I Bohmian mechanics and special relativity

[kurt101]

[Moderator's note: Thread spun off from previous thread due to topic change.]

Good luck but as soon as you posit that there are nonlocal effects and real waves, you run into problems with relativity.
If it weren't for this, nearly everyone would have been onboard with you and BM would have been standard quantum curriculum(it isn't).

Thanks! I am working on understanding why non-local effects violate the special theory of relativity. I am not there yet, but maybe when I finally get there I will leave the BM train.

 

[PeterDonis]

I am working on understanding why non-local effects violate the special theory of relativity.

They don't as long as you're careful about what you take "non-local effects" to mean. Quantum field theory is consistent with special relativity and predicts violations of the Bell inequalities, which is what "non-local effects" means experimentally. So violations of the Bell inequalities are perfectly consistent with special relativity.

 

[Demystifier]

If the Bohmian interpretation is simply that at some level a particle of sorts is guided by a wave of sorts then the Bohmian interpretation is an interpretation I subscribe to.

Yes, that's what Bohmian interpretation is.

I will try reading your paper, but I suspect it won't provide any additional explanation over Quantum Mechanics as to what is actually happening.

Why do you think so?

 

[Demystifier]

I am working on understanding why non-local effects violate the special theory of relativity. I am not there yet, but maybe when I finally get there I will leave the BM train.

BM seems incompatible with fundamental relativity, but is compatible with emergent relativity.

 

[kurt101]

Why do you think so?

The discussions on this forum and my initial skim of your paper gave me that impression. However reading it again, has me very interested. I look forward to learning more about it.

BM seems incompatible with fundamental relativity, but is compatible with emergent relativity.

Is there a good link or paper that explains emergent relativity?

 

[EPR]

[Moderator's note: Thread spun off from previous thread due to topic change.]Thanks! I am working on understanding why non-local effects violate the special theory of relativity. I am not there yet, but maybe when I finally get there I will leave the BM train.

As someone suggested earlier, you need to study the basics and get a good grasp of the notions. At some point you'll notice that most of the 'mysteries' arise not out what and how the fundamental particles are, but from their 'motion'. You'll notice the different approaches to amend the paradoxes, the subtleties, the arguments and the controversies and why some of them remain as deep as ever. But you need a better grasp of what is happening. You can't just jump start from the middle and make up an interpretation.

 

[A. Neumaier]

Is there a good link or paper that explains emergent relativity?

For a discussion of potentials and problems of this approach (including references) see the threads Is QED nonrelativistic? and Lattice QED, and posts #73 and the subsequent discussion from the thread How to derive Born's rule for arbitrary observables from Bohmian mechanics?

 

[martinbn]

BM seems incompatible with fundamental relativity, but is compatible with emergent relativity.

What is emergent relativity!?

 

[Demystifier]

Is there a good link or paper that explains emergent relativity?

What is emergent relativity!?

Loosely speaking, emergent relativity is any theory in which relativistic laws are approximately valid at large distances but not at small distances. Some ideas in that direction can be found in my "Bohmian mechanics for instrumentalists" (linked in my signature below) and references therein.

 

[martinbn]

Loosely speaking, emergent relativity is any theory in which relativistic laws are approximately valid at large distances but not at small distances. Some ideas in that direction can be found in my "Bohmian mechanics for instrumentalists" (linked in my signature below) and references therein.

What about space-time? What is it in such an emergent relativity? It seems that you want space-time to be the one from classical/nonrelativistic physics but somehow to appear to be the one from relativity. Is that even possible?

 

[A. Neumaier]

What about space-time? What is it in such an emergent relativity? It seems that you want space-time to be the one from classical/nonrelativistic physics but somehow to appear to be the one from relativity. Is that even possible?

It might be possible in principle, but so far no one came up with something reproducing QED, let alone the standard model. In view of the nontrivial ''triviality'' issues of lattice approximations it is extremely unlikely that someone ever will. See the discussions referred to in post #7.

 

[Demystifier]

In view of the nontrivial ''triviality'' issues of lattice approximations it is extremely unlikely that someone ever will.

