What makes the interpretations of Quantum Mechanics so important?

[vanhees71]

In all pictures the wave function is time-dependent. See my summary of the formalism in an arbitrary picture here:

https://www.physicsforums.com/threads/realism-from-locality.974177/page-9#post-6209720
Now the position eigenvectors obey
$$|\vec{x},t \rangle_{j} = \hat{A}^{(j)}(t) |\vec{x},t=0 \rangle$$
and the state ket
$$|\psi(t) \rangle_{j}=\hat{C}^{(j)}(t) |\psi(t=0) \rangle.$$
The wave function by definition is
$$\psi(t,\vec{x})={_{j} \langle}\vec{x},t|\psi(t) \rangle_{j} = \langle \vec{x},t=0|\hat{A}^{(j) \dagger}(t) \hat{C}^{(j)}(t)|\psi,t=0 \rangle.$$
In the above linked posting, I've shown that the unitary operator
$$\hat{G}(t)=\hat{A}^{(j) \dagger}(t) \hat{C}^{(j)}(t)$$
is independent of the picture of time evolution, and so is
$$\psi(t,\vec{x})=\langle \vec{x},0|hat{G}(t)|\psi(t=0).$$
Note that this must be so, because nothing with physical, i.e., measurable meaning can dependent on the arbitrary choice of the picture of time evolution, and now
$$P(t,\vec{x})=|\psi(t,\vec{x})|^2$$
is the probability distribution for the particle's position, which is a measurable quantity and thus must be picture independent.

 

[DarMM]

Of course, while its not generally talked about in pop-sci accounts, its likely all our theories are

Effective in the sense of inaccurate at some point or effective in the sense of requiring a cutoff to be well defined?

 

[DarMM]

Nothing.

Is this content free remark going to be justified at some point?

Clearly the work on interpretations has lead to progress. This seems to disparage attempts to understand QM.

 

[bhobba]

Effective in the sense of inaccurate at some point or effective in the sense of requiring a cutoff to be well defined?

Cutoff. Considering the re-normalization group not sure that cutoff's are necessarily well defined in the sense of one only.

Thanks
Bill

 

[A. Neumaier]

The wave function is independent of the picture of time evolution!

It depends on how the wave function is defined. Yours is not the only definition.

For example, with your definition, photons have no wave function, but the Silberstein vector is a perfectly valid 1-photon wave function in the same sense as the solutions of the Dirac equation with positive energy Fourier transform are wave functions for the electron. See this paper by Tamburini and Vicino.

 

[Quanundrum]

One exception is David Deutsch's idea of universes /worlds / minds already being "pre-split" - but he is explicit that he has augmented Everett's pared down QM with an extra assumption.

And yes, there is some disagreement about how to recover the Born rule, but that is hardly a fundamental disagreement - more like a work in progress.

I strongly disagree with this perspective. Let's first take a deeper look at the Split vs Divergence. David Wallace and Simon Saunders has also argued in favor of Divergence in the seminal The Emergent Multiverse: Quantum Theory according to the Everett Interpretation book. It's also recognized by Sean Carroll as undecided in his and Sebens much cited paper from 2014 wherein they attempt to derive the Born Rule from Self-Location Uncertainty. Jenna Ismael has touched on it, Max Tegmark as well and so have numerous others, including as you point out David Deutsch. So this is not a fringe topic, it's just one of the least discussed fundamental topics of MWI. For the most in-depth work on this topic I would point to the work by Alastair Wilson.

However, it doesn't stop with the Born Rule and Splitting vs Divergence, one can also make the same case for the very ontology of Everettian QM. Some just blurt out: "The wavefunction is real and all there is", whereas others such as David Wallace is strong opponents of this view. He and Chris Timpson has in recent years dug into State Space Time and reject Wavefunction realism. Valia Allori has done extensive work on this as well. If you asked someone like Wojciech Zurek you would get an answer that tries to derive ontology through decoherence via quantum darwinism. If you ask Dieter Zeh you would get a very different derivation attempt via decoherence. If you go even further and ask the likes of Max Tegmark you would again get a wildly different perspective.

Finally I will mention the issue of local vs non-local in MWI. On the surface it seems to be a local theory, but then again when you dig deeper you realize that it's largely undecided and the main proponents hold very different views.

