# I Classicality and the Correspondence Principle

#### jlcd

If the pointer state is "cat alive", it will stay that way until the cat dies. If the pointer state is "cat dead", as far as we know, that's permanent.
Ok. If the pointer state is "cat alive", it will stay that way until the cat dies. If the pointer state is "cat dead", as far as we know, that's permanent. No problem.

But how about the pointer state of your mood (related to brain dynamics). Now since it varies hour to hour or even minute to minute (for very moody people). Then it is right to say the pointer states of mood in brain can change hour to hour, minute to minute?

Of course, this just reflects the fact that "cat alive"and "cat dead" are not single quantum states of the cat; they are huge subspaces of the cat's state space. This will be true for any "pointer state" since any such state will involve a macroscopic object and any macroscopically distinguishable state of a macroscopic object (which is pretty much what "pointer state" means) will be some subspace of the object's full state space that contains a huge number of microstates. And the pointer states behave classically to a very, very good approximation, so the way to answer your question is to look at classical-type dynamics of typical macroscopic objects. And such objects don't "re-prepare" themselves.

#### PeterDonis

Mentor
Then it is right to say the pointer states of mood in brain can change hour to hour, minute to minute?
Yes. Obviously there are various kinds of "pointer states" (states of macroscopic objects) that behave in various ways. We didn't need quantum mechanics to tell us that.

Nor did we need it to tell us that we can partition the states of macroscopic objects in various ways at various levels of detail. For example, cats have moods too; saying that a cat is "alive" is a very, very coarse way of specifying its "pointer state". It could be happy, sad, hungry, thirsty, etc., etc., and its mood can change hour to hour, minute to minute. All of these are just ways of partitioning the "cat alive" subspace of all cat states into finer-grained distinctions. All of this is obvious, and none of it has anything to do with QM, as far as I can see.

I think you're making this much harder than it needs to be, and I'm not sure why.

#### Mentz114

Gold Member
I found that the Ehrenfest approach (dynamics) helps a lot to get rid of the confusion caused by introducing the wave function without any context.

There does not appear to be a much literature but there is a chapter in Ballentine and it is used in the thermal interpretation arXiv:1902.10779v2 [quant-ph] 24 Apr 2019 and possibly made rigorous in this arXiv:0907.1877v1 [math-ph] 10 Jul 2009.

The wiki article is also informative https://en.wikipedia.org/wiki/Ehrenfest_theorem

#### jlcd

Yes. Obviously there are various kinds of "pointer states" (states of macroscopic objects) that behave in various ways. We didn't need quantum mechanics to tell us that.

Nor did we need it to tell us that we can partition the states of macroscopic objects in various ways at various levels of detail. For example, cats have moods too; saying that a cat is "alive" is a very, very coarse way of specifying its "pointer state". It could be happy, sad, hungry, thirsty, etc., etc., and its mood can change hour to hour, minute to minute. All of these are just ways of partitioning the "cat alive" subspace of all cat states into finer-grained distinctions. All of this is obvious, and none of it has anything to do with QM, as far as I can see.

I think you're making this much harder than it needs to be, and I'm not sure why.
When I mentioned "pointer states" and macroscopic object. I was using the context of Zurek quantum darwinism where he didn't use the born rule and states are the primitive (versus copenhagen where observations are the primitive). I understood a lot about fragments and how objectivity is derived. No problem about that. I just want to know whether using quantum darwinism without born rule can give macroscopic object more degrees of freedom (meaning is the dynamics using quantum darwinism without born rule and orthodox using born rule exactly identical or does the former has more degrees of freedom in some aspects?).

Another issue. We humans use born rule so we can get eigenstates and see the results with our eyes or the via the instruments.

But let's take an object like iron bar. When a molecule inside it is "measuring" other molecules inside it. Does the latter have to be in eigenstates before the molecule can interact with it? Or can molecules interact with one another within being in any eigenstates? And can molecules interact with one another without using any born rule? Let's not use MWI or BM analysis. Point is. Only we humans need born rule in measurement (where we need to see the eigenstates)? Can molecules self measure each other without using any born rule or doesn't need to be in any eigenstates? I'm thinking they may not because they have the wave function to see through each other while we are not wave function aware so need to see the particles via detector using born rule (?).

