I Danger for the Many-Worlds Interpretation?

[PeterDonis]

If in this context consciousness is not a matter of the physical state of your brain, then MWI is not an interpretation of QM but rather philosophical theory that just as well could be incompatible with philosophical basis of science, don't you agree?

If consciousness is not a matter of the physical state of your brain, then no theory of physics can account for your conscious experience. But this has nothing to do with whether the MWI is an interpretation of QM or a different theory; that depends only on whether the MWI makes different experimental predictions from standard QM. As MWI is currently formulated, it doesn't.

 

[Minnesota Joe]

The difficulty is in justifying why she would adjust her credences to be $|a(\omega)|^2$ in the non-uniform case. That's essentially what all proofs of the Born Rule in MWI attempt to do, but none manage it in a way that is considered generally convincing in the Foundations community.

If you want to look into the attempted proofs fall into three rough classes. Frequency type proofs offered by de Witt and others that build on the work of Everett. Proofs based on a certain type of invariance under swapping of environmental states due to Zurek and the proofs based on decision theoretic arguments by Wallace and Deutsch.

Yeah, I think the frequentist approach has all the problems that frequentism has in probability theory. I don't know anything about the invariance approach, thanks! The last approach is what Hossenfelder complains about at the end of her video and what David Albert addresses in his interview with Sean Carroll.

Meanwhile it seems like Carroll argues that, in the non-uniform case, you are forced to make the $|a(\omega)|^2$ choice to prevent future branching from changing the probabilities on parallel branches.

 

[Minnesota Joe]

But you are still going to have to "collapse" the wave function you use to predict your own probabilities for future measurement results. You might say the other branches of the wave function exist, but you won't actually calculate as if they exist; you will calculate as if your own branch is the only one that exists, because that's the only one you will use in your calculations. And doing that is not compatible with the Schrodinger Equation, because there is no "collapse" in the SE. The SE does not tell you to calculate using only one branch. It tells you to calculate using all the branches.

In other words, when the MWI says "collapse is only apparent and there is no contradiction with the Schrodinger Equation", it doesn't really mean that literally; it can't, since taking it literally leads you to calculate things that are contrary to what you actually observe.

Hmm. It doesn't mean literally? You mean they would agree that really there is Schrodinger equation violating collapse? Because they don't seem to, but I could be misunderstanding them.

It seems to me there is a distinction between the universal wave function collapsing and a branch-bound observer ignoring other current, non-interacting branches in order to describe future branches. But I don't know. And it's making my head hurt. An example would be appreciated.

 

[DarMM]

Yeah, I think the frequentist approach has all the problems that frequentism has in probability theory.

It's problems are separate technical ones related to certain limiting operators not converging.

I don't know anything about the invariance approach, thanks! The last approach is what Hossenfelder complains about at the end of her video and what David Albert addresses in his interview with Sean Carroll.

Carroll's proof is in the same class as the invariance proofs. The decision theoretic proofs are a bit different.

Meanwhile it seems like Carroll argues that, in the non-uniform case, you are forced to make the $|a(\omega)|^2$ choice to prevent future branching from changing the probabilities on parallel branches.

It doesn't work out as even other Many Worlds advocates have noted. See here:
http://philsci-archive.pitt.edu/14389/

 

[PeterDonis]

It seems to me there is a distinction between the universal wave function collapsing and a branch-bound observer ignoring other current, non-interacting branches in order to describe future branches.

There is according to the MWI, yes: the MWI says the universal wave function never collapses, and it also says that an observer who observes a particular measurement result can ignore all other branches except his own. The MWI has to say that because it's the only way to make it consistent with experiment.

The problem Hossenfelder is pointing out, as I understand it, is that admitting the latter claim, that an observer can ignore all branches other than his own, means the former claim, that the universal wave function never collapses, is untestable by definition, which makes it unscientific.

 

[Minnesota Joe]

Carroll's proof is in the same class as the invariance proofs. The decision theoretic proofs are a bit different.

That helps!

It doesn't work out as even other Many Worlds advocates have noted. See here:
http://philsci-archive.pitt.edu/14389/

Ugh, what a mess. I'm not sure what to think about that or what should be granted or challenged in Sean's description.

I'm just trying to understand the outline of the MWI argument and the motivation right now.

