Stephen Weinberg on Understanding Quantum Mechanics

[stevendaryl]

In solid state physics, already a small bar of metal is treated very successfully as an infinite system. We know that even for simple model systems e.g. an infinite system of spin 1/2 particles, it isn't possible to write down neither a wavefunction nor a Schroedinger equation.

Do you mean that such a system doesn't have a wave function, or that the wave function is too complex to reason about?

 

[bhobba]

You seem to think it's a personal choice which physical theory to accept, but I disagree.

Ok - we disagree.

Thanks
Bill

 

[stevendaryl]

There is at least one counterexample, namely non-linear quantum-like theory without instantaneous communication. It is classical mechanics itself:
https://arxiv.org/abs/0707.2319

Good point. But Weinberg may have meant that adding nonlinearity to the rest of the rules for quantum mechanics (in particular, that measuring a quantity yields an eigenvalue with a probability given by the Born rule) produces FTL effects. Your rewriting of classical mechanics as nonlinear quantum mechanics doesn't preserve this rule. That would probably be okay if the rule were preserved as an approximation, but for classical mechanics, it's not even approximately true (I don't think).

 

[Demystifier]

Your rewriting of classical mechanics as nonlinear quantum mechanics doesn't preserve this rule.

My rewriting of classical mechanics as nonlinear QM does preserve the Born rule.

 

[stevendaryl]

My rewriting classical mechanics as nonlinear QM does preserve the Born rule.

Really? That a measurement of angular momentum must yield a multiple of $\hbar$? But that's not a prediction of classical physics.

 

[Demystifier]

Really? That a measurement of angular momentum must yield a multiple of $\hbar$? But that's not a prediction of classical physics.

No, it preserves the Born rule only in the preferred basis, which turns out to be the position basis.

 

[stevendaryl]

No, it preserves the Born rule only in the preferred basis, which turns out to be the position basis.

Okay. Then that makes my original point correct (or at least, more plausible): A nonlinear generalization of Schrödinger's equation that preserves the rule that a measurement always yields an eigenvalue of the operator corresponding to the observable being measured would allow FTL influences.

 

[Demystifier]

A nonlinear generalization of Schrödinger's equation that preserves the rule that a measurement always yields an eigenvalue of the operator corresponding to the observable being measured would allow FTL influences.

Yes, but this assumes some version of the "collapse" postulate for QM (even if "collapse" is nothing but an update of knowledge), and in my paper I have explained why such a "collapse" postulate is totally unjustified for non-linear theories. Classical mechanics as non-linear QM works precisely because there is one part of the wave function which satisfies a linear equation, so one can use a "collapse" postulate for that part.

 

[stevendaryl]

Yes, but this assumes some version of the "collapse" postulate for QM (even if "collapse" is nothing but an update of knowledge), and in my paper I have explained why such a "collapse" postulate is totally unjustified for non-linear theories. Classical mechanics as non-linear QM works precisely because there is one part of the wave function which satisfies a linear equation, so one can use a "collapse" postulate for that part.

Okay, but I think what Weinberg was talking about was the possibility of a theory that is approximately the same as current quantum mechanics, except for the small nonlinearity. In cases like EPR, I'm guessing that the slight nonlinearity would allow the weird correlations to be used for FTL communications.

 

[RockyMarciano]

@stevendaryl , if you think that classical mechanics predicts possible instantaneous communication of information, why don't you correct the following assertion by Demystifier. A thread was closed just for saying that classical mechanics doesn't imply instantaneous information sending.

There is at least one counterexample, namely non-linear quantum-like theory without instantaneous communication. It is classical mechanics itself:
https://arxiv.org/abs/0707.2319

 

[Demystifier]

Okay, but I think what Weinberg was talking about was the possibility of a theory that is approximately the same as current quantum mechanics, except for the small nonlinearity. In cases like EPR, I'm guessing that the slight nonlinearity would allow the weird correlations to be used for FTL communications.

Yes, and I am saying that the way how Weinberg formulated this theory involves some kind of "collapse" postulate, which makes his formulation unjustified for even a small non-linearity.

 

[bhobba]

@stevendaryl , if you think that classical mechanics predicts possible instantaneous communication of information, why don't you correct the following assertion by Demystifier. A thread was closed just for saying that classical mechanics doesn't imply instantaneous information sending.

Hmmmmm. I think I know what Dymystifyer means, but its best if he expands on it.

But it must be said that because Newtonian Classical Mechanics is based on the Galilean transformations it is explicitly non-local. I have mentioned this many times and its quite obvious but for some reason Landau - Mechanics is the only text I know that goes into it. Don't know why.

Thanks
Bill

 

[Demystifier]

I think I know what Dymystifyer means, but its best if he expands on it.

That's why I write papers, to avoid explaining the same thing several times.

 

[Auto-Didact]

Okay, but I think what Weinberg was talking about was the possibility of a theory that is approximately the same as current quantum mechanics, except for the small nonlinearity. In cases like EPR, I'm guessing that the slight nonlinearity would allow the weird correlations to be used for FTL communications.

To quote Feynman and paraphrase Penrose, you can't add imperfections to a perfect thing, you need another perfect thing. In a similar vein, merely tinkering with the structure of QM by adding small nonlinearities to the Schrödinger equation is an unlikely route of arriving at a theory to which QM is an approximation; mere tinkering by adding nonlinearities is not what Einstein did, he did something much more radical, yet his theory is reducible to Newtonian gravity in appropriate limits. The resulting theory of gravity going from Newton to Einstein was from a mathematical point of view completely different. This is what is meant by a non-linear extension of a theory.