#### PeterDonis

Mentor
We humans use born rule so we can get eigenstates
We humans don't directly observe any quantum system that is simple enough that we can even write down a specific operator and its eigenstates. By the time we observe anything, a huge number of degrees of freedom are involved and a lot of decoherence has occurred. So this statement is not correct as a statement about what we humans actually do.

In some very special cases, we can ignore the above and treat our observation as if we were directly observing eigenstates of some simple operator of some simple quantum system. But that's a convenient simplification that has practical uses. It is not in any way a claim about how things "really are".

let's take an object like iron bar.
What I said above applies to any macroscopic object.

When a molecule inside it is "measuring" other molecules inside it.
This is not a useful way of viewing what is going on. The iron bar has something like $10^{25}$ atoms in it. All of them are continually interacting; the bar as a whole is continually decohering. This is not a "measurement" in any useful sense. It's just part of being a macroscopic object.

Does the latter have to be in eigenstates before the molecule can interact with it?
Individual atoms can interact with each other regardless of what states they are in.

Also, you keep saying "eigenstates" as though they were properties of the atom. They're not. They're properties of an operator, i.e., some measurement you can make on a quantum object. For example, if you measure an electron's spin in the $z$ direction, there are two states of the electron that are eigenstates of that measurement. But if you measure the spin in the $x$ direction, there are two different states of the electron that are eigenstates of that measurement.

But even the above, as I said before, assumes that you have a quantum system and a measurement that are simple enough that you can write them down. We can't do that for macroscopic objects.

(Also, saying that we "measure" the electron's spin, if all that's involved is the electron's spin degree of freedom, is really a misnomer. We pass the electron through a magnetic field that entangles its spin with its linear momentum. This is a unitary interaction and does not require anything to be observed at all. When we say we "measure" the spin, what we mean is that we have a detector, such as a piece of photographic film--a macroscopic object--placed so that we will see a spot at one of two points on the detector: which point depends on which direction the electron was moving when it came out of the magnetic field, i.e., what its linear momentum was, which the magnetic field entangled with the spin. The "result" of the measurement is actually the position of the spot--that's what we observe. We infer from that observation that the electron's spin was up or down, because we follow the chain of reasoning backwards from the spot position to the electron's momentum coming out of the magnetic field to the electron's spin that got entangled with the momentum.)

#### jlcd

We humans don't directly observe any quantum system that is simple enough that we can even write down a specific operator and its eigenstates. By the time we observe anything, a huge number of degrees of freedom are involved and a lot of decoherence has occurred. So this statement is not correct as a statement about what we humans actually do.

In some very special cases, we can ignore the above and treat our observation as if we were directly observing eigenstates of some simple operator of some simple quantum system. But that's a convenient simplification that has practical uses. It is not in any way a claim about how things "really are".

What I said above applies to any macroscopic object.

This is not a useful way of viewing what is going on. The iron bar has something like $10^{25}$ atoms in it. All of them are continually interacting; the bar as a whole is continually decohering. This is not a "measurement" in any useful sense. It's just part of being a macroscopic object.

Individual atoms can interact with each other regardless of what states they are in.
I kept saying "eigenstates" as though they were properties of the atom because of position. We tried to measure positions as in the spots in the double slit. Position is the most common preferred basis. Even without any measurements by humans. Objects naturally have positions, although one can say the environment is continuously doing measurement causing decoherence. Yet still postion basis is singled out. This is why I seemed to be saying eigenstates are properties of the atom because of the self decoherence of objects.
Now refer to this.

Why do you think Einstein mean?

Since atoms don't have to form position eigenstates first before they can interact. And as you say "Individual atoms can interact with each other regardless of what states they are in.". This means they can even interact without position basis? So it's like the position basis is so we can see them? If there were no position basis, as Einstein was contemplating, does it mean atoms and molecules can still exist? They can become shapeless and not visible? This is in context to Einsteins reflections and not personal speculations.

Also, you keep saying "eigenstates" as though they were properties of the atom. They're not. They're properties of an operator, i.e., some measurement you can make on a quantum object. For example, if you measure an electron's spin in the $z$ direction, there are two states of the electron that are eigenstates of that measurement. But if you measure the spin in the $x$ direction, there are two different states of the electron that are eigenstates of that measurement.