So in summary, not a single topic within the Everettian Interpretation enjoys consensus even among its supporters.

I personally hold the opinion that the Everettian programme has been way too simplified. "What happens during measurement? EVERYTHING!" is very simple for the mind to comprehend, so it has this intellectual lazy appeal, but then when you try to make sense of what that would actually mean you open Pandora's Box of inconsistencies, unresolved issues, additional assumptions etc. which render the simplicity selling point of the interpretation nonexistent.

 

[Michael Price]

I strongly disagree with this perspective. Let's first take a deeper look at the Split vs Divergence. David Wallace and Simon Saunders has also argued in favor of Divergence in the seminal The Emergent Multiverse: Quantum Theory according to the Everett Interpretation book. It's also recognized by Sean Carroll as undecided in his and Sebens much cited paper from 2014 wherein they attempt to derive the Born Rule from Self-Location Uncertainty. Jenna Ismael has touched on it, Max Tegmark as well and so have numerous others, including as you point out David Deutsch. So this is not a fringe topic, it's just one of the least discussed fundamental topics of MWI. For the most in-depth work on this topic I would point to the work by Alastair Wilson.

However, it doesn't stop with the Born Rule and Splitting vs Divergence, one can also make the same case for the very ontology of Everettian QM. Some just blurt out: "The wavefunction is real and all there is", whereas others such as David Wallace is strong opponents of this view. He and Chris Timpson has in recent years dug into State Space Time and reject Wavefunction realism. Valia Allori has done extensive work on this as well. If you asked someone like Wojciech Zurek you would get an answer that tries to derive ontology through decoherence via quantum darwinism. If you ask Dieter Zeh you would get a very different derivation attempt via decoherence. If you go even further and ask the likes of Max Tegmark you would again get a wildly different perspective.

Finally I will mention the issue of local vs non-local in MWI. On the surface it seems to be a local theory, but then again when you dig deeper you realize that it's largely undecided and the main proponents hold very different views.

So in summary, not a single topic within the Everettian Interpretation enjoys consensus even among its supporters.

I personally hold the opinion that the Everettian programme has been way too simplified. "What happens during measurement? EVERYTHING!" is very simple for the mind to comprehend, so it has this intellectual lazy appeal, but then when you try to make sense of what that would actually mean you open Pandora's Box of inconsistencies, unresolved issues, additional assumptions etc. which render the simplicity selling point of the interpretation nonexistent.

Can you give me the page numbers for Divergence in The Emergent Multiverse? Deutsch was the one who started the Divergence buisiness, by explicitly introducing an extra axiom. And Zurek is no longer a MW - I wouldn't like to say what he is nowadays, but he rejects the MWI.

 

[timmdeeg]

The wavefunction is real.

How do you explain that the wavefunction isn't just a mathematical construct?

 

[Michael Price]

What does "just a mathematical construct" mean? If it gives physical results it is clearly more.
Do you accept the existence of atoms?

 

[DarMM]

Cutoff. Considering the re-normalization group not sure that cutoff's are necessarily well defined in the sense of one only.

Cutoffs don't seem to be necessary though. There's no strong argument that they are in QED and for Yang Mills we know a continuum limit exists.

 

[A. Neumaier]

Cutoffs don't seem to be necessary though. There's no strong argument that they are in QED and for Yang Mills we know a continuum limit exists.

QED in [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] has no cutoffs.

 

[timmdeeg]

What does "just a mathematical construct" mean?

A calculation which yields probabilities. If I remember correctly Zeilinger was arguing this way somewhere.

 

[Michael Price]

A calculation which yields probabilities. If I remember correctly Zeilinger was arguing this way somewhere.

In that case it clearly applies to a wavefunction - by definition. But is highly contrived.

 

[Quanundrum]

Can you give me the page numbers for Divergence in The Emergent Multiverse? Deutsch was the one who started the Divergence buisiness, by explicitly introducing an extra axiom. And Zurek is no longer a MW - I wouldn't like to say what he is nowadays, but he rejects the MWI.

From page 281 and onwards. David Deutsch has been the most explicit, however due to the fact that there is nothing in EQM that says splitting is happening it is undetermined by the maths. I would like to ask you, since you seem to have a strong opinion on it definitely splitting in EQM. What makes you think that?