Yes, and I am saying that the way how Weinberg formulated this theory involves some kind of "collapse" postulate, which makes his formulation unjustified for even a small non-linearity.

That's why I write papers, to avoid explaining the same thing several times.

Link please.

 

[DrDu]

It is mathematically ill defined.

 

[Auto-Didact]

It is mathematically ill defined.

What is?

 

[George Jones]

It is mathematically ill defined.

What is?

I, too, would like to know what DrDu meant by this. I do know that many of standard pertunbation series produced in non-relativistic quantum mechanics (almost certainly) are divergent (but probably asymptotic) series, but I don't think that I would count this as "mathematically ill defned." For example, see section 21. 4 "Divergences of perturbation series" in the text "Quantum Mechanics: A New Introduction" by Konishi and Paffuti,

https://www.amazon.com/dp/0199560277/?tag=pfamazon01-20

 

[PAllen]

Link please.

See post #19 of this thread.

 

[A. Neumaier]

We know that even for simple model systems e.g. an infinite system of spin 1/2 particles, it isn't possible to write down neither a wavefunction nor a Schroedinger equation.

But there are density operators describing states. These encode the true reality.

 

[A. Neumaier]

http://www.nybooks.com/articles/2017/01/19/trouble-with-quantum-mechanics/

Weinberg's formal proposal about a possible resolution of the troubles is discussed in another thread here.

 

[DrDu]

But there are density operators describing states. These encode the true reality.

There are still states as functionals on the algebra of local operators. But there is no longer a clear distinction between pure states and mixtures and this is precisely the point. In an infinite system, we have from the outset no possibility to tell a pure state from a mixture. Therefore, the question how a pure state evolves into a mixture during measurement is also pointless.

 

[DrDu]

I, too, would like to know what DrDu meant by this.

Sorry, I was trying to reply to post #31 from my smartphone, but somehow, the reference was missing.

 

[Demystifier]

Link please.

As Pallen said, see post #19.

 

[A. Neumaier]

There are still states as functionals on the algebra of local operators. But there is no longer a clear distinction between pure states and mixtures and this is precisely the point. In an infinite system, we have from the outset no possibility to tell a pure state from a mixture. Therefore, the question how a pure state evolves into a mixture during measurement is also pointless.

I fully agree. In the case of interacting relativistic quantum field theories, there are no pure states at all! This is the correct level on which foundations must be discussed. Treating instead pure states and Born's rule as God-given foundations is very questionable! There is also no concept of a superposition of general states; so the alleged problems with Schroedinger's cat disappear!

 

[stevendaryl]

There are still states as functionals on the algebra of local operators. But there is no longer a clear distinction between pure states and mixtures and this is precisely the point. In an infinite system, we have from the outset no possibility to tell a pure state from a mixture. Therefore, the question how a pure state evolves into a mixture during measurement is also pointless.

Understanding is pointless?

 

[stevendaryl]

I fully agree. In the case of interacting relativistic quantum field theories, there are no pure states at all! This is the correct level on which foundations must be discussed. Treating instead pure states and Born's rule as God-given foundations is very questionable! There is also no concept of a superposition of general states; so the alleged problems with Schroedinger's cat disappear!

This attitude seems bizarre to me. It's not that the use of density matrices provides any new answers, it just makes it more difficult to rigorously formulate the question.

 

[A. Neumaier]

This attitude seems bizarre to me. It's not that the use of density matrices provides any new answers, it just makes it more difficult to rigorously formulate the question.

On the deepest level (where factors are of type III_1) there are no pure states, so starting with pure states (pretending that factors have type I) is introducing artifacts that are not present on the underlying level. Taking these artifacts as the full truth produces strange things. In particular, whatever is rigorously formulated at that level, is nonrigorous (and indeed meaningless) on the more fundamental level.

 

[stevendaryl]

On the deepest level (where factors are of type III_1) there are no pure states, so starting with pure states (pretending that factors have type I) is introducing artifacts that are not present on the underlying level. Taking these artifacts as the full truth produces strange things. In particular, whatever is rigorously formulated at that level, is nonrigorous (and indeed meaningless) on the more fundamental level.

I disagree, because we can understand mixed states in terms of pure states with uncertainty (or in terms of pure states in which some of the degrees of freedom have been traced out). There is nothing conceptually new about mixed states that changes anything, as far as I can see.

 

[DrDu]

Understanding is pointless?

I am sorry I can't formulate this any better, but I think at least A. Neumaier understood what I wanted to say. I tried to grasp a little bit of AQFT some years ago, and think I got some intuition, but not sufficient to explain myself clearly.

 

[A. Neumaier]

I disagree, because we can understand mixed states in terms of pure states with uncertainty (or in terms of pure states in which some of the degrees of freedom have been traced out).

You are thinking only in terms of type I representations (in the classification of von Neumann). For these, which adequately describe the quantum mechanics of finitely many degrees of freedom, your statement is correct. However, the real world is occupied by macroscopic bodies, which need quantum field theory and infinitely many degrees of freedom for their description. Already a laser, which generates the quantum states with which Bell-type experiments are performed, is such a system. Once the number of degrees of freedom is infinite, the other types in von Neumann's classification play a role. In particular, in relativistic QFTs one has always representations of type III_1; see the paper by Yngvason cited in the link given above.

There is nothing conceptually new about mixed states that changes anything, as far as I can see.

Type III_1 representations behave conceptually very differently, as no pure states exist in these representations. In these representations one cannot rigorously argue about states by considering partial traces in nonexistent pure states! This shows that pure states are the result of a major approximating simplification, and not something fundamental.