But even the above, as I said before, assumes that you have a quantum system and a measurement that are simple enough that you can write them down. We can't do that for macroscopic objects.

(Also, saying that we "measure" the electron's spin, if all that's involved is the electron's spin degree of freedom, is really a misnomer. We pass the electron through a magnetic field that entangles its spin with its linear momentum. This is a unitary interaction and does not require anything to be observed at all. When we say we "measure" the spin, what we mean is that we have a detector, such as a piece of photographic film--a macroscopic object--placed so that we will see a spot at one of two points on the detector: which point depends on which direction the electron was moving when it came out of the magnetic field, i.e., what its linear momentum was, which the magnetic field entangled with the spin. The "result" of the measurement is actually the position of the spot--that's what we observe. We infer from that observation that the electron's spin was up or down, because we follow the chain of reasoning backwards from the spot position to the electron's momentum coming out of the magnetic field to the electron's spin that got entangled with the momentum.)

#### PeterDonis

Mentor
Position is the most common preferred basis.
That has everything to do with human convenience, and nothing to do with the physics (as far as we know).

Objects naturally have positions
Macroscopic objects have such a small uncertainty in position that we can treat their positions classically to a very good approximation. But this does not mean your statement is true.

postion basis is singled out
No, as above, we treat macroscopic objects as having classical positions and behaving classically to a good approximation. But this has nothing to do with the "position basis" at the quantum level.

People seem to have a strong desire to interpret wave functions in the position basis, i.e., as functions on ordinary 3-dimensional space, even though, as soon as you have more than one particle, this interpretation is no longer valid. The wave function of a two-particle system is not a function on ordinary 3-dimensional space; it's a function on a 6-dimensional space, and there is no "position operator" on this space. (There are operators for "position of the first particle" and "position of the second particle", but those aren't the same thing, and even they are problematic.) The wave function on a $10^{25}$ particle system is a function on a $3 \times 10^{25}$ dimensional space.

Why do you think Einstein mean?
I would have to see the entire letter to know. Anyway, it was a letter, not a peer-reviewed paper, so I would be very careful in putting any weight on what it says.

This means they can even interact without position basis?
Interactions between atoms have nothing whatever to do with a human's choice of basis. The choice of basis is a human convenience for doing the math. The atoms don't know what basis the humans have chosen, and don't care because the basis is not there in the physics, it's only there in the math the humans do to model the physics. Don't confuse the map with the territory.

#### PeterDonis

Mentor
If there were no position basis, as Einstein was contemplating
I don't see how what Einstein was saying in what you quoted has anything to do with the position basis, or indeed any basis.

#### jlcd

No, as above, we treat macroscopic objects as having classical positions and behaving classically to a good approximation. But this has nothing to do with the "position basis" at the quantum level.
Is the above statement colored by intepretation?

I have thought for that puzzling statement the whole day even when in the mall at starbucks or at midnight when waking up.

I then consulted a Ph.D. physicist about it. He thought of classical physics as an approximation derived from quantum physics. This means if for sake of discussion an object had no position basis, objects won't have any position or invisible. In your case, he thought you answered differently because you thnk of classical physics as independent of quantum physics, is this right?

Interactions between atoms have nothing whatever to do with a human's choice of basis. The choice of basis is a human convenience for doing the math. The atoms don't know what basis the humans have chosen, and don't care because the basis is not there in the physics, it's only there in the math the humans do to model the physics. Don't confuse the map with the territory.
I searched for "map territory" under your name in PF to get more details of it and I came up with the following passage in the thread "Hawking believes "God confuses us throwing dice...", why? started by someone called mario rossi. You wrote:

The reason the map-territory distinction is made is to make it clear that our physical models are distinct from the things they are trying to model. That lets us take a step back, so to speak, and keep ourselves from making commitments based on our models that might be too broad. For example, quantum physics is a model--a map: it can be used to make very accurate predictions about the results of experiments. But there are multiple, mutually inconsistent interpretations of QM, which amount to multiple, mutually inconsistent claims about exactly what kind of territory the map is modeling. Those claims can't be resolved by experiment (because all of the interpretations make the same predictions for all experimental results)
So for those who think the wave function or state vectors are the objects themselves such as Many worlders, quantum physics is both map and territory? Territory in the sense that objects are really wave functions and state vector themselves (where they are models of).

