Does QED Originate from Non-Relativistic Systems?

[RockyMarciano]

No. The latter is about proving rigorously the existence of quantum Yang-Mills theory and is still an open problem. This problem has nothing to do with QED, which is the topic of the present discussion.

True but irrelevant for QED, which (in the version under discussion) by definition is only about electromagnetic fields, electrons, and positrons. We are not discussing the standard model.

You are permanently shifting your meaning of QED, but the posts are there to be read. We are discussing the theory that gives extremely accurate predictions, in your own words "in each finite loop approximation". This is not asymptotic and divergent QED, which you seem to refer to by QED at times to deliberately confuse the discussion, but the renormalized one that is indeed part of the predictions of the standard model (wich is indeed Yang-Mills), in other words, in the absence of a non-perturbative QED, perturbative QED rests on the renormalizable gauge quantum field theory $U(1)×SU(2)×SU(3)$, the mathematical base of the standard model . This is the topic of the present discussion since it is about accurate predictions in renormalized perturbative QED versus nonrelativistic lattice QED. The actual physics cannot be separated in independent compartments with QED separated from QCD or weak interaction theories, at enough precision different interactions concur in a given high energy particle physics process.

 

[vanhees71]

But that is not really what is being discussed. The question being discussed is:

In the Wilsonian viewpoint, is lattice QED at fine but finite lattice spacing a conceptually adequate starting point for obtaining perturbative Poincare invariant QED as a low energy effective QFT?

As I said in my previous posting in this thread, lQFT is a regularized version of continuum QFT, not more and not less.

 

[RockyMarciano]

This is important since indeed the standard model is invariant only under the continuous part of the Poincare group, which is a semidirect product of the proper orthochronous Lorentz group with the group of translations.

Exactly, mathematically is called the Poincare algebra, althought it is common among physicists to ignore the difference between Lie groups and Lie algebras.

 

[RockyMarciano]

What is a spacetime anomaly?

For instance the chiral anomaly leading to baryonic charge non-conservation and to violations of lepton number conservation.

 

[A. Neumaier]

For physics at low energies, it is not relevant to know what happens in the limit of zero lattice spacing.

But to get the predictions of the anomalous magnetic moment (a low energy quantity) agree to experimental accuracy one needs already the covariant version of QED. And to get the covariant version one needs a continuum limit. But the continuum limit of lattice QED is probably not covariant QED but as free theory.

Lattice QED as it exists is an extremely poor approximation to real QED. It has not given a single accurate contribution to the prediction of low energy physics.

 

[A. Neumaier]

You are permanently shifting your meaning of QED, but the posts are there to be read. We are discussing the theory that gives extremely accurate predictions, in your own words "in each finite loop approximation". This is not asymptotic and divergent QED, which you seem to refer to by QED at times to deliberately confuse the discussion, but the renormalized one that is indeed part of the predictions of the standard model (wich is indeed Yang-Mills).

This is the last time I discuss your confusion related to this thread.

The title and the OP define what is discussed in this thread. We mainly discuss textbook QED, which is a Poincare invariant theory at few loops, lattice QED, which is a nonrelativistic caricature of true QED (unlike lattice QCD which because of asymptotic freedom has a respectable status as an approximation to true QCD), and rigorous QED, of which it is unknown whether it exists. We discuss other theories such as the standard model or QCD not as topic in itself, but only only to obtain contrasting statements. (The standard model makes slightly different predictions than QED since it incorporates corrections from the interaction with the other fields.)

 

[vanhees71]

Exactly, mathematically is called the Poincare algebra, althought it is common among physicists to ignore the difference between Lie groups and Lie algebras.

Of course, usually physicists treat the Lie algebra, but the exponential map lifts it to unitary representations of the group. Since for QT it's sufficient to have unitary ray representations, instead of the classical propoer orthochronous Lorentz group you consider the covering group, which means to substitute $\mathrm{SL}(2,\mathbb{C})$ instead of the $\mathrm{SO}(1,3)^{\uparrow}$. There are no non-trivial central charges in the Poincare algebra. So at the end everything is deduced from unitary irreps. of the covering group.

 

[atyy]

As I said in my previous posting in this thread, lQFT is a regularized version of continuum QFT, not more and not less.

In other words, you agree that lQFT is a good starting point for continuum QFT (without taking the lattice spacing to zero)?

 

[atyy]

Because the hypothesis of your question is invalid. Your argument from belief means nothing; it is a mere belief.

The question of the existence of QED is widely open, even the question of existence of $\Phi^4$ theory (see Section 8 of this paper, which discusses the state of the art from a rigorous point of view), for which the triviality arguments (i.e., the impossibility to construct it as a scaling limit from the corresponding lattice theory) are overwhelming.

The causal construction of QED avoids all these triviality arguments as it doesn't construct QED as a limit of theories with a cutoff but dirsectly from a mathematical characterization by means of covariant axioms by Bogoliubov - not the Wightman axioms. Each loop order is valid at all energies. The only unsolved question is how to resum the series to make the construction nonperturbative. There is not the slightest argument indicating that this is impossible. And as mentioned, the constructive question is wide open.

