QM and Determinism: Can We Predict the Future?

[Jano L.]

Of course, everything in physics is (asssumed to be) causal...
...
the state evolution is causal. The Schrödinger equation is a causal differential equation, as it must be, because it describes a physical dynamical process.

Initial value problem with the Schroedinger equation may have unique solution. As I understand, you call that "causal".
When we calculate the spin wave function this way, we obtain unique result $\boldsymbol \psi_1$ giving probability density in space symmetrical in $z$.

But this calculated $\boldsymbol\psi_1$ is appropriate only before the measurement of the z coordinate takes place; after the measurement, we know the appropriate pair of wave functions in space is no longer that calculated in the above way. Based on the result of the measurement, the best choice is asymmetric pair where one component carries most of the probability and its density is localized asymmetrically in z.

This new pair of wave functions $\boldsymbol \psi_2$ cannot be obtained from the Schroedinger equation in a "causal" way. It is chosen based on the result of the measurement, which is random, not causal in quantum theory.

Another theory (not quantum theory in the usual sense of this name) may explain this change of the wave functions in a "causal" way (as a result of some evolution equation), but I do not think that is what you meant.

 

[BruceW]

You may walk away with an incorrect conclusion depending on how you interpret some of the statements above.

1. Classical causality has been soundly refuted and there are no standing interpretations generally accepted otherwise.

2. Deterministic interpretations today are all non-local.
...

This looks very interesting, thanks for replying. But still I don't understand what is meant by several things. "Classical Causality" - no idea. "Deterministic interpretations are non-local" - This is Bell's theorem, right? We can't have a local theory of hidden variables. So now I'm guessing that "Classical Causality" means "local and deterministic", which is not allowed because of Bell's theorem.

Classically, one would think that correlations between distant events would be due to correlations prepared in the common past light cone of these events. This is often called local causality, or EPR-Bell locality. But quantum mechanics has wave function collapse, and violates local causality. Surprisingly, we still cannot use it to send information faster than light, and quantum mechanics is consistent with relativity. So there are at least two notions of causality - classical local causality, and the wider notion of relativistic causality, as discussed eg. by http://arxiv.org/abs/quant-ph/9508009.

Right, so the classical local causality is what Bell refers to as "local causality". And the standard QM is not locally causal. And what is the other idea of causality you mention? Is it just the idea that "QM does not give us a method for faster-than-light communication" ? I've skim-read a couple of papers that describe why this is true, given that the classical phenomena itself travels at the speed of light or less. Still, it does seem quite surprising. I would hope (someday) that there will be some overarching principle which automatically explains both ideas.

 

[vanhees71]

Initial value problem with the Schroedinger equation may have unique solution. As I understand, you call that "causal".
When we calculate the spin wave function this way, we obtain unique result $\boldsymbol \psi_1$ giving probability density in space symmetrical in $z$.

But this calculated $\boldsymbol\psi_1$ is appropriate only before the measurement of the z coordinate takes place; after the measurement, we know the appropriate pair of wave functions in space is no longer that calculated in the above way. Based on the result of the measurement, the best choice is asymmetric pair where one component carries most of the probability and its density is localized asymmetrically in z.

This new pair of wave functions $\boldsymbol \psi_2$ cannot be obtained from the Schroedinger equation in a "causal" way. It is chosen based on the result of the measurement, which is random, not causal in quantum theory.

Another theory (not quantum theory in the usual sense of this name) may explain this change of the wave functions in a "causal" way (as a result of some evolution equation), but I do not think that is what you meant.

Sure, the "measurement" here is done by filtering out one beam (it's the paradigmatic example for what's called an ideal von Neumann filter measurement which is at the same time a preparation procedure to produce a beam with determined spin-z component). Of course, this is not described by the simple Schrödinger equation, because I didn't inclue the filter. If you'd include the whole apparatus, you'd have to solve a complicated many-body quantum problem, but in principle then the entire dynamics is described by causal equations.

Analogously, even in a classical description, you wouldn't describe the measurement, if you wouldn't include the interaction of the particle with the measurement apparatus or filter in this case.