Also on Zurek, his exact position is notoriously hard to pin down. I remember ~10 years ago when he released his Existential Interpretation; it seems like a mixture of Copenhagen, Decoherent Histories, QBism and Everett all in one. However, in this interview from 5-10 years ago he seems quite pro-MWI (https://www.closertotruth.com/series/why-the-quantum-so-mysterious#video-3689) is there any more recent data that makes you declare that he is "no longer MWI"?

 

[DarMM]

If work on interpretations has lead to the tower of Bell inqualities, advancement in quantum computing, the discovery of supraclassical resources in information theory and motivated many results in quantum information theory, how can it be worth nothing?

So "skeptical" is the response. I wonder did these results not occur or were they unmotivated by interpretational issues despite the comments of those who discovered them.

 

[Michael Price]

From page 281 and onwards. David Deutsch has been the most explicit, however due to the fact that there is nothing in EQM that says splitting is happening it is undetermined by the maths. I would like to ask you, since you seem to have a strong opinion on it definitely splitting in EQM. What makes you think that?

Also on Zurek, his exact position is notoriously hard to pin down. I remember ~10 years ago when he released his Existential Interpretation; it seems like a mixture of Copenhagen, Decoherent Histories, QBism and Everett all in one. However, in this interview from 5-10 years ago he seems quite pro-MWI (https://www.closertotruth.com/series/why-the-quantum-so-mysterious#video-3689) is there any more recent data that makes you declare that he is "no longer MWI"?

Thank you for the Zurek video, which I've just watched. I'm looking forward to watching the Polkinghorne and Dyson interviews. Thanks for the page number - I can see I shall actually have to read the book (it has been in my library for awhile). ☺

Yes, I am definitely a "splitter". First, I don't like Deutsch's approach, which is to impose divergence by fiat. Uuurrgh. Second, I would ask how many classical descriptions of the measuring apparatus are there? Before measurement there is one. After measurement there are a number - one for each possible outcome, or more if the apparatus has extra degrees of freedom. You could combine all these states into one pure state - but you could not give it a single classical description. I prefer my use of language to reflect classical structures, so we are led to conclude the original apparatus state has split.

 

[bhobba]

Cutoffs don't seem to be necessary though. There's no strong argument that they are in QED and for Yang Mills we know a continuum limit exists.

I accept what you and Dr Neumaier says but then we have the problem of gravity - it requires an actual cutoff about the Plank scale. But most these days seem to interpret it in the way of Wilson with an actual cutoff.

I still have something I am not quite sure of - zeta function regularization. The answer comes from analytic continuation when the cutoff is taken to the correct value - does it really have a cutoff? Maybe for another thread.

Thanks
Bill

 

[DarMM]

I accept what you and Dr Neumaier says but then we have the problem of gravity - it requires an actual cutoff about the Plank scale. But most these days seem to interpret it in the way of Wilson with an actual cutoff

I see what you mean, but it's not as if the cut off Wilsonian quantum field theory approach to gravity is known to be a sensible working model. It could just be that a QFT of gravity is a nonsense idea. The forces that are modeled correctly by QFT are either known not to need a cutoff or there is no strong argument they need one. Though probably for another thread as you said.

 

[vanhees71]

Also at least in QED we have a Landau pole, which restricts the theory to be valid only below some energy scale. I'd also say as far as realistic QFTs are concerned, aka the Standard Model, is concerned, it's an effective theory. We only don't know the actual energy scale, where it breaks down.

 

[A. Neumaier]

Also at least in QED we have a Landau pole, which restricts the theory to be valid only below some energy scale.

The Landau pole is a serious problem only for approaches with cutoff, as one cannot move the cutoff through the pole. In causal perturbation theory, the Landau pole just says that the perturbative construction works only for some range of the energy scale chosen in the renormalization conditions but not close to the Landau pole. For QED, the scale can be chosen at low (even zero) energy.

Note that QCD also has a Landau pole due to infrared effects, even at physically realizable energies. But this does not restricts its validity, but only its perurbative tractbility at these energies. See https://www.physicsoverflow.org/21329 for details.

