Danger for the Many-Worlds Interpretation?

[A. Neumaier]

the purity of even global states is an open question for theories with massless particles.

This needs qualification since the free Maxwell field is massless but has pure global states.

 

[DarMM]

Are you saying that in the finite-volume model, i.e., imposing periodic spatial boundary conditions with a cube as "quantization condition", there are no pure states?

No. It's that finite volume states in QFT with no cutoffs are always mixed states. The "finite volume" here does not refer to cutoffs but the physical volume of the system under consideration.

Any system with finite volume in QFT is mixed. This isn't really anything to do with Haag's theorem or the need to regularize.

This needs qualification since the free Maxwell field is massless but has pure global states.

Yes indeed. I should say it is an open problem as to whether theories with interacting massless spin-1 particles have global pure states, e.g. QED

 

[vanhees71]

I'm really puzzled. I talk about free fields. Take for simplicity Klein-Gordon particles in a finite cubic volume (length ##L##) with periodic boundary conditions. Then the plane-wave modes are
$$u_{\vec{p}}(x)=\frac{1}{\sqrt{2 E_{\vec{p}}} L^{3/2}} \exp(\mathrm{i} \vec{p} \cdot \vec{x}-\mathrm{i} E_{\vec{p}} t), \quad E(\vec{p})=\sqrt{m^2+\vec{p}^2}, \quad \vec{p} \in \frac{2 \pi}{L} \mathbb{Z}^3.$$
The quantum field then is
$$\hat{\phi}(x)=\sum_{\vec{p}} [\hat{a}(\vec{p}) u_{\vec{p}}(x) + \hat{b}^{\dagger}(\vec{p}) u_{\vec{p}}^*(x)].$$
The Fock states for particles and anti-particles
$$|N_a(\vec{p}),N_b(\vec{p}) \rangle \propto \prod_{\vec{p}} \hat{a}^{\dagger N_a(\vec{p})} \hat{b}^{\dagger N_a(\vec{p})} |\Omega \rangle, \quad \sum_{\vec{p}} N_a(\vec{p}), \quad \sum_{\vec{p}} N_b(\vec{p}) \quad \text{finite},$$
are proper pure states with definite numbers of particles and anti-particles.

Other examples for proper pure states are the coherent states.

 

[DarMM]

I'm really puzzled. I talk about free fields. Take for simplicity Klein-Gordon particles in a finite cubic volume (length ##L##) with periodic boundary conditions

As I said this concerns infinite volume QFT with no cutoffs.

However there is an analogue in the finite volume case. Try to construct a state that only takes up a portion of the cutoff volume. Your plane waves span the entire volume.

 

[Michael Price]

As I said this concerns infinite volume QFT with no cutoffs.

However there is an analogue in the finite volume case. Try to construct a state that only takes up a portion of the cutoff volume. Your plane waves span the entire volume.

And how does this prove there are mixed states?

 

[DarMM]

And how does this prove there are mixed states?

The reference I gave in #190 contains the full details.

In essence one proves that for QFT the algebra of observables for a finite region is a Type III C*-algebraic factor. It is then a known result in the theory of C*-algebras that Type III factors have no pure states.

 

[Michael Price]

The reference I gave in #190 contains the full details.

In essence one proves that for QFT the algebra of observables for a finite region is a Type III C*-algebraic factor. It is then a known result in the theory of C*-algebras that Type III factors have no pure states.

In other words the reference does not make your claim, but it is your inference. And how does this prove that dead and alive cats (or elephants) only appear as mixtures? If that were proven it would be headline news - somehow I think it does not follow.

 

[DarMM]

In other words the reference does not make your claim

It makes my claim directly in section 4.1.2 as I pointed you to in #190

 

[Michael Price]

It makes my claim directly in section 4.1.2 as I pointed you to in #190

I note you left unanswered the relevance to the cats in a mixture issue.

 

[DarMM]

I note you left unanswered the relevance to the cats in a mixture issue.

I cannot believe I have to spell this out and this will probably be one of strangest sentences I have ever written here in earnest:
Cats are not infinite volume systems.

 

[Michael Price]

I cannot believe I have to spell this out and this will probably be one of strangest sentences I have ever written here in earnest:
Cats are not infinite volume systems.

Still waiting for your answer.

In the absence of that, I conclude there is no reason to suppose that dead and alive cats always appear as mixtures. So no danger to MWI from QFT.

 

[DarMM]

Still waiting for your answer.

What? Are you serious?

In QFT finite volume systems are mixed. Cats are finite volume systems. Thus cats in QFT are in mixed states.

 

[Michael Price]

What? Are you serious?