The claim seems to be that the Schrodinger equation describes a branching structure that introduces an uncertainty in your epistemic situation so that it makes sense to apply elementary probability theory to the branch structure at some stage so that you can derive the Born Rule which would justify your claim to having derived the Born Rule from only the Schrodinger equation and simpler stuff that everyone agrees with.

Sound right?

 

[timmdeeg]

The problem Hossenfelder is pointing out, as I understand it, is that admitting the latter claim, that an observer can ignore all branches other than his own, means the former claim, that the universal wave function never collapses, is untestable by definition, which makes it unscientific.

Sabine Hossenfelder claims [s. the OP]:

To "evaluate the probability relative to the detector in one specific branch at a time" is "logically entirely equivalent to the measurement postulate."

But isn't this claim not just Kopenhagen view? And if yes, so what? On the other side this reasoning seems too simple, so how would you comment on that?

 

[DarMM]

I'm just trying to understand the outline of the MWI argument and the motivation right now.

The claim seems to be that the Schrodinger equation describes a branching structure that introduces an uncertainty in your epistemic situation so that it makes sense to apply elementary probability theory to the branch structure at some stage so that you can derive the Born Rule which would justify your claim to having derived the Born Rule from only the Schrodinger equation and simpler stuff that everyone agrees with.

That is roughly claim as it was in the late 1990s. Problems include the fact that to derive the branching structure you need the Born rule, so to take a branching structure as given and then use it to derive a Born rule is already quite circular.

Thus most MWI people now try to get the branch structure out without using the Born rule. This has not succeeded yet. Thus it's not yet an interpretation of QM strictly speaking as it is unknown if it can real replicate the predictions.

 

[Minnesota Joe]

That is roughly claim as it was in the late 1990s. Problems include the fact that to derive the branching structure you need the Born rule, so to take a branching structure as given and then use it to derive a Born rule is already quite circular.

Thus most MWI people now try to get the branch structure out without using the Born rule. This has not succeeded yet. Thus it's not yet an interpretation of QM strictly speaking as it is unknown if it can real replicate the predictions.

Perhaps my phrasing was poor, because I don't understand this. The only thing MWI has is the Schrodinger equation. It's just a real wave equation to them. The branching comes through entanglement and decoherence which are features of the Schrodinger equation and matter. It isn't necessary to assume probability at this stage, just interaction. So the branching structure doesn't assume the Born Rule.

 

[DarMM]

Perhaps my phrasing was poor, because I don't understand this. The only thing MWI has is the Schrodinger equation. It's just a real wave equation to them. The branching comes through entanglement and decoherence which are features of the Schrodinger equation and matter. It isn't necessary to assume probability at this stage, just interaction. So the branching structure doesn't assume the Born Rule.

Decoherence requires the Born rule, it's not purely a feature of the Schrodinger equation .

 

[Minnesota Joe]

Sabine Hossenfelder claims [s. the OP]:

To "evaluate the probability relative to the detector in one specific branch at a time" is "logically entirely equivalent to the measurement postulate."

But isn't this claim not just Kopenhagen view? And if yes, so what? On the other side this reasoning seems too simple, so how would you comment on that?

This is why I muddied up your nice thread to show that there is different content to the claims of the MWI people. They are not evaluating the probability relative to the detector in a single branch and saying: the wave function collapses. So how is it "logically equivalent"?

I mean, maybe she is correct. But she doesn't elaborate enough for me in her video, post, or comment thread. She doesn't show how, necessarily, "evaluate the probability relative to the detector in one specific branch at a time" entails "the measurement problem".

And how can she? The measurement problem is that the Copenhagen Interpretation doesn't explain why we update our probability to 100%. It just says: do it. MWI tells you why this occurs.

I'm not defending MWI by the way, despite what it might seem like. I'm trying to understand the bloody thing. I just don't want to dismiss it as the same thing as the Copenhagen interpretation too hastily and not give MWI its due.

 

[DarMM]

And how can she? The measurement problem is that the Copenhagen Interpretation doesn't explain why we update our probability to 100%

I would say it doesn't say "how" the outcome of a measurement comes about. Why you update your probabilities is obvious, i.e. because that's the outcome you witnessed.

 

[Minnesota Joe]

Decoherence requires the Born rule, it's not purely a feature of the Schrodinger equation .

I don't think so. It is when macroscopic things (your detector) become entangled with everything else in its environment. So the wave evolves into a superposition of terms involving your detector. That evolution is described by the Schrodinger equation. No Born Rule is required, but we are talking about many particles and interaction potentials,etc.

