Stephen Weinberg on Understanding Quantum Mechanics

[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 Schrodinger'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 Schrodinger'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 Schrodinger 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.

 

[A. Neumaier]

Understanding is pointless?

Dr. Du said the question is pointless, not the understanding. Understanding must be based on asking and answering questions that can be meaningfully formulated in the framework in which the theory is described. If a theory contains no notion of pure states, asking questions involving the latter is not meaningful.

 

[stevendaryl]

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.

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.

First of all, I don't agree that any of the conceptual problems with quantum mechanics are resolved by using density matrices.

Second, I'm not sure I understand the claim about the nonexistence of pure states. I thought that in QFT, you can still work with pure states. At least, in perturbation, you can think of the pure states as being of the form of a superposition of states with zero, one, two, etc. applications of creation operators on the vaccuum.

 

[DrDu]

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.

I just had a look at the nice article by Yngvarson A. Neumaier cited. There are examples of how these type III theories arise for infinite tensor products of spin 1/2 which I mentioned earlier. The point is that the concept of a state as used in this article is not simply the infinite product of single particle spin wavefunctions.

 

[stevendaryl]

The recent (this year) article by Weinberg seems to me to be claiming that he considers there to still be unresolved conceptual problems with quantum mechanics. So if people are citing Weinberg as evidence that the use of density matrices resolves everything, it seems to me that they are disagreeing with Weinberg.

 

[A. Neumaier]

First of all, I don't agree that any of the conceptual problems with quantum mechanics are resolved by using density matrices.

I am not talking about density matrices. In QFT, states are described by positive linear functionals, in the simplest case given by density operators, infinite-dimensional versions of what you like to play with. Moreover, I didn't claim that all conceptual problems are resolved when working with density operators. Only that working with pure states cannot solve the conceptual problems since pure states are themselves an approximation.

Second, I'm not sure I understand the claim about the nonexistence of pure states. I thought that in QFT, you can still work with pure states. At least, in perturbation, you can think of the pure states as being of the form of a superposition of states with zero, one, two, etc. applications of creation operators on the vacuum.

I am sure you don't understand. Please read Yngvason's paper. The use of pure states is restricted to free quantum field theory, which is described in Fock representations, which have type I. One can do perturbation theory about a free theory, but only approximately (which makes rigorous arguments impossible) and only after renormalization (which destroys the Fock structure and causes the change of type).

Pure states are from a fundamental point of view useful approximations, nothing more. If one runs into conceptual problems when using them, the problems may well be caused by the approximations involved, especially if the arguments used assume that everything holds exactly without error.

 

[RockyMarciano]

I don't think density matrices solve anything, but how does that prevent us from realizing that pure states are just fictional approximations?

 

[A. Neumaier]

if people are citing Weinberg as evidence that the use of density matrices resolves everything, it seems to me that they are disagreeing with Weinberg.

You are talking about the empty set. I neither claimed that the use of density matrices resolves everything, nor was I citing Weinberg - the material I referred to is in Yngvason's paper linked to in my discussion of Weinberg's paper. Weinberg does not refer to him and may well be unaware of these matters. Finally, disagreement with Weinberg is no argument against truth.

 

[stevendaryl]

I took a look at the paper by Yngvason here:

https://arxiv.org/abs/1401.2652

It's very interesting, but I'm not sure I understand the point about the Type III states for which there are no pure states. Yngvason is defining a "pure" state $\omega$ as one that cannot be written in the form $\omega = \frac{1}{2} \omega_1 + \frac{1}{2} \omega_2$ with $\omega_1 \neq \omega_2$. I don't understand the motivation for this definition.

In the case of density matrices for non-relativistic quantum mechanics, a density matrix $\rho$ can always be written in the form:

$\rho = \sum_j p_j |\phi_j\rangle \langle \phi_j |$

where $p_j \geq 0$ and $\sum_j p_j = 1$.

Then we can define a pure state to be one that can be written using only one vector:

$\rho = |\phi\rangle \langle \phi |$

That has the interpretation in terms of classical probability that $\rho$ represents the situation in which the system is in state $|\psi_j\rangle$ with probability $p_j$.

Yngvason is using a different definition of "pure state" (which presumably reduces to the same thing in the case of NRQM), and is saying that

• There are types of systems for which there are no pure states.
• (Therefore) density matrices cannot be interpreted as classical probabilities for being in this or that pure state.

I don't understand the definition or the conclusion, so I need to think about it a little more.

 

[stevendaryl]

You are talking about the empty set. I neither claimed that the use of density matrices resolves everything, nor was I citing Weinberg - the material I referred to is in Yngvason's paper linked to in my discussion of Weinberg's paper. Weinberg does not refer to him and may well be unaware of these matters. Finally, disagreement with Weinberg is no argument against truth.

That's certainly true, but I thought you were citing Weinberg's paper on Quantum Mechanics Without State Vectors
as evidence that the issues were resolved by using density matrices instead of state vectors.

 

[stevendaryl]

I notice that in the paper by Yngvason , he concludes with a warning that this discussion does not resolve any of the foundational issues with QM:

On the other hand, the framework of LQP does not per se resolve all “riddles” of
quantum physics. Those who are puzzled by the violation of Bell’s inequalities in EPR
type experiments will not necessarily by enlightened by learning that local algebras are
type III. Moreover, the terminology has still an anthropocentric ring (“observables”,
“operations”) as usual in Quantum Mechanics. This is disturbing since physics is concerned
with more than designed experiments in laboratories. We use quantum (field)
theories to understand processes in the interior of stars, in remote galaxies billions
of years ago, or even the “quantum fluctuations” that are allegedly responsible for
fine irregularities in the 3K background radiation. In none of these cases “observers”
were/are around to “prepare states” or “reduce wave packets”! A fuller understanding
of the emergence of macroscopic “effects” from the microscopic realm, without
invoking “operations” or “observations”, and possibly a corresponding revision of the
vocabulary of quantum physics is still called for.