Of course not - I agree with you that the construction problem is wide open - and that is the point! Since the construction problem is wide open, we cannot at the moment use a truly Poincare invariant QED as a starting point for Wilsonian renormalization.

Since the construction problem is open, the claim that Poincare invariant QED exists is not substantiated. Only when the construction problem is solved can one claim that Poincare invariant QED exists.

 

[vanhees71]

In other words, you agree that lQFT is a good starting point for continuum QFT (without taking the lattice spacing to zero)?

Hm, I don't know, whether it helps much in the case of QED. In QCD it's of course a great way to explore QCD (both in vacuo as well as finite temperature) from a first-principles numerical approach. For QED, there seems not to be gained much compared to the standard perturbative methods. Particularly when you use the modern techniques as dim. reg. to regularize the theory and then renormalize it. The Wilsonian point of view of the renormalization group is indeed equivalent to the older techniques developed by Stueckelberg&Petermann, Gell-Mann and Low, et al. and they are also better suited for practical calculations using the RG method as a way to resum leading log contributions.

The modern functional renormalization-group approaches (aka Wetterich equation) are closer in spirit to the Wilsonian and right now a hot topic in thermal-QFT applications in heavy-ion theory to explore the phase diagram of strongly interacting matter.

 

[atyy]

Hm, I don't know, whether it helps much in the case of QED. In QCD it's of course a great way to explore QCD (both in vacuo as well as finite temperature) from a first-principles numerical approach. For QED, there seems not to be gained much compared to the standard perturbative methods. Particularly when you use the modern techniques as dim. reg. to regularize the theory and then renormalize it. The Wilsonian point of view of the renormalization group is indeed equivalent to the older techniques developed by Stueckelberg&Petermann, Gell-Mann and Low, et al. and they are also better suited for practical calculations using the RG method as a way to resum leading log contributions.

The modern functional renormalization-group approaches (aka Wetterich equation) are closer in spirit to the Wilsonian and right now a hot topic in thermal-QFT applications in heavy-ion theory to explore the phase diagram of strongly interacting matter.

Yes, but the idea is not to use lattice for practical calculation. The idea is to be able to define a finite quantum theory that at least conceptually leads to the usual covariant perturbative QED as a low energy effective theory. Since at present we don't know how to make a Poincare invariant QEF that exists at all energies, if we want a well-defined quantum theory from which to start, we have to go with a quantum theory with a high energy cutoff such as lattce QED.

 

[Demystifier]

But to get the predictions of the anomalous magnetic moment (a low energy quantity) agree to experimental accuracy one needs already the covariant version of QED.

I have no idea why do you think that non-covariant (lattice) version cannot give the anomalous magnetic moment which also agrees to experimental accuracy. Reference?

 

[A. Neumaier]

I don't know, whether it helps much in the case of QED. In QCD it's of course a great way to explore QCD (both in vacuo as well as finite temperature) from a first-principles numerical approach.

It does not, because unlike QCD which is asymptotically free and hence presumably has a good continuum limit, the continuum limit of lattice QED is unlikely to be nontrivial because of the Landau pole.

For QED, there seems not to be gained much compared to the standard perturbative methods.

There is nothing to even seemingly gain but a lot is actually lost. There a no good predictions at all of lattice QED.

The modern functional renormalization-group approaches (aka Wetterich equation) are closer in spirit to the Wilsonian

But the Wetterich equation and other exact RG equations are based on the continuum version and not the lattice.

 

[A. Neumaier]

a finite quantum theory that at least conceptually leads to the usual covariant perturbative QED as a low energy effective theory.

This has nowhere be done; it is wishful thinking.

 

[vanhees71]

Yes, but the idea is not to use lattice for practical calculation. The idea is to be able to define a finite quantum theory that at least conceptually leads to the usual covariant perturbative QED as a low energy effective theory. Since at present we don't know how to make a Poincare invariant QEF that exists at all energies, if we want a well-defined quantum theory from which to start, we have to go with a quantum theory with a high energy cutoff such as lattce QED.

As I said, I consider it also not very advantageous to use the lattice regularization to define QFT. There are better ways to regularize perturbative QFT like Pauli Villars or dim. reg. or a stupid old cutoff (in Euclidean QFT maybe the most simple idea). Conceptually regularization is even a bit unintuitive for my taste, and my great "aha feeling" came when I learned the BPHZ formalism, which uses always physical masses, couplings and fields order by order in PT (either loop-wise, i.e., in powers of $\hbar$ or in some coupling constant(s) or any other counting scheme appropriate to the definite model under investigation) and the counter-term approach. Then you simply subtract the divergences, introducing the renormalization scale etc. Then you get the RG equation from the independence of S-matrix elements on the choice of the renormalization scale (and even the renormalization scheme). This more conventional approaches are all well defined ways to work in renormalized perturbation theory with only finite quantities at any step of the calculation. For me it's way more intuitive than an artificial space-time lattice with quite nasty properties (fermion doublers and other kinds of artifacts, the lost Poincare, Lorentz, rotation invariance etc. ect.) and very delicate convergence issues if you want to get to the continuum limit. I'm also not familiar with the calculational techniques to evaluate even a simple one-loop diagram as vacuum polarization and electron self-energy on the lattice. With the other techniques it's not such a big deal. For a nice introduction also to calculational techniques see P. Ramon, QFT - A Modern Primer.

