Effective Dynamics of Open Quantum Systems: Stochastic vs Unitary Models

[A. Neumaier]

Why do classical objects such as cats or pointers have approximately definite positions (and approximately definite momenta, etc.)?

This is answered by the law of large numbers and statistical mechanics. It is very well-known that the standard deviations of all macroscopic variables of interest in physics scale like $O(N^{-1/2})$, where $N$ is the conserved number of particles involved, and the mean number if there is no conservation. Metastability answers why in the case of a binary measurement one of these actually comes out.

All of this is completely unrelated to MWI.

 

[stevendaryl]

This is answered by the law of large numbers and statistical mechanics. It is very well-known that the standard deviations of all macroscopic variables of interest in physics scale like $O(N^{-1/2})$, where $N$ is the conserved number of particles involved, and the mean number if there is no conservation. Metastability answers why in the case of a binary measurement one of these actually comes out.

All of this is completely unrelated to MWI.

No, I think you're completely wrong about both paragraphs above. Metastability has nothing to do with it. [edit] I shouldn't say nothing, but it doesn't explain definite outcomes. I think you're completely wrong about this.

 

[stevendaryl]

No, I think you're completely wrong about both paragraphs above. Metastability has nothing to do with it. [edit] I shouldn't say nothing, but it doesn't explain definite outcomes. I think you're completely wrong about this.

An example of a metastable system might be a lattice of 1000 magnetic dipoles. They tend to line up; the state with all dipoles pointing in the same direction is lower energy than the state with them pointing in different directions. So if you start with an unmagnetized state (the dipoles pointing in all sorts of different directions), then a small perturbation will likely result in most dipoles pointing in the same direction. But that does not mean that you can't have a superposition of one state with all dipoles pointing up, and another state with all dipoles pointing down. If you started in such a superposition, it would not ever evolve into a state with all pointing one way, or all pointing the other way. If the initial state is symmetric under parity, then the final state will be.

I know what you're going to say: Couple it to an environment--a thermal bath of some sort. But I think that that would not make any difference. The same argument holds: If the thermal bath + lattice is initially symmetric under parity, then it will never evolve into a state that is not symmetric. It will never evolve into a state with a nonzero magnetic moment. Metastability just does not explain definite outcomes.

 

[A. Neumaier]

I think you're completely wrong about this.

I cannot argue about your subjective beliefs.

But what I stated is the reason why practitioners of QM don't feel a need to investigate the foundations of quantum mechanics, except in as far as there are challenging experiments to perform. It is very clear to them that statistical mechanics explains the gradual emergence of classicality, due to the law of large numbers to an ever increasing accuracy as the object size grows, and that the quantum dynamics morphs as gradually to classical dynamics. There are even all sorts of intermediate stages modeled by quantum-classical dynamics, used a lot in situations where the quantum regime is important for some degrees of freedom but not for others. Thus there is a continuum from the fully quantum to the fully classical, and the only role of observers is to select from this spectrum the model that is most tractable computationally given a desired resolution.

A measurement problem arises only if one ignores all this and insists on the rigid, far too idealized framework in which quantum mechanics was introduced historically and is typically introduced in textbooks.

 

[Mentz114]

An example of a metastable system might be a lattice of 1000 magnetic dipoles. They tend to line up; the state with all dipoles pointing in the same direction is lower energy than the state with them pointing in different directions. So if you start with an unmagnetized state (the dipoles pointing in all sorts of different directions), then a small perturbation will likely result in most dipoles pointing in the same direction. But that does not mean that you can't have a superposition of one state with all dipoles pointing up, and another state with all dipoles pointing down. If you started in such a superposition, it would not ever evolve into a state with all pointing one way, or all pointing the other way. If the initial state is symmetric under parity, then the final state will be.

I know what you're going to say: Couple it to an environment--a thermal bath of some sort. But I think that that would not make any difference. The same argument holds: If the thermal bath + lattice is initially symmetric under parity, then it will never evolve into a state that is not symmetric. It will never evolve into a state with a nonzero magnetic moment. Metastability just does not explain definite outcomes.

A small enough collection of dipoles ( eg quantum magnetic dot) may be in a superposition, but if the object was large enough then at some point it becomes fixed and irreversibly in one outcome. What else could possibly happen ? Your argument is based on Platonic ideals. "Every quantum state has fluctuations" - Ballentine ( says it twice actually)

 

[atyy]

I cannot argue about your subjective beliefs.

But what I stated is the reason why practitioners of QM don't feel a need to investigate the foundations of quantum mechanics, except in as far there are challenging experiments to perform. It is very clear to them that statistical mechanics explains the gradual emergence of classicality, due to the law of large numbers to an ever increasing accuracy as the object size grows, and that the quantum dynamics morphs as gradually to classical dynamics. There are even all sorts of intermediate stages modeled by quantum-classical dynamics, used a lot in situations where the quantum regime is important for some degrees of freedom but not for others. Thus there is a continuum from the fully quantum to the fully classical, and the only role of observers is to select from this spectrum the model that is most tractable computationally given a desired resolution.