For Bohr Copenhagen who think the wave functions or state vectors are just tools to make statistical predictions. quantum physics is only map and not the territory? Here the territory is something unknown.

So when you believe positon basis has nothing to do with classical positions of objects. You were thinking in terms of the latter or Copenhagen context. While the Ph.d expert was thinking of the former where the state vectors are the objects themselves hence the classical positions are the actual position basis themselves?

#### Mentz114

Gold Member
Is the above statement colored by intepretation?

[..]
He thought of classical physics as an approximation derived from quantum physics. This means if for sake of discussion an object had no position basis, objects won't have any position or invisible. In your case, he thought you answered differently because you thnk of classical physics as independent of quantum physics, is this right?
I don't know what your current interlocutor thinks but the extent to which classical mechanics is an approximation of quantum mechanics is fruitfully addressed by the Ehrenfest formalism. Interpretaions which adopt it assume that CM is an approximation of QM.

#### PeterDonis

Mentor
Is the above statement colored by intepretation?
No.

He thought of classical physics as an approximation derived from quantum physics.
That's how it is viewed now that we know about quantum physics, yes.

This means if for sake of discussion an object had no position basis
A basis is not a property of an object. It's a choice humans make in the math for convenience. It makes no sense to say an object has or doesn't have a position basis or any other basis.

In your case, he thought you answered differently because you thnk of classical physics as independent of quantum physics, is this right?
No.

So for those who think the wave function or state vectors are the objects themselves such as Many worlders, quantum physics is both map and territory?
No. Saying "the wave function is real" does not mean that the mathematical expression you write down on a piece of paper to describe a quantum system (the map) is the same as the system itself (the territory). Quantum objects don't know or care what math you use.

So when you believe positon basis has nothing to do with classical positions of objects. You were thinking in terms of the latter or Copenhagen context.
No. As I said above, what I said is independent of any particular interpretation.

While the Ph.d expert was thinking of the former where the state vectors are the objects themselves hence the classical positions are the actual position basis themselves?
I don't know what your Ph. D. expert was thinking.

However, perhaps I can help by expanding a bit on what I said about classical positions. Consider a macroscopic object like a baseball. What is its "classical position"? Mathematically, it is the expectation value of the center of mass position of the baseball. In other words, there is a "center of mass position" operator $\hat{P}_{\text{CM}}$ which, given the quantum state $| \psi \rangle$ of the ball, has an expectation value. $\langle \psi | \hat{P}_{\text{CM}} | \psi \rangle$. The Ehrenfest theorem, which @Mentz114 mentioned, shows that this expectation value obeys the classical equations of motion for the center of mass of the ball; that's what justifies interpreting it as the "classical position" of the ball.

Now, what does this operator have to do with the "position basis" at the quantum level? Nothing. Saying that the center of mass of the ball is at some particular position tells you nothing at all about the position of any of the individual atoms in the ball. (More precisely, nothing at all beyond that it's within some spatial region around the center of mass.) It doesn't even tell you that any of the atoms in the ball even has a particular position. All of them can be fluctuating at the quantum level and never be in a position eigenstate at all. In fact, all of the atoms in the ball are entangled with each other, so none of them even has a well-defined wave function by itself, much less a wave function that is in a position eigenstate. (Strictly speaking, exact position eigenstates can't exist physically, but we can read "wave function with a sufficiently narrow spread in position" for "position eigenstate" above and it won't change any of what I said.) And, as I said before, the wave function of the ball as a whole, i.e., the $| \psi \rangle$ that appears in the expectation value above, is a function on a space with $3 \times 10^{25}$ dimensions or so (even more if we include spin degrees of freedom), not a function on ordinary 3-dimensional space.

#### jlcd

A basis is not a property of an object. It's a choice humans make in the math for convenience. It makes no sense to say an object has or doesn't have a position basis or any other basis.
I have given this a lot of thought.

