroms?

Perhaps it is time to talk about unicorns rather than cats.

 

[DaveC426913]

Perhaps it is time to talk about unicorns rather than cats.

I feel like that doesn't address my post, or the thread in general.

It seems the discussion is very much about the fuzzy (or possibly non-existent) line between QM particles and macro objects.

 

[jbergman]

Perhaps it is time to talk about unicorns rather than cats.

I have to agree with @kered rettop 's analysis of the situation.

The article at https://plato.stanford.edu/entries/qm-manyworlds/ is a brilliant primer for anyone wanting to understand the MWI interpretation of QM. In particular, the section on the preferred basis problem talks about the partitioning of hilbert space into worlds in order to represent a quantum state.

 

[DaveC426913]

(Well that's a slightly more useful response...)

 

[kered rettop]

I feel like that doesn't address my post, or the thread in general.

Well DaveC426913's Law says that if we start talking about unicorns, the discussion is over. So yes, unicorns!

It seems the discussion is very much about the fuzzy (or possibly non-existent) line between QM particles and macro objects.

Yes. But it's confounded by the fact that we refer to a cat's state as being alive or dead - which are not defined physically at any scale.

 

[DaveC426913]

But it's confounded by the fact that we refer to a cat's state as being alive or dead - which are not defined physically at any scale

Why? We certainly can tell a dead cat from a live one. In fact, it doesn't even have to be dead at all. All we need to know is that there is poison in its body. That is a macro, classical state that definitively answers the did-it-or-didnt-it question of the experiment. An actual dead act is simply morbid icing on the cake.

In fact, the cat itself is unnecessary. If the air has poison in it, the experiment has collapsed into that macro, classical state.

 

[kered rettop]

Why? We certainly can tell a dead cat from a live one. In fact, it doesn't even have to be dead at all. All we need to know is that there is poison in its body. That is a definitively macro, classical state that proves the experiment. An actual dead act is simply morbid icing on the cake.

Yes. That's what I'm saying.

 

[hutchphd]

Note to self: don't eat the cake at Dave's......
But the question of the exact instant(measurement) of death is fraught. In some ways it is a very "quantum" concept (in the same way that "pregnant" is.....) and we have left the realm of physics now.

 

[DaveC426913]

Yes. That's what I'm saying.

Well OK, but the issue of quantum events deciding the state of classical macro systems is still on the table. So we still need to discuss how that gap is bridged. That's what the topic is.

The cat is simply a convenient tool that we all understand to be a stand-in for any classical system. IOW, removing it changes nothing except eliminating an otherwise useful conceptual tool.

We could instead talk about the little hammer having fallen or not, if that would help simplify things.

 

[kered rettop]

Well OK, but the issue of quantum events deciding the state of classical macro systems is still on the table. So we still need to discuss how that gap is bridged. That's what the topic is.

Is it? I thought it was about unequal probabilities.

The cat is simply a convenient tool that we all understand to be a stand in for any classical system. IOW, removing it changes nothing except eliminating an otherwise useful conceptual tool.

Yep!

 

[kered rettop]

We could instead talk about the little hammer having fallen or not, if that would help simplify things.

If you want simplicity, stick with a beam-splitter. A single particle whose wave function contains the reflected photon and the transmitted photon in equal proportions???? Quite ridiculous!

Now can we talk about unicorns?

 

[PeterDonis]

There just needs to be an operator which resolves the cat-state into |dead> and |alive>

You are missing the point. The point is that this operator you speak of is a different operator if we apply it at different times. In the Heisenberg picture this is obvious. And if you are trying to count "worlds", you have to take that into account; you can't just say "well, the cat is either dead or alive", because different times of death for the cat are macroscopically distinguishable, so they count as different worlds.

You are not addressing this point at all.

 

[PeterDonis]

A single particle whose wave function contains the reflected photon and the transmitted photon in equal proportions???? Quite ridiculous!

Huh? What's ridiculous about it?

 

[PeterDonis]

How does the many worlds interpretation deal with events where there are two possible outcomes with unequal probability. Or worse, when the probabiliy ratio is an irrational number?

I don't know if there is a unique answer to this among MWI proponents. Some view the relative weights of the branches as a "measure" of the worlds, others don't.

 

[PeterDonis]

There is, in any case, no uniquely identifiable state that corresponds to a dead cat. Nor to a live cat. To suppose there are only two states seems to invoke the "magic split" mentioned above.

No, it isn't. You don't need to have just two states. All you need is two orthogonal subspaces of the cat's Hilbert space. Each subspace could contain zillions of states, as long as all of the states in the "alive" subspace are orthogonal to all of the states in the "dead" subspace.

The real issue is whether what I have just described is actually feasible. Discussions of the Schrodinger's cat thought experiment seem to assume without argument that it is, but I don't think anyone has actually tried to prove it, or even to give a plausibility argument for it.

 

[kered rettop]

Huh? What's ridiculous about it?

I was being ironic, for goodness sake!

 

[kered rettop]

You are missing the point. The point is that this operator you speak of is a different operator if we apply it at different times. In the Heisenberg picture this is obvious. And if you are trying to count "worlds", you have to take that into account; you can't just say "well, the cat is either dead or alive", because different times of death for the cat are macroscopically distinguishable, so they count as different worlds.

You are not addressing this point at all.

You can sum the timed dead-cat worlds together to create a single timeless one so I'm not sure what the point is that you think I ought to address.