In QFT finite volume systems are mixed. Cats are finite volume systems. Thus cats in QFT are in mixed states.

I would like to see you publish that claim, or give an explicit reference.

And include the word "always", which you left out by accident, I'm sure.☺

 

[DarMM]

I would like to see you publish that claim, or give an explicit reference.

And include the word "always", which you left out by accident, I'm sure.☺

Okay. Let's try this very slowly.

Step 1: In QFT all finite volume systems are always mixed states. This and links to proofs of it are given in the reference I explicitly cited. This property of QFT has been known for decades so having me write a paper about it is pointless.

Step 2: A cat is a finite volume system

Step 3: So in QFT a cat is always in a mixed state.

 

[Tendex]

Okay. Let's try this very slowly.

Step 1: In QFT all finite volume systems are always mixed states.

Not in QFT, in a version of QFT according to you not regularized, so non-physical, and non- perturbative, so without rigorous mathematical justification.

 

[DarMM]

Not in QFT, in a version of QFT according to you not regularized, so non-physical, and non- perturbative, so without rigorous mathematical justification.

Yang Mills has been proven to obey the necessary conditions for the theorem to go through by Tadeusz Balaban.
As do free theories, which have been rigorously constructed in 4D as well.
Also interacting theories in lower dimensions.

 

[Michael Price]

So you claim that the Schrödinger cat paradox is resolved by QFT?
I am all ears.

 

[DarMM]

So you claim that the Schrödinger cat paradox is resolved by QFT?
I am all ears.

I'm claiming that finite volume systems in QFT have mixed states. Where by "claim", I mean "I have provided links to the proof".

 

[Tendex]

Yang Mills has been proven to obey the necessary conditions for the theorem to go through by Tadeusz Balaban.
As do free theories, which have been rigorously constructed in 4D as well.
Also interacting theories in lower dimensions.

What theorem?

 

[DarMM]

What theorem?

Really? That finite volume states in QFT are mixed.

 

[Minnesota Joe]

Her objection doesn't make sense to me. And would apply to all interpretations. She seems to be objecting to standard nomenclature.

Well your aren't alone if her comment thread is any indication, but I don't think she is complaining about the Born Rule per se, but that MWI needs a postulate beyond the Schrodinger equation.

I could be wrong, but I don't know that her objection would apply to interpretations that add equations or modify the Schrodinger equation in some way. Because those interpretations are paying their own price to explain the measurement problem whereas MWI has to be committed to deriving everything from the Schrodinger equation.

In the original post she phrased the problem as "You should only evaluate the probability relative to the detector in one specific branch at a time". That isn't entailed by the Schrodinger equation presumably. So MWI needs another postulate.

She elaborates in her latest video start at minute 5:50 and mentioning many worlds specifically at 6:35. She is claiming that MWI needs an additional postulate about what a detector does. And not just MWI, but all "neo-Copenhagen" interpretations.

Finally, in the comments for the latest video is perhaps the clearest statement of her argument:

No, the assumption "the probability becomes one" does of course not assume wave-function collapse. The probability that anything happens is always one. The assumption is that the probability *of what you have measured* becomes one. This requires that you update the wavefunction. Just write it down if you do not understand what I mean.

Many worlds doesn't help you with that. Without a wave-function update or equivalent postulate, many worlds predicts that you split into a huge number of universes and observe anything with probability one. This is clearly not what we observe.

http://backreaction.blogspot.com/20...howComment=1571757895358#c2361450303254659859

 

[Tendex]

Really? That finite volume states in QFT are mixed.

That's not a theorem in any mathematically serious sense.

 

[akvadrako]

In QFT finite volume systems are mixed. Cats are finite volume systems. Thus cats in QFT are in mixed states.

Not to sidetrack this fascinating exchange /s, but how do you interpret these mixed states? Is it just that there are always multiple possibilities for the subsystem, evolving in parallel without any interference terms? That one can't prepare a system in a pure state; there is always some uncertainty about which state it is?

 

[DarMM]

That's not a theorem in any mathematically serious sense.

How so? It has been proven from a subset of the Haag-Kastler axioms. Are the Haag-Kastler axioms not serious?

 

[DarMM]

Not to sidetrack this fascinating exchange /s, but how do you interpret these mixed states? Is it just that there are always multiple possibilities for the subsystem, evolving in parallel without any interference terms? That one can't prepare a system in a pure state without some uncertainty about which pure state it is?

It's a bit more than that. There are no pure states in a finite region in QFT due to this result. So you can't consider these mixed states as uncertainty about pure states or as pure states evolving in parallel.

 

[Michael Price]

Well your aren't alone if her comment thread is any indication, but I don't think she is complaining about the Born Rule per se, but that MWI needs a postulate beyond the Schrodinger equation.