 

[DarMM]

I don't think so. It is when macroscopic things (your detector) become entangled with everything else in its environment. So the wave evolves into a superposition of terms involving your detector. That evolution is described by the Schrodinger equation. No Born Rule is required, but we are talking about many particles and interaction potentials,etc.

It does. To derive decoherence you need to use tracing over subsystems. Tracing as an operation assumes the Born rule. Nielsen and Chuang's famous text probably has the best introductory exposition on this.

 

[Minnesota Joe]

I would say it doesn't say "how" the outcome of a measurement comes about. Why you update your probabilities is obvious, i.e. because that's the outcome you witnessed.

MWI explains why we would develop an interpretation of quantum mechanics that just gives up and asserts the collapse postulate. You might need Bohr, Pauli, and Heisenberg and a bad attitude too.

 

[Minnesota Joe]

It does. To derive decoherence you need to use tracing over subsystems. Tracing as an operation assumes the Born rule. Nielsen and Chuang's famous text probably has the best introductory exposition on this.

Decoherence doesn't require that we interpret the square magnitude of amplitudes as giving the probability of a result of measurement. There are amplitudes of course, but that just comes from solutions to the Schrodinger equation.

 

[DarMM]

MWI explains why we would develop an interpretation of quantum mechanics that just gives up and asserts the collapse postulate. You might need Bohr, Pauli, and Heisenberg and a bad attitude too.

The collapse postulate is just a form of Bayesian conditioning. Once you view QM as involving probability theory as Copenhagen does you're going to have the collapse postulate as you always update after witnessing an event.

For example the probability that a given dice roll occurred "collapses" upon learning the outcome was even, i.e. the probabilities update.

It's more the issue of how the outcome occurs rather than "Why collapse?"

 

[DarMM]

Decoherence doesn't require that we interpret the square magnitude of amplitudes as giving the probability of a result of measurement. There are amplitudes of course, but that just comes from solutions to the Schrodinger equation.

That's not related to what I'm saying. I'm saying that decoherence requires the Born rule due to the use of the tracing operation which itself assumes the Born rule.

 

[Minnesota Joe]

That's not related to what I'm saying. I'm saying the decoherence requires the Born rule due to the use of the tracing operation which itself assumes the Born rule.

Okay, maybe we are victims of physics jargon. What Born Rule are you talking about?

ETA: I'm specifically referring to Max Born's 1926 probabilistic interpretation of the wave function.

 

[DarMM]

Okay, maybe we are victims of physics jargon. What Born Rule are you talking about?

ETA: I'm specifically referring to Max Born's 1926 probabilistic interpretation of the wave function amplitudes.

That's what I'm referring to ultimately as well. Though for decoherence you can't just use wave functions you need density matrices. In it's most general form that the probability for some event represented by a POVM element $E$ is $Tr(\rho E)$ with $\rho$ the state.

 

[Minnesota Joe]

That's what I'm referring to ultimately as well. Though for decoherence you can't just use wave functions you need density matrices. In it's most general form that the probability for some event represented by a POVM element $E$ is $Tr(\rho E)$ with $\rho$ the state.

No, that smuggles in probability unnecessarily. That's what I'm saying. It assumes textbook quantum mechanics most likely.

The Schrodinger equation is a wave equation, right? Including waves involving multiple waves (particles) like that make up macroscopic systems. Wave equations in general exhibit properties like coherence and decoherence. For example you get decoherence when you have many sources with different phase relationships. You get coherence in the ripple tank with a double-slit because both slits are emitting waves from a single source and therefore have a well-defined phase relationship (so you get constructive and destructive interference). No Born Rule required.

But Born interpreted the square of the wave function as the probability distribution and that works. So interpretations that have real waves, that don't just assume the wave function is related to probability, have to explain why this works.

 

[vanhees71]

I would say it doesn't say "how" the outcome of a measurement comes about. Why you update your probabilities is obvious, i.e. because that's the outcome you witnessed.

But isn't this a mute argument? Physics never answers "why" and "how" questions like this.

Why in Newtonian or SRT mechanics is it that there exists an inertial frame of reference? Filling this with the details about how space and time is described is all you need to do mechanics, but why this basic assumption works, is not answered. It's just used as an empirical fact to describe as many other observables using it as an input for mathematical deduction.