The triviality issue is a problem if one insists on having a well defined continuum limit. But if the fundamental (as yet unknown) theory is not a continuum field theory, then I don't see any reason to insist on having a well defined continuum limit. This is like insisting that the atomic theory should be well defined in the limit in which the Bohr radius is zero because the continuum fluid mechanics corresponds to the limit in which the size of atoms is zero.

 

[Demystifier]

What about space-time? What is it in such an emergent relativity? It seems that you want space-time to be the one from classical/nonrelativistic physics but somehow to appear to be the one from relativity. Is that even possible?

It's possible. The most suggestive evidence is the fact that from a nonrelativistic theory of atoms one can derive an approximative Lorentz invariant wave equation of sound.

 

[martinbn]

It's possible. The most suggestive evidence is the fact that from a nonrelativistic theory of atoms one can derive an approximative Lorentz invariant wave equation of sound.

That is again about properties of matter. I am asking about the properties of space-time. How do you get relativistic(effectively) space-time if fundametally it is classical!

 

[A. Neumaier]

The triviality issue is a problem if one insists on having a well defined continuum limit. But if the fundamental (as yet unknown) theory is not a continuum field theory, then I don't see any reason to insist on having a well defined continuum limit.

QED is defined only in the continuum limit. According to published numerical experiment, triviality of lattice QED sets in already at numerically realizable lattice spacing. Thus the lattice approximates the free theory rather than QED. Decreasing the lattice spacing will not improve this. Thus no lattice approximation will be able to reproduce the highly accurate predictions of the continuum QED.

This is like insisting that the atomic theory should be well defined in the limit in which the Bohr radius is zero because the continuum fluid mechanics corresponds to the limit in which the size of atoms is zero.

No. You cannot even show that a lattice approximation comes anywhere close to reproducing the experimentally verified results of QED.

 

[Demystifier]

According to published numerical experiment, triviality of lattice QED sets in already at numerically realizable lattice spacing. Thus the lattice approximates the free theory rather than QED. Decreasing the lattice spacing will not improve this. Thus no lattice approximation will be able to reproduce the highly accurate predictions of the continuum QED.

Do you have a reference for this claim?

 

[Demystifier]

That is again about properties of matter. I am asking about the properties of space-time. How do you get relativistic(effectively) space-time if fundametally it is classical!

The effective space-time is defined by its effects on matter. In experiments we never measure the space-time itself, we only measure how matter behaves in it. Hence, from an effective point of view, the effect on matter is all what we really need.

 

[A. Neumaier]

Do you have a reference for this claim?

Do you have a reference for the opposite claim?

I repeat some facts from the other threads which amount to almost a proof of my claim:

All formulas of the renormalized, textbook QED, existing since 1948 as witnessed by a Nobel prize given for its discovery, are fully Poincare invariant at every loop order, and make excellent predictions. 1-loop Lorentz invariant QED is fully (and higher loop QED conceptually) constructed without any cutoff or regularization or lattices in Scharf's book on quantum electrodynamics. There is no divergence in any of the Lorentz invariant 1-loop results provided by Scharf, and these give already much better agreement with experiment than lattice QED. 1-loop QED is the version of QED (though with a different derivation) for which Feynman, Tomonaga and Schwinger got the Nobel prize! That each term is finite is enough to claim local Poincare invariance which is the actual mathematical claim of Poincare invariance in perturbative QED, the one that gives outstanding predictions. One gets the 10-decimal agreement of the anomalous magnetic moment only starting with the covariant version (and then making approximations, but not lattice approximations).

Compare this with the poor accuracy obtained by noncovariant lattice theories. They give finite results at each lattice spacing, but none of the lattice spacings for which computations can be carried out gives results matching experiment, and it is not known whether (or in which sense) the limit exists in which the lattice spacing goes to zero. But numerical lattice studies indicate that the limit is a free theory (i.e., the renormalized charge vanishing in the continuum limit) and that this shows already at quite coarse spacing, with lattice sizes ranging from $8^4$ to $24^4$. See, e.g., (all papers with links to the arXiv)

• M. Göckeler et al., Is there a Landau Pole Problem in QED? Phys.Rev.Lett. 80 (1998), 4119-4122.
• Kogut & Strouthos, The Logarithmic Triviality of Compact QED Coupled to a Four Fermi Interaction, Physical Review D 71 (2005), 094012.