 

[DarMM]

Also at least in QED we have a Landau pole, which restricts the theory to be valid only below some energy scale. I'd also say as far as realistic QFTs are concerned, aka the Standard Model, is concerned, it's an effective theory. We only don't know the actual energy scale, where it breaks down.

As @A. Neumaier says above the Landau pole only demonstrates a problem with the perturbative construction. The Gross-Neveau model has a Landau pole perturbatively, but nonperturbatively has a well defined continuum limit

 

[vanhees71]

I'm not so sure, whether one can proof the absence of a Landau pole in non-perturbative "exact" QED. I'm not even sure, whether there's a complete proof of its existence. For sure @A. Neumaier , has a better overview about the current status of axiomatic QFT than I.

 

[DarMM]

The absence of a Landau Pole nonperturbatively in QED has not been shown. However counterexamples where a perturbative Landau pole is present, but nonperturbatively there are none shows that one cannot use perturbative Landau poles to argue for triviality. This is what I meant by there being no strong argument that our QFTs need cutoffs.

 

[A. Neumaier]

As @A. Neumaier says above the Landau pole only demonstrates a problem with the perturbative construction.

I'm not so sure, whether one can proof the absence of a Landau pole in non-perturbative "exact" QED. I'm not even sure, whether there's a complete proof of its existence.

The Landau pole still exists in causal perturbation theory but is there something harmless not affecting the validity of the resulting expansion. It affects only cutoff-based perturbation theory, not perturbation theory in general.

For sure @A. Neumaier , has a better overview about the current status of axiomatic QFT than I.

The link given in post #55 together with https://www.physicsoverflow.org/32752 summarizes most of what is known about the question of rigorously constructing QED. But in 1+1D, more is known, and DarMM mentioned the rigorous nonperturbative construction of the Gross-Neveau model, which has a Landau pole.

 

[vanhees71]

I don't understand the above statement. What do you mean by "cutoff-based perturbation theory". A cutoff is just a way to regularize the theory. After renormalization there's no cutoff anymore. You can also renormalize the theory without introducing any cutoff by just using BPHZ. Another way to regularize is dim. reg. No matter, what you do, you'll have to introduce in the one or the other way an energy-momentum-mass scale (however you name it), namely the renormalization scale, where you define your coupling constants. To change the scale there are the renormalization-group equations. Perturbation theory is of course only a good approximation, where the effective coupling is small, i.e., at low scales for QED and at large for QCD.

I always thought causal PT is at the end yielding the same result as any other perturbation theory. So how can it prevent that the perturbative QED doesn't break down finaly at energy scales defined by the Landau pole?

As the case of QCD shows, it's sometimes also just a bad choice of field-degrees of freedom which makes perturbation theory fail. At low energies it's rather wise to choose hadron fields in effective hadronic theories, based on some symmetry ideas from QCD like chiral symmetry in the light-quark sector, heavy-quark effective theory in the heavy-quark sector etc. Since these are usually non Dyson-renormalizable formal expansions in powers of coupling constants but rather effective low-energy expansion you have the energy scale built in from scratch, telling you where (hopefully) the effective theory is valid.

 

[timmdeeg]

Some thoughts/questions regarding the physical status of the interpretations of QM.

In GR one can interpret the increasing distances as being due to motion or to expansion of space. As these interpretations are coordinate dependent they are not 'true physics' or not 'real' in a sense.

In contrast do we expect that one of the interpretations of QM is 'true physics' or 'real' and we just don't know which one? If yes is there a slight chance to find the 'real' interpretation e.g. by advanced future technology?

Do you think that a future theory of Quantum Gravity could solve the puzzle?

 

[A. Neumaier]

I don't understand the above statement. What do you mean by "cutoff-based perturbation theory". A cutoff is just a way to regularize the theory. After renormalization there's no cutoff anymore. You can also renormalize the theory without introducing any cutoff by just using BPHZ. Another way to regularize is dim. reg. No matter, what you do, you'll have to introduce in the one or the other way an energy-momentum-mass scale (however you name it), namely the renormalization scale, where you define your coupling constants. To change the scale there are the renormalization-group equations. Perturbation theory is of course only a good approximation, where the effective coupling is small, i.e., at low scales for QED and at large for QCD.

I always thought causal PT is at the end yielding the same result as any other perturbation theory. So how can it prevent that the perturbative QED doesn't break down finaly at energy scales defined by the Landau pole?