There's nothing mysterious about measurement apparati. In contrast to Bohr, I don't think that a cut between a quantum dynamics and classical dynamics makes sense. Anything is quantum, according to our understanding today, and the classical behavior of macroscopic systems (including measurement apparati) is emergent and can be understood by decoherence.

The main difference between quantum theory and classical theory is that the complete possible knowledge about a system (encoded in the quantum theoretical formalism as a ray in an appropriate Hilbert space) is only probabilistic, i.e., not all observables can have determined values (according to the Heisenberg-Robertson uncertainty relation). That's why I call quantum theory causal but indeterministic. Again, I recommend to read the introductory part of

J. Schwinger, Quantum Mechanics, Symbolism for Atomistic Measurements, Springer (2001)

It's the best introduction to a quantum-mechanics text I've ever read, although it's without math, which after that is introduced in a marvelous way. It's an unusual but very illuminating approach to quantum theory. I'd recommend it as a very good read for advanced students who have learned QT from a more conventional approach. The best thing about this book is that it does start with the representation-free formulation. The same approach is followed in Sakurai's textbook, which I'd recommend as a first book on quantum theory.

 

[DrChinese]

This looks very interesting, thanks for replying. But still I don't understand what is meant by several things. "Classical Causality" - no idea. "Deterministic interpretations are non-local" - This is Bell's theorem, right? We can't have a local theory of hidden variables. So now I'm guessing that "Classical Causality" means "local and deterministic", which is not allowed because of Bell's theorem.

All good. "Classical" can mean a variety of things according to context, as it is always a reference to an earlier perspective. In this case, the view in 1935 (when EPR was written) was that nothing could exceed the speed of light. So classical meant local in this context.

The easiest thing is to realize that 2 people must agree on some definition of locality/separability/hidden variables/determinism/causality/whatever you want to call it etc. in order to have Bell's Theorem make sense. The beauty of EPR and Bell is that despite the somewhat arcane language, the argument comes through regardless. Since, many alternate definitions have been floated and many "improved" versions of the arguments have been presented. And yet, in the end, none are accepted over the originals.

 

[naima]

I often read that local theories cannot describe reality.
Could you tell me if Haag's local theory could be disproofed
http://en.wikipedia.org/wiki/Local_quantum_field_theory

 

[bhobba]

I often read that local theories cannot describe reality.

Where do you get that from.

QFT is based on the cluster decomposition property:
https://www.physicsforums.com/threads/cluster-decomposition-in-qft.547574/

But as the link shows its subtle.

Thanks
Bill

 

[atyy]

I often read that local theories cannot describe reality.
Could you tell me if Haag's local theory could be disproofed
http://en.wikipedia.org/wiki/Local_quantum_field_theory

I'll eat my hat if Haag's theory is disproved :D

The sense of local in Haag is the observables at spacelike separation should commute, eg. http://arxiv.org/abs/1303.2849 (footnote 1). This notion of locality can violate the Bell inequalities, so it is nonlocal in that sense.

However, QM has two notions of nonlocality. The first is the typical tensor product idea, and from there we get that the maximum CHSH is $2\sqrt{2}$, which is the Tsirelson bound http://en.wikipedia.org/wiki/Tsirelson's_bound. However, it is not known whether the bound is the same if we define it using the notion of spacelike commutation, which is called Tsirelson's problem http://www.tau.ac.il/~tsirel/Research/bellopalg/main.html.

 

[atyy]

Right, so the classical local causality is what Bell refers to as "local causality". And the standard QM is not locally causal. And what is the other idea of causality you mention? Is it just the idea that "QM does not give us a method for faster-than-light communication" ?

Yes. This is also known as "no signalling".

I've skim-read a couple of papers that describe why this is true, given that the classical phenomena itself travels at the speed of light or less. Still, it does seem quite surprising. I would hope (someday) that there will be some overarching principle which automatically explains both ideas.

Even more surprising to me is that a theory that is even more nonlocal than QM can be consistent with no signalling. This was discovered by Popescu and Rohrlich.

 

[DrChinese]

I often read that local theories cannot describe reality.
Could you tell me if Haag's local theory could be disproofed
http://en.wikipedia.org/wiki/Local_quantum_field_theory

I have never heard of Haag. It originated pre-Bell. Assuming it is also intended to be realistic, Bell would be all you need to disprove it. Keep in mind that we have an excellent standard model, so any alternative model is going to need something spectacular to gain any attention.