At this coarse spacing, the Poincare invariance observed in experiments is badly broken. Hence no high accuracy prediction as in covariant QED is possible. No experimentally verified predictions have ever been made with lattice QED (unlike with lattice QCD, which produces useful predictions and is believed to have a nontrivial continuum limit). In the substantial literature discussing the triviality problem there is not the slightest reason why this should improve at shorter spacing; one expects that one gets even closer to the trivial limit when the lattice spacing is decreased further. See also

• S. Kim et al., On the triviality of textbook quantum electrodynamics, Physics Letters B 502 (2001) 345-349.
• F. Akram et al., Vacuum Polarization and Dynamical Chiral Symmetry Breaking: Phase Diagram of QED with Four-Fermion Contact Interaction, Physical Review D *7 (2013), 01301.
• Section 8, ''IS DESTRUCTIVE FIELD THEORY POSSIBLE?'' of a paper by Gallavotti and Rivasseau from 1984.

For the differences between QED and QCD regarding lattice approximations see this post.

For further information see the discussion and references in my posting at PhysicsOverflow.

 

[Demystifier]

Do you have a reference for the opposite claim?

I don't, but I don't think it's much relevant. I also don't have a reference explaining the flight of airplanes directly from atomic theory (without the use of continuous fluid mechanics) but it doesn't shake my confidence that atomic theory can explain it.

Indeed, it's a common belief in the large part of the community that all known field theories (QED, QCD) are just effective theories valid at large distances, while at smaller distances (such as the Planck scale) they can be completely wrong. The Wilson renormalization theory provides a general justification for such a belief.

The fact that we don't have direct computational evidence that lattice QED gives correct results is widely believed to be a mere artefact of the fact that we still don't have sufficiently strong computers. Current computers can simulate 4-dimensional lattices with $20^4$ lattice points or so, which is a very crude approximation and nobody really expects to get very good agreement with observations within such a crude approximation.

Sure, a common belief is not a proof. But the belief that lattice QED would give results in excellent agreement with observations if only we had sufficiently strong computers that can handle $1.000.000^4$ lattice points or so - is a very justified belief.

 

[martinbn]

The effective space-time is defined by its effects on matter. In experiments we never measure the space-time itself, we only measure how matter behaves in it. Hence, from an effective point of view, the effect on matter is all what we really need.

The question remains. How?

 

[Demystifier]

The question remains. How?

How what? Did you read any of the references I mentioned?

 

[PeterDonis]

Moderator's note: An off topic question about the thermal interpretation and EPR experiments has been moved to a new thread:

https://www.physicsforums.com/threads/thermal-interpretation-and-epr-experiment.981290/

 

[martinbn]

How what? Did you read any of the references I mentioned?

Yes, but there is nothing there. It just says that may be (somehow) it could be that way. But my question is how do you get relativistic space-time out of classical one?

 

[A. Neumaier]

Indeed, it's a common belief in the large part of the community that all known field theories (QED, QCD) are just effective theories valid at large distances, while at smaller distances (such as the Planck scale) they can be completely wrong. The Wilson renormalization theory provides a general justification for such a belief.

Originally, Wilson's renormalization theory was about a lattice theory being approximated at a critical point by a nonrelativistic continuum field theory. But he assumes that the limit actually exists! Thus Wilson's theory provides justification for the belief that lattice QED approximates the free limiting theory with zero charge. Indeed, this is what the computer simulations demonstrate.

In its application to relativistic quantum field theory, Wilson's renormalization theory is always described by the Callan-Symanzik equation, which is about a Poincare invariant effective theory approximating another Poincare invariant theory at lower energies. As far as I know, it was never about lattices approximating a known relativistic continuum theory.

Thus Wilson's theory provides no rational justification for your belief. While you are free to believe anything you like with any justification you like, there are no grounds at all why anyone informed should share your belief.