In QT it's the same with the probabilities. It's the (imho) so far only consistent interpretation of Schrödinger's wave function, and how the heuristics towards the Schrödinger wave function was, is well known, leading from Planck and Einstein right away to de Broglie's idea and then the somewhat ironic remark by Debye (a pupil of Sommerfeld by the way) to Schrödinger that, when you talk about waves you'd better should have a wave equation. At the next meeting Schrödinger presented one ;-)) with no clear idea about its physical meaning. Then Born happily used it to attack the problem of scattering and came to the idea with the probability interpretation (with a missing square first, but Einstein told him to put it in right away ;-)).

So, what I never understood is the obsession to "deriving" Born's rule from something else. Isn't it simply one of the basic empirical facts entering the theory like axioms are used to build a mathematical theory?

 

[DarMM]

So, what I never understood is the obsession to "deriving" Born's rule from something else. Isn't it simply one of the basic empirical facts entering the theory like axioms are used to build a mathematical theory?

Personally I would say yes it is a basic empirical fact entering the theory. Like you the only view of the quantum state that makes much sense to me and conforms with practice is a probabilistic one. I'm simply conveying the "problem" as far as MWI proponents see it.

 

[DarMM]

Wave equations in general exhibit properties like coherence and decoherence

To demonstrate decoherence you need the Born rule. Even MWI people recognise this which is why Zurek is working on Quantum Darwinism, an attempt to derive decoherence without using the Born rule.

Show me a textbook where decoherence is derived without the Born rule.

 

[PeroK]

Okay, maybe we are victims of physics jargon. What Born Rule are you talking about?

ETA: I'm specifically referring to Max Born's 1926 probabilistic interpretation of the wave function.

Even in MWI, you must have probabilities for the following reason:

You can repeatedly carry out an experiment with two outcomes where one outcome occurs, say, 90% of the time. There are lots of examples of this.

If, in MWI, there is one branch for the first outcome and one branch for the second outcome, then why do we end up 90% of the time in the world corresponding to the first outcome?

As has been pointed out several times on this thread, MWI has no good answer to this.

 

[Minnesota Joe]

To demonstrate decoherence you need the Born rule. Even MWI people recognise this which is why Zurek is working on Quantum Darwinism, an attempt to derive decoherence without using the Born rule.

Show me a textbook where decoherence is derived without the Born rule.

Okay, I agree with you on this I think: it appears they haven't derived decoherence without assuming the Born Rule in a way that is generally accepted. At the very least this is a deeper issue than I gave it credit. It appears that so far they are only motivated by having a wave equation. So that is two problems: derive decoherence without the Born Rule and derive the Born Rule after decoherence. Carroll really glosses over some important stuff on this issue. For example, he derives the Born Rule after the decoherence, but that would be viciously circular if it is impossible to derive quantum decoherence without assuming the Born Rule! Irritating. At the very least he should have mentioned this when he wrote about decoherence, because it is very important to what he says later.

 

[PeroK]

Okay, I agree with you on this I think: it appears they haven't derived decoherence without assuming the Born Rule in a way that is generally accepted. At the very least this is a deeper issue than I gave it credit. It appears that so far they are only motivated by having a wave equation. So that is two problems: derive decoherence without the Born Rule and derive the Born Rule after decoherence. Carroll really glosses over some important stuff on this issue. For example, he derives the Born Rule after the decoherence, but that would be viciously circular if it is impossible to derive quantum decoherence without assuming the Born Rule! Irritating. At the very least he should have mentioned this when he wrote about decoherence, because it is very important to what he says later.

You don't need the Born rule, per se, but without probabilities you are just going to have a large number of branches, all with equal weight.

There would be no reason then to have any phenomena consistent with one thing being more likely than another. The physics we see is dependent on the most likely outcomes being favoured.

The Born rule gives a specific outcome distribution. But, you need something; otherwise you are giving equal weight to what - classically at least - would be impossible outcomes.

 

[DarMM]

derive decoherence without the Born Rule and derive the Born Rule after decoherence.

Exactly. That is what people like Zurek are trying to do. However it hasn't worked out yet. Ruth Kastner and others have pointed out that there is circularity even in the Quantum Darwinist program as it is. See her paper here:
https://arxiv.org/abs/1406.4126

Carroll really glosses over some important stuff on this issue.

I would say so yes.

 

[akvadrako]

'm simply conveying the "problem" as far as MWI proponents see it.

I don’t think most MWI proponents consider it a problem. I think it’s mostly a focus of critics.