I could be wrong, but I don't know that her objection would apply to interpretations that add equations or modify the Schrodinger equation in some way. Because those interpretations are paying their own price to explain the measurement problem whereas MWI has to be committed to deriving everything from the Schrodinger equation.

In the original post she phrased the problem as "You should only evaluate the probability relative to the detector in one specific branch at a time". That isn't entailed by the Schrodinger equation presumably. So MWI needs another postulate.

She elaborates in her latest video start at minute 5:50 and mentioning many worlds specifically at 6:35. She is claiming that MWI needs an additional postulate about what a detector does. And not just MWI, but all "neo-Copenhagen" interpretations.

Finally, in the comments for the latest video is perhaps the clearest statement of her argument:

http://backreaction.blogspot.com/20...howComment=1571757895358#c2361450303254659859

Her comment "This [MWI] is clearly not what we observe." suggests that perhaps she doesn't get how decoherence mimics wavefunction collapse.

 

[Tendex]

How so? It has been proven from a subset of the Haag-Kastler axioms. Are the Haag-Kastler axioms not serious?

I hope you can appreciate the difference between theorem and axioms.
As I said that statement needs some important assumptions and it is not valid "in QFT" in general without them.

 

[DarMM]

In which case MWI is okay with QFT. No danger at all.

You originally said that a cat must have a pure state that is a superposition of alive and dead. This shows this isn't true, fundamentally a cat will not be in a pure state. Specifically this:

Yes you can so describe the two coarse grained states in a single pure state.

Is not true.

Whether this is a problem or not for Many Worlds, I don't know. The only real implication I think is that you have to go to mixed state realism, not just pure state realism. There are interesting follow ons from that which one may consider an issue or you might not. For example it might lead toward implying that all uncertainty, not just that from quantum superposition, e.g. "where are my keys", is self-locating uncertainty.

 

[akvadrako]

It's a bit more than that. There are no pure states in a finite region in QFT due to this result. So you can't consider these mixed states as uncertainty about pure states or as pure states evolving in parallel.

Yes, I see it must be something more fundamental. If mixed states aren't "mixtures" of something else, what can they be? For example if your finite volume system is a single qubit that would in NRQM be described as the pure state of spin up?

 

[DarMM]

I hope you can appreciate the difference between theorem and axioms.

Yes I do. As I said, this is a theorem that follows from a subset of the Haag-Kastler axioms. Thus we distinguish the theorem (that finite volume states are mixed) from the axioms (those of Haag and Kastler)

As I said that statement needs some important assumptions and it is not valid "in QFT" in general without them.

Those assumption being that the field theory obeys a subset of the Haag-Kastler axioms (or a subset of the Wightman axioms). If those axioms are broken, the QFT's correlation functions don't even exist.
Can you state the Haag-Kastler axiom you find doubtful? Or the Wightman axiom?

 

[DarMM]

Yes, I see it must be something more fundamental. If mixed states aren't "mixtures" of something else, what can they be? For example if your finite volume system is a single qubit that would in NRQM be described as the pure state of spin up?

I know this sounds like an empty answer initially, but they are just a state then. Not classical uncertainty about some smaller set of "real" states, i.e. pure states. Really you might characterize a pure state as one for which there is a single measurement (of some observable ##\mathcal{O}##) for which one of its outcomes is certain, i.e. a pure state has a single determined experiment.

This result would then say for actual real finite volume systems there is no such experiment, i.e. there is no operation that has utterly determined results.

 

[Minnesota Joe]

Her comment "This [MWI] is clearly not what we observe." suggests that perhaps she doesn't get how decoherence mimics wavefunction collapse.

Maybe, but in that same video she is explicit that decoherence by itself doesn't solve the measurement problem. And that is true as I understand it. Decoherence leaves all the approximate eigenstates in the sum.

 

[PeterDonis]

@Michael Price, you have earned yourself a warning and a thread ban. Your conduct in this thread has become indistinguishable from trolling.

 

[PeterDonis]

As I said that statement needs some important assumptions and it is not valid "in QFT" in general without them.

Every theorem requires assumptions. If you want to claim that a theorem is not relevant physically, you need to describe some actual physical situation in which key assumptions of the theorem are not met. Just pointing out that the theorem requires assumptions is not enough and contributes nothing useful to the discussion. Please take note.

 

[vanhees71]

Really? That finite volume states in QFT are mixed.

I think now I understand, where our mutual misunderstanding comes from: You discuss the impossibility of pure states with finite support in QFT in the infinite-volume limit, while I understood the whole time you mean the finite-volume regularized model... That of course resolves the issue.