Current computers can simulate 4-dimensional lattices with $20^4$ lattice points or so, which is a very crude approximation and nobody really expects to get very good agreement with observations within such a crude approximation.

I fully agree. But one sees it already converge to the trivial theory, hence there is no reason to expect that nonfree QED results would be obtained by refining the lattice. Thus one will never have very good agreement with observations.

that lattice QED would give results in excellent agreement with observations if only we had sufficiently strong computers that can handle $1.000.000^4$ lattice points or so - is a very justified belief.

No, because the only justification of that belief would be that taking the number of lattice points to infinity would produce exact QED. But all evidence points to that taking the number of lattice points to infinity would produce QED with zero renormalized charge. The numerical evidence for this is already visible at the lattice sizes that can be simulated today, and the observed closeness to triviality grows with lattice size. Extrapolating to $1.000.000^4$ lattice points or so we would see an almost perfect free theory!

 

[Demystifier]

Yes, but there is nothing there. It just says that may be (somehow) it could be that way. But my question is how do you get relativistic space-time out of classical one?

I think I already answered.

 

[Demystifier]

But one sees it already converge to the trivial theory, hence there is no reason to expect that nonfree QED results would be obtained by refining the lattice. Thus one will never have very good agreement with observations.

My point is that agreement of lattice QED with observations is possible (and in fact expected) despite the fact that lattice QED is trivial in the continuum limit. Obviously you disagree. We already exchanged arguments on that issue several times and never came close to an agreement, so maybe it's counterproductive to further argue about that issue.

 

[martinbn]

I think I already answered.

What you said is this.

It's possible. The most suggestive evidence is the fact that from a nonrelativistic theory of atoms one can derive an approximative Lorentz invariant wave equation of sound.

Surely you can understand why I am confused. The phenomenon of sound and the wave equation are entirely classical/nonrelativistic. So I don't know what any if this has to do with relativity as emergent/effective theory!

 

[Demystifier]

The phenomenon of sound and the wave equation are entirely classical/nonrelativistic. So I don't know what any if this has to do with relativity as emergent/effective theory!

Please read Sec. 5.3 of my paper, especially the paragraph around Eq. (29).

 

[martinbn]

Please read Sec. 5.3 of my paper, especially the paragraph around Eq. (29).

I did, it says exactly what you wrote in post #13.

I suppose there is no point in me asking repeatedly. Either I just don't get it or you have a very strange understanding of what relativity is. In any case if I continue I will only be flooding the thread, so I will stop.

 

[Demystifier]

Either I just don't get it or you have a very strange understanding of what relativity is.

For me, special relativity is nothing but measurable consequences of Lorentz symmetry. Do you find it strange?

 

[Demystifier]

No, because the only justification of that belief would be that taking the number of lattice points to infinity would produce exact QED. But all evidence points to that taking the number of lattice points to infinity would produce QED with zero renormalized charge. The numerical evidence for this is already visible at the lattice sizes that can be simulated today, and the observed closeness to triviality grows with lattice size. Extrapolating to $1.000.000^4$ lattice points or so we would see an almost perfect free theory!

I think you misunderstood triviality. "Almost perfect free theory" means that all effects of interactions are small if the bare coupling constant is $g\sim 1$. But the bare coupling constant does not need to be $g\sim 1$. Instead, it can be $g\gg 1$. By taking $g$ large enough, one can always get a theory which is not almost free. Moreover, one can fit the exact value of $g$ such that the lattice theory predictions agree with the long-distance (low-energy) experiments. (By long distance, I mean distance much larger than the lattice spacing $a$.) It turns out that $g$ must be taken larger when $a$ is smaller, but it is not a problem as long as $a>0$, because then the required bare coupling constant is $g<\infty$.

The problem only appears when $a\rightarrow 0$, because then the required bare coupling constant diverges $g\rightarrow\infty$. In other words, the effects of interaction vanish for any finite value of $g$, which is called triviality.

But from the effective theory point of view, there is no any physical reason to consider $a\rightarrow 0$. One does not expect QED to be a right theory at the Planck distance, but only at distances much larger than that. Hence there is no need to take $a$ smaller than the Planck distance. Hence the triviality is not a problem.