A measurement problem arises only if one ignores all this and insists on the rigid, far too idealized framework in which quantum mechanics was introduced historically and is typically introduced in textbooks.

Landau, Dirac, Bell, Adler, Weinberg, Haroche, Raimond, Laloe, Susskind, Zurek, Zeilinger, Hartle, Gell-Mann - are these not practioners of quantum mechanics?

 

[A. Neumaier]

Landau, Dirac, Bell, Adler, Weinberg, Haroche, Raimond, Laloe, Susskind, Zurek, Zeilinger, Hartle, Gell-Mann - are these not practitioners of quantum mechanics?

Who of these thinks that there is an unsolved measurement problem? The unsolved problems Landau, Dirac, and Weinberg are concerned about are the problematic mathematical basis of relativistic quantum field theory, not the measurement problem.

 

[stevendaryl]

If the initial state is symmetric under parity, then the final state will be.

A small enough collection of dipoles ( eg quantum magnetic dot) may be in a mixed starte, but if the object was large enough then at some point it becomes fixed and irreversibly in one outcome. What else could possibly happen ? Your argument is based on Platonic ideals. "Every quantum state has fluctuations" - Ballentine ( says it twice actually)

I think you're completely wrong about that. The evolution of the wave function is linear. So if initial state I_1 leads to final state F_1, and initial state I_2 leads to final state F_2, then the superposition of I_1 and I_2 will lead to a superposition of F_1 and F_2. It will not lead to a random pick between F_1 and F_2. The same thing is true if you want to do density matrices.

Metastability cannot explain definite outcomes.

I have no idea what you mean by my argument being based on "Platonic ideals". It's based on quantum mechanics.

 

[stevendaryl]

I cannot argue about your subjective beliefs.

Then let me put it more strongly: You are wrong about this.

 

[A. Neumaier]

The evolution of the wave function is linear.

The evolution of the Fokker-Planck equation is also linear. Nevertheless it describes classical nonlinear stochastic processes.

 

[Mentz114]

I think you're completely wrong about that. The evolution of the wave function is linear. So if initial state I_1 leads to final state F_1, and initial state I_2 leads to final state F_2, then the superposition of I_1 and I_2 will lead to a superposition of F_1 and F_2. It will not lead to a random pick between F_1 and F_2. The same thing is true if you want to do density matrices.

Metastability cannot explain definite outcomes.

Linear evolution is your Platonic ideal. It can only exist in very small highly-isolated systems. It only takes energy to leak out to make the sub-system non-conservative and lose normalization. This will drive a stochastic process to a definite result.

We must agree to disagree about this.

 

[A. Neumaier]

Then let me put it more strongly: You are wrong about this.

I cannot argue about your subjective beliefs. Repeating variations on them doesn't improve the situation.

 

[atyy]

Who of these thinks that there is an unsolved measurement problem? The unsolved problems Landau, Dirac, and Weinberg are concerned about are the problematic mathematical basis of relativistic quantum field theory, not the measurement problem.

All believed there was an unsolved measurement problem (eg. Dirac, Weinberg) or that a classical/quantum cut is needed (eg. Landau).

 

[A. Neumaier]

that a classical/quantum cut is needed

The cut is just the decision at which description level the quantum corrections (that decay like $O(N^{-1/2})$) can be neglected. It is not a bigger problem than the problem of whether or not to include into the classical description of a pendulum the surrounding air and the way it is suspended, or whether taking it into account with a damping term is enough.

 

[atyy]

The cut is just the decision at which description level the quantum corrections (that decay like $O(N^{-1/2})$) can be neglected. It is not a bigger problem than the problem of whether or not to include into the classical description of a pendulum the surrounding air and the way it is suspended, or whether taking it into account with a damping term is enough.

Not in Landau's view.

 

[georgir]

Linear evolution is your Platonic ideal. It can only exist in very small highly-isolated systems. It only takes energy to leak out to make the sub-system non-conservative and lose normalization. This will drive a stochastic process to a definite result.

We must agree to disagree about this.

What if you look at the whole universe? Where does energy leak out to?

 

[A. Neumaier]

Not in Landau's view.

In Volume IX (Statistical physics, Part 2) of their treatise on theoretical physics, Landau and Lifshits derive the hydrodynamic equations without needing any cut. The cut is mentioned only in the introduction to quantum mechanics and nowhere used - thus recognizable as a purely pedagogical device.