 

[atyy]

I have never heard of Haag. It originated pre-Bell. Assuming it is also intended to be realistic, Bell would be all you need to disprove it. Keep in mind that we have an excellent standard model, so any alternative model is going to need something spectacular to gain any attention.

Haag is one of the axiomatic formulations of QFT. So it is not realistic, because it is just quantum theory. The sense of locality in Haag is that spacelike observables commute, which is different from the local realism addressed by Bell. So Haag is local quantum theory in the sense of relativistic quantum theory, which is non-local in the sense of Bell.

http://ncatlab.org/nlab/show/Haag-Kastler axioms
"Therefore this translates into the axiom: quantum fields on a spacetime form an isotonic copresheaf of algebras such that the algebras assigned to any two spacelike separated regions commute with each other inside the algebra assigned to any larger region containing these two regions."

A remaining open issue is that when we say QM maximally violates CHSH with $2\sqrt{2}$ (Tsirelson's bound), this is proved using tensor products and finite dimensional Hilbert spaces. However, the Hilbert space of interacting QFT is infinite dimensional, and it is not known whether the QM bound is still $2\sqrt{2}$. This is Tsirelson's problem http://www.tau.ac.il/~tsirel/Research/bellopalg/main.html.

 

[Jano L.]

Of course, this is not described by the simple Schrödinger equation, because I didn't inclue the filter. If you'd include the whole apparatus, you'd have to solve a complicated many-body quantum problem, but in principle then the entire dynamics is described by causal equations.

Describing the apparatus and the atom by one wave function only makes the description more removed from experimental physics. Some dynamics would be described by such complicated differential equations. But it is not very clear that the result would be that the atom actually gets definite spin state and gets deflected towards one of the two z coordinates. The result of such computation would most probably be that the whole system atom + apparatus has wave function where there is some superposition of exclusive possibilities which is never observed. This is not physics. The fact that we get definite result for the z coordinate still needs to be put by hand.

Anything is quantum, according to our understanding today, and the classical behavior of macroscopic systems (including measurement apparati) is emergent and can be understood by decoherence.

I do not think this has ever been accomplished. But I could be wrong - can you suggest a paper that does that?

complete possible knowledge about a system (encoded in the quantum theoretical formalism as a ray in an appropriate Hilbert space) is only probabilistic, i.e., not all observables can have determined values (according to the Heisenberg-Robertson uncertainty relation).

It seems you are contradicting yourself here. Above you said that everything, even the apparatus, can be described by causal equations. Now you imply some states of the apparatus do not have determined values...?

 

[vanhees71]

Well, the deduction of the classical laws for many-body systems is standard. A very good introduction to this is Landau+Lifshitz vol. X, where the kinetic (Botlzmann) equation is derived from (non-relativistic) quantum-field theory. In our community (theory relativistic heavy-ion collisions) everybody learns this from the excellent paper

Danielewicz, P.: Quantum Theory of Nonequilibrium Processes I, Ann. Phys. 152, 239 (1984)

 

[Johan0001]

"Chaos: When the present determines the future, but the approximate present does not approximately determine the future"

When observing the properties of microscopic "particles" we currently have no practical way of completely defining the initial conditions of the closed system. This is inevitable.

It follows that we could never say (with 100 % certainty) that the state of the system in an experiment , is exactly the same state, as the same experiment done many times over previously.

Would I be correct in saying that this is a direct result of decoherence with the environment at the time of observation?

In fact would this not apply to all Theories whether QM or Classical. All theories are just an approximation of repeated observations.
Yet QM still defines the notion of mixed states before observation. How does this improve/advance science as we know it.

If we do not know whether the cat is dead or alive before actually observing. Why does it make sense to say it is in a mixed superposition of states i.e. alive and dead , until we look?

.

 

[vanhees71]

According to quantum theory a completely determined state is something else than an exact point in phase space. That's why it is so important to distinguish causality and determinism.

 

[BruceW]

There's nothing mysterious about measurement apparati. In contrast to Bohr, I don't think that a cut between a quantum dynamics and classical dynamics makes sense. Anything is quantum, according to our understanding today, and the classical behavior of macroscopic systems (including measurement apparati) is emergent and can be understood by decoherence.