 

[A. Neumaier]

I think you misunderstood triviality. "Almost perfect free theory" means that all effects of interactions are small if the bare coupling constant is $g\sim 1$. But the bare coupling constant does not need to be $g\sim 1$. Instead, it can be $g\gg 1$. By taking $g$ large enough, one can always get a theory which is not almost free.

Please provide a reference for your claim that by taking $g$ large enough, one can always get a theory which is not almost free!

I think you misunderstood triviality. Lattice QED has three free parameters, two coupling constants (the bare electron mass $m_0$ and the bare electron charge $e_0$) and the lattice spacing $a$. To get a physical approximation to QED, one would have to adjust for given lattice spacing $a$ both bare parameters such that the predicted electron mass $m(m_0,e_0,a)$ and the predicted electron charge $e(m_0,e_0,a)$ reproduce the physical electron mass $m_{phys}$ and the physical electron charge $e_{phys}$. These renormalization conditions require that the equations $m(m_0,e_0,a)=m_{phys}$ and $e(m_0,e_0,a)=e_{phys}$ have a solution. Triviality is the statement that
$$e_{max}(a):=\max_{m_0,e_0} e(m_0,e_0,a)\to 0 \mbox{ for }a\to 0,$$
so that for sufficiently small $a>0$ it is impossible to renormalize to nontrivial values of the charge. This is the meaning of the statement

For the renormalized charge and mass we find results which are consistent with the renormalized charge vanishing in the continuum limit.

in the abstract of the Göckeler paper quoted earlier. And the consistency is visible already at the lattice spacings used in their simulation!

Moreover, one can fit the exact value of $g$ such that the lattice theory predictions agree with the long-distance (low-energy) experiments. (By long distance, I mean distance much larger than the lattice spacing $a$.)

Here you assume without proof what has to be shown! Where is the paper or book that shows that at macroscopic distances (much larger than the lattice spacing realized by current computer simulations), lattice QED predictions agree with the long-distance (low-energy) Maxwell equations?

It is unscientific behavior to claim as facts what you don't have proofs for.

It turns out that $g$ must be taken larger when $a$ is smaller

Where is this shown? I think you mean that you hope that this turns out, but the papers I quoted show that this hope is most likely an illusion only.

but it is not a problem as long as $a>0$, because then the required bare coupling constant is $g<\infty$.

Please provide a reference proving this remarkable claim!

But from the effective theory point of view, there is no any physical reason to consider $a\rightarrow 0$. One does not expect QED to be a right theory at the Planck distance, but only at distances much larger than that. Hence there is no need to take $a$ smaller than the Planck distance. Hence the triviality is not a problem.

The problems with your scenario don't begin at a Planck scale $a$ but are already visible at the values of $a$ currently realizable in simulations!

 

[Demystifier]

Please provide a reference for your claim that by taking $g$ large enough, one can always get a theory which is not almost free!

By $g$ I meant the set of all coupling constants, including the mass. (Such a language is common within the general abstract theory of renormalization of field theories.) Is it OK now?

These renormalization conditions require that the equations $m(m_0,e_0,a)=m_{phys}$ and $e(m_0,e_0,a)=e_{phys}$ have a solution.

Do you agree that those equations have a solution for any small but non-zero $a>0$?

 

[A. Neumaier]

By $g$ I meant the set of all coupling constants, including the mass. (Such a language is common within the general abstract theory of renormalization of field theories.)

? The common language has a number of counterterms called Z with various indices.

Is it OK now?

No, not even with $g=(m_0,e_0)$. See my previous post; please read the details!

Do you agree that those equations have a solution for any small but non-zero $a>0$?

No; I stated in my previous post why not. If you have grounds to think that they have, please provide a reference containing a proof.

 

[Demystifier]

If you have grounds to think that they have, please provide a reference containing a proof.