 

[DarMM]

I don’t think most MWI proponents consider it a problem. I think it’s mostly a focus of critics.

What do you mean? It is one of the more commonly cited issues with Copenhagen.

 

[akvadrako]

What do you mean?

I think many MWI proponents think that it's possible to derive the Born rule from unitary evolution and considered that an advantage of the theory. But that's not why they are proponents, so if it can't be derived it's not a problem. Vaidman is the most clear on this point.

Others like Carroll seem to consider the existing derivations sufficient. They do require some assumptions, which perhaps are just ways of rephrasing the Born rule, but they find them acceptable.

I think the most relevant point is that most Born rule derivations seem to be just as relevant to all interpretations and don't really have anything to do with MWI. Either it's a redundant (perhaps approximate) assumption or it must be postulated, but MWI has no advantage in this aspect.

The only really relevant point is in MWI there isn't objective collapse — so the Born rule needs to be interpreted differently.

 

[DarMM]

I think many MWI proponents think that it's possible to derive the Born rule from unitary evolution and considered that an advantage of the theory. But that's not why they are proponents, so if it can't be derived it's not a problem.

I wasn't talking about derivations of the Born rule in MWI in that post, I was talking about Copenhagen.

 

[akvadrako]

I wasn't talking about derivations of the Born rule in MWI in that post, I was talking about Copenhagen.

I see; then I would say (perhaps more in reply to vanhess's concern) the issue many worlders have with the Born rule in Copenhagen isn't the lack of derivation, but that non-unitary evolution (collapse) is incompatible with unitary evolution.

 

[PeroK]

I see; then I would say (perhaps more in reply to vanhess's concern) the issue many worlders have with the Born rule in Copenhagen isn't the lack of derivation, but that non-unitary evolution (collapse) is incompatible with unitary evolution.

Even the Copenhagen-ers have that issue with Copenhagen!

 

[DarMM]

I see; then I would say (perhaps more in reply to vanhess's concern) the issue many worlders have with the Born rule in Copenhagen isn't the lack of derivation, but that non-unitary evolution (collapse) is incompatible with unitary evolution.

Why is that a problem exactly? In Stochastic theories in general Bayesian updating is not "compatible" with the general Stochastic evolution operator.

 

[DarMM]

The only really relevant point is in MWI there isn't objective collapse — so the Born rule needs to be interpreted differently.

What's an example of one of these interpretations of the Born rule?

 

[akvadrako]

Why is that a problem exactly? In Stochastic theories in general Bayesian updating is not "compatible" with the general Stochastic evolution operator.

That seems like a problem of any theory to me. If you only apply one form of evolution at a time you need a rule about which to apply. If QM is assumed complete then there can't be any rule like that.

 

[DarMM]

That seems like a problem of any theory to me. If you only apply one form of evolution at a time you need a rule about which to apply. If QM is assumed complete then there can't be any rule like that.

What I'm saying is that in an probabilistic theory in physics or elsewhere one applies Bayesian updating after an observation and this cannot be derived from the dynamical laws that apply otherwise. Why can't there be a rule like that. Unless you essentially mean a fundamental theory cannot be probabilistic.

 

[akvadrako]

What's an example of one of these interpretations of the Born rule?

As a measure of the "stuff" that makes up reality, or in a diverging view of MWI, the number of worlds. As Vaidman says it:

Probability Postulate: An observer should set his subjective probability of the outcome of a quantum experiment in proportion to the total measure of existence of all worlds with that outcome.

 

[akvadrako]

What I'm saying is that in an probabilistic theory in physics or elsewhere one applies Bayesian updating after an observation and this cannot be derived from the dynamical laws that apply otherwise. Why can't there be a rule like that. Unless you essentially mean a fundamental theory cannot be probabilistic.

Even in a probabilistic theory, I don't see how this is consistent. You need a rule that tells you when to apply which kind of evolution.

 

[DarMM]

Even in a probabilistic theory, I don't see how this is consistent. You need a rule that tells you when to apply which kind of evolution.

Statistical Mechanics is the same though. When you make an observation the Liouville distribution collapses on observation. In any stochastic theory is there a "rule" for when to apply Bayesian updating?

 

[DarMM]

As a measure of the "stuff" that makes up reality, or in a diverging view of MWI, the number of worlds. As Vaidman says it:
Probability Postulate: An observer should set his subjective probability of the outcome of a quantum experiment in proportion to the total measure of existence of all worlds with that outcome.