There must be a cut, or we would not be able to assign a certain probability to our measurement apparatus giving one answer or another. If we had decoherence only, and no cut, then there would be no probabilities in quantum mechanics. (Unless we made some other changes to the standard QM, like using hidden variables, or many-worlds, e.t.c.)

 

[BruceW]

"Chaos: When the present determines the future, but the approximate present does not approximately determine the future"

When observing the properties of microscopic "particles" we currently have no practical way of completely defining the initial conditions of the closed system. This is inevitable.

It follows that we could never say (with 100 % certainty) that the state of the system in an experiment , is exactly the same state, as the same experiment done many times over previously.

Would I be correct in saying that this is a direct result of decoherence with the environment at the time of observation?
...

uh... not quite. For chaotic behaviour, you have something like a stream for example, and if you drop a pebble in the stream, even though you measure where you dropped it, you can't predict where it will end up, because small errors in measurement will be magnified.

On the other hand, decoherence for example, is when an electron goes into a detector, the state of the detector gets entangled with the electron and then the signal goes from the detector to a computer, and eventually the whole room gets entangled with the electron state. From here, it is very difficult to get back to the original state of electron and computer being not entangled, because the computer contains a large number of molecules which have an extremely large number of degrees of freedom.

So, in the decoherence example, the huge number of degrees of freedom of the environment mean that getting back the original electron state is practically impossible. In the chaotic stream example, if we pick up the pebble at the end of its journey, we can't work out where the pebble was initially dropped, because the chaotic behaviour of the stream will have drowned out any knowledge of where the pebble was initially placed. So, I guess the two concepts are similar, but not quite the same.

 

[Johan0001]

[QUOTE
All theories are just an approximation of repeated observations.
Yet QM still defines the notion of mixed states before observation. How does this improve/advance science as we know it.
If we do not know whether the cat is dead or alive before actually observing. Why does it make sense to say it is in a mixed superposition of states i.e. alive and dead , until we look?
][/QUOTE]

I guess this is the crux of my original question.
Why would we want to say the cat is in a super position of all states before decoherence/or an observation is made.
Why can we not just repeat the experiment many times and say the probability of finding the cat alive is X and finding the cat dead is
(1-X).
We will never , ever find the cat alive and dead!
No disrespect to the cat of course.

 

[DrChinese]

I guess this is the crux of my original question.
Why would we want to say the cat is in a super position of all states before decoherence/or an observation is made.
Why can we not just repeat the experiment many times and say the probability of finding the cat alive is X and finding the cat dead is
(1-X).
We will never , ever find the cat alive and dead!
...

A superposition acts differently in some cases. Although it is true that we never see a "cat" (originally in a superposition) other than dead or alive, that does not imply that those cases cannot be demonstrated. For example, it is possible to swap the superposition from object A to object B. And there are objective measures of entanglement, which is always a form of superposition.

 

[Jano L.]

Well, the deduction of the classical laws for many-body systems is standard. A very good introduction to this is Landau+Lifshitz vol. X, where the kinetic (Botlzmann) equation is derived from (non-relativistic) quantum-field theory. In our community (theory relativistic heavy-ion collisions) everybody learns this from the excellent paper

Danielewicz, P.: Quantum Theory of Nonequilibrium Processes I, Ann. Phys. 152, 239 (1984)

I am afraid your answer and the paper by Danielewicz do not address the point of our discussion above. Kinetic equations or other probabilistic models seem to contribute very little to the problem of derivation of common-language description of an apparatus which would be based on purely quantum-theoretical formalism. If you do not like to discuss that further, it's OK - I did not expect to get as far with this anyway.

I still think that your statement "all physics is causal" is misleading, on multiple accounts. I think you would better say something like "initial value problem with the evolution equation in quantum theory has unique solution".

 

[bhobba]

the problem of derivation of common-language description of an apparatus which would be based on purely quantum-theoretical formalism.

First I think we need to be careful with language. In the following by causal I mean deterministic - but not necessarily local determinism.