I don't know a reference with a rigorous proof, but I know references which make claims similar to mine. For instance http://www.scholarpedia.org/article/Triviality_of_four_dimensional_phi^4_theory_on_the_lattice
says the following:
"Quantum field theories are only formally defined by their Lagrange density. To extract physical predictions, the theory is regularized (modified) at short distance, an ultraviolet cutoff is imposed. ... A quantum field theory is called trivial, if the above limit invariably leads to a non-interacting renormalized theory ... It is possible that such a theory can still describe Nature if ... the cutoff is not fully removed, but its effects are invisible on a certain limited range of scales to a certain precision. ...Trivial theories can still be useful as effective theories, if there are parameter values with enough interaction to match experiments and at the same time small enough that effects of the unremoved lattice (or other UV cutoff) are so small as to not contradict experiment...In D=4 one can argue that the coupling dies out only logarithmically, gR∝c/|ln(amR)| and the effective scenario is quite viable. Presumably the status of being an effective theory in the above sense is also true for the Standard Model of particle physics, which contains the scalar Higgs sector. ...Historically, all our theories seem to be eventually superseded by a next more comprising theory revealing itself beyond a certain domain of validity. Triviality may now even be seen as a bonus rather than a defect."

The author is an expert in the field:
http://inspirehep.net/search?ln=en&p=find+author+wolff,+u&of=hcs&action_search=Search&sf=earliestdate&so=d

In particular, note the statement above: "Presumably the status of being an effective theory in the above sense is also true for the Standard Model of particle physics". If we ignore the word "presumably", it says that I was essentially right. The word "presumably" indicates that I could have been wrong (mea culpa!), but that there is no proof that I was wrong and that there is a good reason to think that I was right.

Perhaps we need a new thread about whether trivial theories can serve as effective theories, because it seems that there are arguments for both "yes" and "no".

 

[A. Neumaier]

I don't know a reference with a rigorous proof

I'd be satisfied with a proof at the level of rigor of Peskin and Schroeder, say.

but I know references which make claims similar to mine.

Only superficially similar! Wolff first considers a much more general context of QFTs that are somehow regularized by a cutoff. This includes standard continuum QED with the usual textbook cutoff (i.e., not the causal approach). The specialization to the lattice comes later (in scenario B of $\phi^4$ theory). There he says

in D⩾4 we encounter the phenomenon of triviality: we find that $g_R\to 0$ whenever we tune for the continuum limit $am_R→0$. Thus only free non-interacting theories are reached in the strict continuum limit.

which means (unlike your earlier claim) that the tuning of the constants is already taken into account.

For instance http://www.scholarpedia.org/article/Triviality_of_four_dimensional_phi^4_theory_on_the_lattice
says the following:
"Trivial theories can still be useful as effective theories, if there are parameter values with enough interaction to match experiments and at the same time small enough that effects of the unremoved lattice (or other UV cutoff) are so small as to not contradict experiment.

This carefully formulated statement is true since it is only a conditional statement. The problem is that the condition can apparently not be satisfied for lattice QED, while you assume without proof that it can. You mix things up with standard renormalized continuum QED, which is trivial when renormalized with an explicit cutoff, but nevertheless matches experiment quite well. But one may not apply it to lattice QED without first having shown that it also matches experiments quite well!

The paper cited by Wolff for details for $\phi^4$ is

• U. Wolff, Precision check on the triviality of the $\phi^4$ theory by a new simulation method. Physical Review D, 79(2009), 105002.

Triviality sets in already for quite large lattice constants. In the paper just cited one can see from Fig. 1 (which displays $g_{max}(a)$) that one cannot reach $g=20$ at the lattice sizes where the simulation was done. At least for this value of $g$, this clearly disproves your claim that the renormalization equations can be solved for any lattice spacing a>0. From the curve drawn, one actually sees that for any $g>0$ there is a limiting value of the lattice spacing below one cannot solve the equation!

To get a positive result for QED one would need to do similar simulations and check in which range of lattice spacings one can realize the physical values of electron mass and charge, and which accuracy limit this entails.

In particular, note the statement above: "Presumably the status of being an effective theory in the above sense is also true for the Standard Model of particle physics". If we ignore the word "presumably", it says that I was essentially right. The word "presumably" indicates that I could have been wrong (mea culpa!), but that there is no proof that I was wrong and that there is a good reason to think that I was right.

''presumably'' means ''it is conjectured'' or "I believe", not "it is justified by solid arguments".