Why would one take this kind of view of the coefficients in Quantum Theory, but not in classical probabilistic theories like Wiener processes? What aspect of QM makes one not view the coefficients in the same way?

 

[akvadrako]

Statistical Mechanics is the same though. When you make an observation the Liouville distribution collapses on observation. In any stochastic theory is there a "rule" for when to apply Bayesian updating?

I don't think so. If there was a rule then I would consider both forms of evolution to be contained within that rule.

Why would one take this kind of view of the coefficients in Quantum Theory, but not in classical probabilistic theories like Wiener processes? What aspect of QM makes one not view the coefficients in the same way?

I am not sure. Maybe because of the assumption of completeness. If QM is incomplete and reality is non-linear, then linear evolution is just an approximation that needs periodic corrections.

 

[Minnesota Joe]

You don't need the Born rule, per se, but without probabilities you are just going to have a large number of branches, all with equal weight.

There would be no reason then to have any phenomena consistent with one thing being more likely than another. The physics we see is dependent on the most likely outcomes being favoured.

The Born rule gives a specific outcome distribution. But, you need something; otherwise you are giving equal weight to what - classically at least - would be impossible outcomes.

Are you talking about just entanglement without decoherence here?

My understanding was that MWI requires decoherence in order to get branches that are separate and non-interacting. This seems absolutely crucial because we don't observe superpositions.

Exactly. That is what people like Zurek are trying to do. However it hasn't worked out yet.

Zurek himself seems to acknowledge the circularity in earlier work.

Can you elaborate on the role the Born Rule plays in the density derivation of decoherence? What happens if you don't apply the Born Rule assumption?

 

[DarMM]

I don't think so. If there was a rule then I would consider both forms of evolution to be contained within that rule.

So really this is a problem with a fundamental theory being Stochastic? Since any Stochastic theory will have two such "evolution" processes.

 

[DarMM]

Can you elaborate on the role the Born Rule plays in the density derivation of decoherence? What happens if you don't apply the Born Rule assumption?

This is a bit of a boring answer, but basically you can't derive decoherence at all since you have no way to pass from the state of the system to the state of a subsystem.

 

[akvadrako]

So really this is a problem with a fundamental theory being Stochastic? Since any Stochastic theory will have two such "evolution" processes.

I can't make sense of such a theory unless there is a rule telling you which form of evolution to apply.

 

[DarMM]

I can't make sense of such a theory unless there is a rule telling you which form of evolution to apply.

So you similarly can't make sense of classical statistical mechanics and Wiener processes or other Stochastic processes as they similarly have no rule* for when to apply Bayesian updating?

*Although I would say the do, i.e. when you make an observation and obtain a result.

 

[DarMM]

As a measure of the "stuff" that makes up reality, or in a diverging view of MWI, the number of worlds. As Vaidman says it:
Probability Postulate: An observer should set his subjective probability of the outcome of a quantum experiment in proportion to the total measure of existence of all worlds with that outcome.

This introduces "worlds" directly into the basic postulates of the theory even though "worlds" only really emerge after decoherence. How does decoherence work in such a picture exactly?

 

[PeroK]

Are you talking about just entanglement without decoherence here?

My understanding was that MWI requires decoherence in order to get branches that are separate and non-interacting. This seems absolutely crucial because we don't observe superpositions.

In my view, you have misunderstood decoherence and, especially, "non-interacting". We don't observe superpositions not because for some physical reason they cannot happen; but, because (after a certain period of time evolution) the probability of a significant superposition is vanishingly small.

There is, quite fundamentally, no hard and fast physical division between branches, but an increasingly low probability of the significant superpositions between the two.

If we take the example of the infamous cat. After a period of time evolution, there is a huge number of states that are largely grouped around the concept of a "live" cat - and between them, they have a significant probability of approx 50%; and, there is another huge array of states that are grouped around the concept of a "dead" cat - and, again, the combined probability is 50%. There are at least as many states again that represent a half-live, half-dead cat, but these states combined have approx 0% probability.

There are not two cats. There is either one cat or an uncountable number of cats, depending on how you define the term "cat". And, these states are constantly evolving. But, the laws of physics - implied by QM and the Born rule, if you like - keep the two sets of states apart. For example:

Cells continue to develop in a live cat; but cells cannot be rejuvenated in a dead cat (or if they can, in such small numbers and with such a low probability that you won't notice). That's decoherence.