Causal PT is essentially BPHZ made rigorous.

QCD shows that a Landau pole is irrelevant when you avoid its vicinity. This holds for every renormalized theory. That the Landau pole was considered a fault of QED at high energies was due to the fact that when working with a cutoff, one obtained cutoff dependent summed contributions to the S-matrix that diverged at some large value of the cutoff.

But with cutoff-independend renormalization schemes this argument becomes empty. It gives a valid perturbative formula for the S-matrix $S(E,\Lambda)$ at any energy $E$ and any energy scale $\Lambda$ used in the renormaization condition. If one could resum the series and adjust the coupling constants appropriately, one would get something independent of $\Lambda$. Different choices of the latter therefore just amount ot chopping in different ways the nonperturbative values into the terms of an infinite asymptotic series.

That a pole appears in these schemes just means that the perturbative expansions resulting for a choice of $\Lambda$ close to the pole gives poor approxmations. But the perturbative expansions at other (higher or lower) energy scales are still well-defined. Extrapolating the results from these to any wanted energy with suitable resumming schemes therefore gives sensible results at all energies.

In particular, the Landau pole is not a sign of that a theory is only effective, only a sign of that the perturb<tion series cannot always be trusted.

 

[vanhees71]

Well, does this imply that after all non-perturbative QED is shown to have no problems at whatever scale? I thought there's still some unsolved (IR/collinear?) problem left. Has this been resolved in recent years?

 

[A. Neumaier]

Well, does this imply that after all non-perturbative QED is shown to have no problems at whatever scale? I thought there's still some unsolved (IR/collinear?) problem left. Has this been resolved in recent years?

For standard QED, no problems are expected anywhere. On a nonrigorous level, everything is well understood for QED; even the infrared problem in QED is understood (Kulish-Faddeev) since QED (with photons and electrons only) has no bound states. QED with 2 species of charged particles is already much harder and poorly understood, as even the nonrigorous techniques for handling the probably infinitely many bound states are poorly developed.

What is unsettled for standard QED is the rigorous nonperturbative construction, i.e., the problem of how to give the perturbative asymptotic series definite values. In general, an asymptotic power series is the Taylor expansion of uncountably many different functions. What is missing is finding a recipe that selects the correct one. This is a very hard and unsolved problem in functional analysis.

 

[vanhees71]

I see. Thanks!

What I don't understand is your statement that QED with just electrons (and then of course necessarily also positrons) has no "bound states". What about positronium? Or is this simply because after all it's unstable due to pair annihilation?

 

[A. Neumaier]

What I don't understand is your statement that QED with just electrons (and then of course necessarily also positrons) has no "bound states". What about positronium? Or is this simply because after all it's unstable due to pair annihilation?

Yes, it decays already in pure QED. Hence mathematically, positronium is only a resonance, i.e., it corresponds to a pole of the analytically continued Green's function, described by the continuous part of the mass spectrum. In contrast, a bound state is, by definition, a discrete eigenvalue of the mass operator, corresponding to a pole of the physical Green's function.

This is different from the neutron, say, which is a bound state in QCD. It is also unstable, but not as a particle in QCD, only as a particle in the standard model, due to the weak interaction.

 

[ftr]

This is different from the neutron, say, which is a bound state in QCD. It is also unstable, but not as a particle in QCD, only as a particle in the standard model, due to the weak interaction.

I thought that QCD was part of the standard model, so what exactly do you mean by this statement?

 

[DarMM]

I thought that QCD was part of the standard model, so what exactly do you mean by this statement?

If QCD alone is considered it is stable, but including the rest of the standard model such as the weak interactions allows it to decay.

 

[ftr]

If QCD alone is considered it is stable, but including the rest of the standard model such as the weak interactions allows it to decay.

Thanks, so can we say that QCD is part of the standard model an approximate statement. How would you like to describe the standard model, say in a wiki.

 

[DarMM]

Thanks, so can we say that QCD is part of the standard model an approximate statement. How would you like to describe the standard model, say in a wiki.

The QCD langragian is a component of the Standard Model Lagrangian. However conclusions about particles in QCD alone are not strictly accurate for the full Standard Model. This is basic enough QFT however, I'd suggest reading an introductory textbook.