The word "presumably" indicates that I could have been wrong (mea culpa!), but that there is no proof that I was wrong and that there is a good reason to think that I was right.

You could have looked up the references instead of believing such crucial things upon a word of authority phrased in the subjunctive! And you shouldn't have generalized to statements that are simply not there - making claims for arbitrarily small lattice spacings...

Perhaps we need a new thread about whether trivial theories can serve as effective theories, because it seems that there are arguments for both "yes" and "no".

Since the viability of relativistic Bohmian lattice model depend on the answer of this question, this fits the present thread; so we don't need a new one.

What are the arguments for "yes"? So far you haven't given any, only assertions that it is so.

I only know arguments for "no", except in the asymptotically free case. The latter applies to QCD, where lattice models are indeed heavily used, but neither to QED nor to the standard model, where they are not used at all, except to demonstrate triviality.

 

[PhilDSP]

The effective space-time is defined by its effects on matter. In experiments we never measure the space-time itself, we only measure how matter behaves in it. Hence, from an effective point of view, the effect on matter is all what we really need.

The question remains. How?

Assuming you have valid and accurate classical field equations, couldn't you essentially apply the Lorentz Transformation with the gamma factor separated out?

It seems to be a mystery to many how both the derivation and application of the de Broglie relations can be done without invoking a Lorentz Transformation or space-time. In his monograph "Beyond the Electron" J. J. Thomson sought to understand and resolve that mystery. The first appendix contains a derivation of classical field equations for a moving electron that appear to be valid at all velocities of the electron. One can easily see how the gamma factor of the Lorentz Transformation is "generated" by the relationship of the particle to the fields. It seems a simple step to factor the gamma factor from the field equations and substitute it with a factored Lorentz Transformation so that space-time emerges.

 

[Elias1960]

And the consistency is visible already at the lattice spacings used in their simulation!

This sounds like you have not understood their use of lattice theory.

They have made computations on $12^4$ and $16^4$ lattices (look simply at pic. 1). They can use them for arbitrary small lattice spacings. Because what they have to compute are not particular solutions, so that they don't have to care about the real size of the region where something nontrivial happens in a particular scattering process, but they have to compute how the renormalization works. Renormalization works from the start with different lattices, and they do this too. But to remain on a $16^4$ lattice once they halve the lattice size, they also halve the cube where they make the computations. So, very high momentum will be simply not present if a is yet big. But once the renormalization has reached a state where they become important, it is the low momentum parts will be no longer represented on that fine lattice, on a $16^4$ lattice with very small lattice size they have no place to live. The region of space they would be able to compute with that $16^4$ lattice shrinks together with the lattice size. This is unproblematic for what they do. But that means they have no boundary for the lattice spacings used. They can with this method, if they like, go to $10^{-100}l_{Pl}$ if there would be a point doing such things. In this particular case, they have done what is natural - to go up to the point where the Landau pole looks problematic.

In a real lattice computation for some physical process you have no such freedom. You cannot reduce the lattice computation to a $16^4$ lattice with, say, lattice size $l_{Pl}$ and periodic boundary conditions. At least, this computation would not give you anything interesting for any real scattering.

Here you assume without proof what has to be shown! Where is the paper or book that shows that at macroscopic distances (much larger than the lattice spacing realized by current computer simulations), lattice QED predictions agree with the long-distance (low-energy) Maxwell equations?
It is unscientific behavior to claim as facts what you don't have proofs for.

I would not expect that people would write such papers. Last but not least, something non-trivial should be shown, and such things look like trivialities for physicists.

What you name "unscientific behavior" is simply the difference in standards of mathematical rigorousness between mathematicians and physicists.

In fact, about the cases where some lattice discretization of some equation gives, in the continuum limit, something different than the original equation, physics know too. This is the case of fermion doubling. Nothing similar is known for QED (beyond the fermion doubling itself, of course).

Where is this shown? I think you mean that you hope that this turns out, but the papers I quoted show that this hope is most likely an illusion only.

No, in this case your plausibility argument that this paper would show something is simply based on misunderstanding the method of using the lattice in that paper.