If treating macroscopic objects as having classical positions and behaving classically to a good approximation has nothing to do with the "position basis" at the quantum level. How come there is a version of MWI where people debate how the preferred basis or factorization occured. You yourself mentioned this:

"If all you have is the pure state vector of the entire universe, how do you pick out the "cat" subspace? If your answer is, "well, I pick some particular basis...", then what justifies picking out that particular basis? If your answer to that is "well, that's the basis in which we have cats that are either dead or alive, instead of a superposition of dead and alive", then you're arguing in a circle."

From numerous threads and discussions and papers. Without a preferred basis in Many worlds, there is not even a position basis. Meaning all classical objects won't have positions. And classical objects were the same macroscopic objects which you somehow said has nothing to do with position basis. This is very puzzling and seem contradictory.

Unless you mean MWI is just a toy model.. tongue in cheek thing where one could say without preferred basis, objects couldn't exist. Yet when they go back to real world. They would state the opposite that objects have no relation to the position basis in the quantum?

But then decoherence is already proven to be true. And decoherence needs preferred basis.

Or maybe your arguments have the assumption decoherence already occured and preferred basis was already chosen. And classical objects don't need to be related to the initial preferred basis? What would happen if the universe wouldn't have any position basis at all. Then there would be no decoherence in the position basis. If so then objects as we know it wouldn't even exist or wouldn't have shapes and invisible, and the Ph.D expert argument that without position basis, objects won't even exist hold?

Also if objects were not really state vectors or ray in hilbert spaces. Then you mean our QM and QFT is just model that only capture a small bit of reality. And we are still missing a lot about nature which future models can describe better and more completely?

#### PeterDonis

Mentor
How come there is a version of MWI where people debate how the preferred basis or factorization occured.
Precisely because the basis is a human choice made for convenience, not anything built into the physics.

Without a preferred basis in Many worlds, there is not even a position basis.
Wrong. Saying that there is no preferred basis is not at all the same as saying there is no position basis.

decoherence is already proven to be true. And decoherence needs preferred basis.
No, it doesn't. Decoherence is an entanglement interaction between the measuring apparatus and the environment. Entanglement is basis independent.

No.

What would happen if the universe wouldn't have any position basis at all.
Why do you keep talking as if a basis were a real thing, in the universe, when I've already said many times now that a basis is a human choice, made for convenience, not anything built into the physics? It's as if you kept saying that latitude and longitude lines were somehow already there on the Earth before humans established them by convention.

if objects were not really state vectors or ray in hilbert spaces.
We already know they aren't. That's confusing the map with the territory again.

we are still missing a lot about nature which future models can describe better and more completely?
I think this is true. But it's not anything specific to QM or QFT. All of our models of nature are missing a lot, which I expect that future models will describe better.

#### jlcd

Precisely because the basis is a human choice made for convenience, not anything built into the physics.

Wrong. Saying that there is no preferred basis is not at all the same as saying there is no position basis.

No, it doesn't. Decoherence is an entanglement interaction between the measuring apparatus and the environment. Entanglement is basis independent.

No.

Why do you keep talking as if a basis were a real thing, in the universe, when I've already said many times now that a basis is a human choice, made for convenience, not anything built into the physics? It's as if you kept saying that latitude and longitude lines were somehow already there on the Earth before humans established them by convention.

We already know they aren't. That's confusing the map with the territory again.

I think this is true. But it's not anything specific to QM or QFT. All of our models of nature are missing a lot, which I expect that future models will describe better.
About missing a lot. You are right. And all your explanations and arguments are consistent with the fact our present physics are missing a lot.

In the famous Letter to Virginia...

"You may break apart the atoms and molecules and see what created the fields inside, but there is a veil covering the unseen world which not the brightest physicist, nor even the united strength of all the brightest physicists that ever lived, could tear apart. Only us seers, sensitives, etc. can push aside that curtain and view and picture the supernal beauty and glory beyond. And guide physics to the next step. Is it all real? Ah, Virginia, in all this world there is nothing else real and abiding."

Thanks for all the help and assistances, Peterdonis. I'd try to digest all your thoughts for the next few weeks and read and study peer reviewed books, papers, articles, etc.

"Classicality and the Correspondence Principle"

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