 

[georgir]

The cut is just the decision at which description level the quantum corrections (that decay like $O(N^{-1/2})$) can be neglected. It is not a bigger problem than the problem of whether or not to include into the classical description of a pendulum the surrounding air and the way it is suspended, or whether taking it into account with a damping term is enough.

The air surrounding the pendulum works to disrupt the macroscopic behavior I expect to observe, not to actually explain it. So I'm not finding this comparison fair or convincing.

 

[Mentz114]

What if you look at the whole universe? Where does energy leak out to?

The whole universe only has one possible outcome

I assume you're joking.

 

[georgir]

The whole universe only has one possible outcome

I assume you're joking.

I'm not sure if you are now. The whole point of MWI etc is many possible outcomes. Or you could call it one, but it could still be a superposition of steven both in Seattle and in New York.

 

[atyy]

In Volume IX (Statistical physics, Part 2) of their treatise on theoretical physics, Landau and Lifshits derive the hydrodynamic equations without needing any cut. The cut is mentioned only in the introduction to quantum mechanics and nowhere used - thus recognizable as a purely pedagogical device.

Sorry, I cannot agree. You, vanhees71, Ballentine, and Peres are wrong.

 

[A. Neumaier]

The air surrounding the pendulum works to disrupt the macroscopic behavior I expect to observe, not to actually explain it. So I'm not finding this comparison fair or convincing.

In both cases, the explanation is in the derivation of the approximations. One needs the surrounding to explain why the pendulum is damped (as observed) rather than ideal (as the ideal Hamiltonian dynamics would suggest). Notice the complete similarity with the collapse (observed in a continuous measurement) rather than the unitary evolution (as the ideal Hamiltonian dynamics would suggest).

 

[A. Neumaier]

Sorry, I cannot agree. You, vanhees71, Ballentine, and Peres are wrong.

I cannot argue about your subjective beliefs. As stevendaryl, you simply call wrong what differs from your preferences.

 

[atyy]

I cannot argue about your subjective beliefs. As stevendaryl, you simply call wrong what differs from your preferences.

You are wrong because your thermal interpretation contradicts Bell's theorem.

 

[Mentz114]

I'm not sure if you are now. The whole point of MWI etc is many possible outcomes. Or you could call it one, but it could still be a superposition of steven both in Seattle and in New York.

Why invoke MWI when a much simpler explanation is available ?

There is no measurement problem. People make measurements and get results. The only quibble is from those who insist that something weird an inexplicable is happening. Maybe they have a need for spookiness.

(I am not being disparaging. I respect other people's freedom to hold any views they wish to)

 

[A. Neumaier]

your thermal interpretation contradicts Bell's theorem.

?
Bell's theorem is a theorem about classical local hidden variable theories.
How can it possibly contradict an interpretation of quantum mechanics?

 

[georgir]

Why invoke MWI when a much simpler explanation is available ?

Because I sort of understand the idealized model but do not yet understand your "much simpler" explanation?
[Though I prefer you to be right instead of MW or other models that allow macroscopic superposition]

 

[Mentz114]

Because I sort of understand the idealized model but do not yet understand your "much simpler" explanation?
[Though I prefer you to be right instead of MW or other models that allow macroscopic superposition]

Fair enough, that is rational.

It is not my explanation, and I don't understand all the details myself but I'm studying the issue at present via Lindblad and classical/quantum dynamics.

 

[georgir]

Ok, I know this is somewhat offtopic, but not entirely - it will help me understand why this whole discussion is not purely philosophical but actually matters.
Can you point me to a simple experiment (or quantum gate circuit or something similar) that distinguishes a superposition state from a "normal" state. Or is there no such thing possible for a single instance of a quantum state and only multiple repetitions with the same preparations can reveal it?

 

[A. Neumaier]

Ok, I know this is somewhat offtopic, but not entirely - it will help me understand why this whole discussion is not purely philosophical but actually matters.
Can you point me to a simple experiment (or quantum gate circuit or something similar) that distinguishes a superposition state from a "normal" state. Or is there no such thing possible for a single instance of a quantum state and only multiple repetitions with the same preparations can reveal it?

Ordinary light is unpolarized, in a mixed state with $2\times 2$ density matrix $1/2$ times the identity matrix. After passing it trough a linear polarizer, it will be in a pure state, described by a superposition of an up and a down state whose relative coefficients are real and depend on the orientation of the polarizer. With up and down defined by the most natural basis of vertical and horizontal polarization, one particular orientation will produce vertically polarized light, which is a ''normal'' up state, and with the orthogonal orientation it will produce horizontally polarized light, which is a ''normal'' down state. But what is ''normal'' depends on the basis assumed. Note that any state can be made ''normal'' by looking at an appropriate basis. Things get more interesting (and more confusing) when looking at tensor products of states...