I don't know what you mean by common language. But what's going on at is well known. When the observational apparatus becomes entangled with what is being observed the mathematics of QM shows it is decohered and behaves exactly the same as if a collapse has occurred.

I still think that your statement "all physics is causal" is misleading, on multiple accounts. I think you would better say something like "initial value problem with the evolution equation in quantum theory has unique solution".

State evolution in QM is completely causal. If observations are causal or not is a matter of interpretation.

I know Vanhees, like me, holds to the ensemble interpretation. That interpretation is actually ambivalent on if an observation is causal or not - most certainly it is compatible with the view of being causal.

Thanks
Bill

 

[bhobba]

So, in the decoherence example, the huge number of degrees of freedom of the environment mean that getting back the original electron state is practically impossible. In the chaotic stream example, if we pick up the pebble at the end of its journey, we can't work out where the pebble was initially dropped, because the chaotic behaviour of the stream will have drowned out any knowledge of where the pebble was initially placed. So, I guess the two concepts are similar, but not quite the same.

That's it - I don't know why others don't get it.

The real issue is it only explains apparent collapse - actual collapse is another matter and requires further assumptions.

Thanks
Bill

 

[atyy]

We will never , ever find the cat alive and dead!
No disrespect to the cat of course.

http://www.nobelprize.org/mediaplayer/index.php?id=1873
http://www.nobelprize.org/mediaplayer/index.php?id=1871

:)

 

[bhobba]

There must be a cut, or we would not be able to assign a certain probability to our measurement apparatus giving one answer or another. If we had decoherence only, and no cut, then there would be no probabilities in quantum mechanics. (Unless we made some other changes to the standard QM, like using hidden variables, or many-worlds, e.t.c.)

Not necessarily. Decoherence is a purely quantum phenomena independent of a cut. The probabilities in the mixed state after decoherence (of course we need a precise definition of when decoherence has occurred) are not dependant on that cut. The simplest solution, and in fact the one I adhere to, is simply to say the improper state after decoherence is a proper one - problem solved.

Well I am being a bit cheeky - the problem of deciding when an observation has occurred without a cut has been solved. The issue of exactly how the improper mixed state becomes a proper one hasnt - that's the problem of outcomes - probably the most difficult problem in QM, being the modern version of collapse. Just to see more of the difficulty decoherence doesn't actually produce an improper mixed state - simply one way below our ability to detect - which throws an even bigger spanner in the works of exactly how it becomes a proper one.

Thanks
Bill

 

[.Scott]

4. Although physicists often debate the "true" meaning of determinism, causality, etc. there are no generally accepted definitions that provide a *useful* difference. They are most often used interchangeably, and when given different definitions, it is usually for a specific purpose and not something accepted all around.

If the QM notion of "conservation of information" is accepted, then there is certainly a very tight tie between one moment and the next. Moreover, the QM notion that one can run time in either direction while maintaining the same information base further suggests that "fully developed" QM rules could be used to describe any moment in time as a function of any other moment in time.

As I see it, QM argues against local determinism but strongly for absolute determinism.

 

[DrChinese]

As I see it, QM argues against local determinism but strongly for absolute determinism.

Even in time reversed interpretations, outcomes of individual observations are statistical. So I don't see that there is ANY strong argument for absolute determinism. It is merely a possibility in some interpretations.

 

[.Scott]

Even in time reversed interpretations, outcomes of individual observations are statistical. So I don't see that there is ANY strong argument for absolute determinism. It is merely a possibility in some interpretations.

That QM only provides statistical results does not indicate either determinism or non-determinism. If we knew the state of the entire universe in one cross section of time, QM, as we know it, limits what might come next - and in time reversal, what may have come just before.

But the conservation of information is much more interesting. If the changing of states is not exclusively dependent on the initial state and on the passage of time, then what else is it dependent on and how does that "what else" work in time reversal. In this case, breaking determinism appears to break the notion of time itself. If there is another parameter, beyond time, that determines how "now" turns into "next", then that other parameter would be another time-like dimension and our common perception of time as one-dimensional is at odds with physics.

 

[DrChinese]

1. That QM only provides statistical results does not indicate either determinism or non-determinism.

2. ...and our common perception of time as one-dimensional...

