This is a distortion of facts. One looks at lattice QCD only because it is derived from the Poincare invariant QCD; without the latter there would be no motivation at all to consider the former. And lattice QED is hardly ever pursued. The successes of QED, both historically and today, come solely from the Poincare invariant version.

I agree with this. Lattice QED was specifically designed to have the right continuum limit. So it's not a good example if you're wanting to show that Lorentz invariance can arise natural as a continuum approximation to a non-invariant theory. To be convincing you would have to have an independent motivation for lattice QED that did not rely on having the right continuum limit.

Any standard textbook of quantum field theory discusses Poincare invariant QED. The renormalized results at any loop order are Poincare invariant. And the experimental tests are about these results at low order (up to six).

That there are unresolved problems about the limit of suitably resummed nonperturbative version, which you seem to allude to, is a completely different matter that has at preesent no experimental consequences.

On the other hand, lattice QED is extremely inaccurate, very few computations have been performed, and if all other QED were erased, QED would immediately lose its status as an excellent physical theory.

Well, it is the resummed nonperturbative version that makes it a quantum theory. In the absence of that, there is no QED as a quantum theory. It is the lattice QED that is a proper quantum theory.

The usual "Poincare invariant" QED has a landau pole, so it cannot exist as a quantum theory at all energies. If QED at high energies does not exist, the Poincare invariance is broken. So in effect, Poincare invariant QED is only a low energy effective field theory.

The existence question for nonperturbative QED is wide open. The Landau pole has not been proved to exist, it may well be an artifact of low order perturbation theory. Your statement is therefore only a belief.

It is the latter that is Poincare invariant and responsible for all successes of QED.

Lattice theories are also only low energy effective field theories; so if you think the latter is a defect of a theory then lattice QED is as defective and far less predictive. There is no reason at all to give it the status you wish it to have.

You may have this opinion but it is not shared by anybody else, as far as I can tell. There are many textbooks that have QED as one of the main examples of an excellent quantum theory, and none that says that there is no QED as a quantum theory.

They all agree QED is only a low energy effective theory. It is only in this region that we need Poincare invariance. So we can think of lattice QED and Poincare invariant QED as the same in the sense that both give the same low energy effective theories. In other words, there is no need for true Poincare invariance, only effective.

Sure, but this is not a defect. All our theories in physics (with possible exception of string theory) are only low energy effective theories.

No we cannot. They don't give the same low energy effective theory. Everything is different about them.

Lattice QED in the form it exists makes not the same predictions but far weaker ones. Please cite a demonstration that lattice QED derives the anomalous magnetic moment to high accuracy!

In addition, lattice QED has one additional parameter, the lattice spacing, and all results depend on it. One gets the experimental results only in the limit where the lattice spacing goes to infinity and the coupling constants are highly tuned functions of the lattice spacing, and the fine tuning must be chosen exactly such that the results approach the QED limit - which presupposes it! The fine-tuning has no other justification!

There is of course fine tuning (without taking the lattice spacing to zero) so that the right low energy limit occurs, but all of this fine tuning is needed in the standard view of QED, and has nothing to do with Bohmian Mechanics.

Once one realizes that QED is non-relativistic in the standard view, there is no special problem for Bohmian Mechanics.

Can Lorentz invariance be maintained if there is an energy cutoff?

https://arxiv.org/abs/hep-lat/0211036 mentions other regularizations like dimensional regularization, but they are not gauge invariant. Also, it is not clear tha the other regularization construct a quantum theory. On the other hand, the lattice regularization does construct a quantum theory, from which Lorentz invariant QED can in principle emerge as a low energy effective theory.

Certainly, asymptotic safety might still be possible. But till then, if we take the Wilsonian viewpoint, lattice QED gives us a secure conceptual starting point (although it is of course impractical for calculations).

A much better example is classical sound and corresponding quantum phonons. Sound satisfies a continuous Lorentz-invariant wave equation (with velocity of sound instead of velocity of light). Yet, it emerges from non-relativistic discrete theory of atoms. At the quantum level it illuminates particle creation/destruction in QFT, in the sense that creation and destruction of phonons really originates from processes in which no actual particles (atoms) are created or destructed.

and ends with a covariant theory. You take the start for the end. None of the references stops at the lattice. Their goal is always to get the covariant, physical theory, not the regularized one.

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).

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. And his book on a true ghost story does the same for other gauge theories.

atyy has a very nonstandard understanding of the meaning of the word ''standard''. He means by it ''his personal standards'', not the standard of the current state of the art in physics.