1. Statistical results is certainly an indication of in-determinism and is certainly NOT an indication of determinism. You would certainly expect a statistical distribution of results from a truly random set of processes.

It is a proof of neither. It is possible to have a sequence of results that appears random but is not. But you wouldn't specifically expect that.

2. One-directional might be a better description, as one dimensional allows time reversal.

 

[.Scott]

One-directional might be a better description, as one dimensional allows time reversal.

The directionality of time, that we see the past as inherently different than the future, is another topic.

My attack is against anything "inherently random" in how the universe changes states as time passes. If there is such a phenomena, you could describe it as in-determinism, but you would be more consistent with science in describing the apparent randomness as the result of another time-like parameter. After all, calling it "inherently random" means that you have decided not to address it any further as an concern of science. Whereas calling it a function of an independent time parameter (a second dimension of time) allows it to be treated more broadly.

 

[bhobba]

My attack is against anything "inherently random" in how the universe changes states as time passes.

Its a meaningless issue.

There is no way to tell the difference between something that is genuinely random and something that simply looks random.

That's why we have interpretations of that are inherently random, some that are deterministic, and even some like many worlds that are a bit unclear with a number of sometimes very long threads on this forum discussing it.

After all, calling it "inherently random" means that you have decided not to address it any further as an concern of science.

Any scientist knows that any assumption is up for grabs as science progresses. In fact this the hallmark of science.

Whereas calling it a function of an independent time parameter (a second dimension of time) allows it to be treated more broadly.

Exactly how such a strange idea has anything to do with the issue has me beat.

Thanks
Bill

 

[Johan0001]

If the QM notion of "conservation of information" is accepted, then there is certainly a very tight tie between one moment and the next.

This seems logical to me. However how does one quantify Information WRT Quantum mechanics?
Is it perhaps related to entropy?
If information is continuously conserved at least we have a history of some sorts.

If a change from one state to the next is not determined by the previous history of states , then QM cannot be complete , just an approximation.
Not chaos like a pebble in the stream as mentioned above , but a more accurate version, given a good statistical approximation.
But still just an approximation.

 

[vanhees71]

I still think that your statement "all physics is causal" is misleading, on multiple accounts. I think you would better say something like "initial value problem with the evolution equation in quantum theory has unique solution".

But that's causality expressed in a mathematically concise way! The evolution equation for the probability amplitudes, and thus the observable probabilities, tells you uniquely the state, given the initial state (provided you have complete knowledge of the Hamiltonian of the system).

It's still not a deterministic theory, because the exact knowledge of the states implies only the knowledge of probabilities for the outcome of measurements of observables for which the state is not a ray in the eigenspace of the corresponding self-adjoint operator. So not all observables are determined although the state is completely known.

Thus I stay with the definition given by Schwinger: Quantum theory is causal but indeterministic.

 

[DrChinese]

If a change from one state to the next is not determined by the previous history of states , then QM cannot be complete , just an approximation.
Not chaos like a pebble in the stream as mentioned above , but a more accurate version, given a good statistical approximation.
But still just an approximation.

Whether or not QM is complete is dependent on interpretation and definition. Is there a "more complete" version out there?

 

[Jano L.]

But that's causality expressed in a mathematically concise way! The evolution equation for the probability amplitudes, and thus the observable probabilities, tells you uniquely the state, given the initial state (provided you have complete knowledge of the Hamiltonian of the system).

Then your notion of causality is just that - property of the evolution equation. Quantum theory has more parts than that - in some events like measurement the equation is not applicable and the new ket vector has to be chosen based on the results of measurement, which as you say, cannot be found from the evolution equation.

So not all observables are determined although the state is completely known.

The state is known only before the measurement occurs. After the measurement of the atom position, the state vector has to be changed manually into new value which can be only found experimentally, or one can consider all possible results and pass on to probabilistic description of the new ket vector. Either way the new ket is not uniquely determined by the initial conditions and the evolution equation.

 

[Johan0001]

Whether or not QM is complete is dependent on interpretation and definition. Is there a "more complete" version out there?

I doubt it , and many years of advancement has taken place in QM , but I suspect we are far from ideal.
It took 200 years before a revised theory replaced classical mechanics.
Thanks for your input guys.