In my opinion, the best way to understand the quite involved ideas behind renormalized perturbation theory is to sit down and do calculations like the one-loop structure of QED to the end. For a beginner, I'd recommend to use the modern approach, using dim. reg. as regulator and then discussing various renormalization schemes like minimal subtraction and/or modified minimal subtraction, the usual on-shell scheme.

 

[A. Neumaier]

I have no idea why do you think that non-covariant (lattice) version cannot give the anomalous magnetic moment which also agrees to experimental accuracy. Reference?

A reference is needed for the claim that it can!

For all alternatives to established successful theories, the alternatives must be proved to be at least as good in order to be taken seriously. This is not the task of the mainstream physicist but that of the promotors of the alternative as a viable way to go! As long as no such proof is given, the alternative is left to rest in peace and ignored.

 

[vanhees71]

I have no idea why do you think that non-covariant (lattice) version cannot give the anomalous magnetic moment which also agrees to experimental accuracy. Reference?

To the contrary, I'd like to ask you for a reference, where even only the famous one-loop result by Schwinger has been achieved using the lattice regularization. I've no clue, how one would do such a calculation, let alone to get analytical results as with the standard continuum approaches like Pauli-Villars or (my favorite) dim. reg. as an intermediate step leading finally to a way to renormalize and get the physical result first derived by Schwinger (I think using a kind of Pauli-Villars reg., but I'd have to look that up, and Schwinger's papers are not easy to read ;-)).

 

[Demystifier]

A reference is needed for the claim that it can!

So from the fact that nobody did it so far, you conclude that it cannot be done?

 

[vanhees71]

But the Wetterich equation and other exact RG equations are based on the continuum version and not the lattice.

Of course. I'd never recommend to use the lattice approach except for where it is used extensively, namely in QCD as a way to numerically address the non-perturbative theory, where of course also the physics only comes out in the continuum limit, which to get is an art of its own. I think, in QED nobody ever has gotten anything via the lattice approach, not even the one-loop corrections, which are available even analytically with continuum-PT methods.

 

[Demystifier]

I'd like to ask you for a reference,

I'm not an expert in lattice QFT, so I cannot give a reference. I know that perturbative analytic calculations at lattice are more difficult than those in the continuum, but this in no way indicates that lattice perturbative results don't agree with observations.

 

[vanhees71]

So from the fact that nobody did it so far, you conclude that it cannot be done?

I strongly doubt it too. I'm not aware of a single paper about lattice QED. Only lattice QCD is of course a standard method for several decades by now, and there are still serious problems to be solved (e.g., getting S-matrix elements, solving the finite-temperature case at finite baryo-chemical potential satisfactorily etc. etc., but indeed it's a very active field of research, while about lattice QED, I've not even seen a single paper).

 

[vanhees71]

I'm not an expert in lattice QFT, so I cannot give a reference. I know that perturbative analytic calculations at lattice are more difficult than those in the continuum, but this in no way indicates that lattice perturbative results don't agree with observations.

Well, as long as there don't exist some lattice perturbative results, we can't check, right?

 

[A. Neumaier]

Of course not - I agree with you that the construction problem is wide open - and that is the point! [...] Since the construction problem is open, the claim that Poincare invariant QED exists is not substantiated.

Only the construction problem for a QED satisfying the Wightman axioms or equivalents is wide open. The construction problem for a Poincare invariant QED at 1-loop that agrees with experiment was fully solved in 1948 and this solution was honored by a Nobel price for Feynman, Tomonaga and Schwinger. In fact, few physicists care about constructive QFT at all; it is a problem only for the small minority of mathematical physicists.

Only when the construction problem is solved can one claim that Poincare invariant QED exists.

This is the view of mathematical physicists. But for them, lattice QED is definitely not QED, and a mere effective theory is no theory at all. Thus to claim their view as the authoritative view completely defeats your goal to promote lattice QED.

Since the construction problem is wide open, we cannot at the moment use a truly Poincare invariant QED as a starting point for Wilsonian renormalization.

Wilsonian renormalization is virtually useless in QED. It is not responsible for any its successes. So not having this starting point is irrelevant in practice.

 

[Demystifier]

I strongly doubt it too.

Doubt what? It's technically more difficult, but I see no reason to expect that, if someone could solve the technical problems, one would not give results that agree with observations.

 

[vanhees71]

Well, I don't believe a calculation that doesn't exist. Why should I bother myself with a tedious calculation leading to results which I can get much easier with the well established calculational methods developed over the past 40 years to evaluate relativistic QFTs perturbatively right away in the continuum?

 

[Demystifier]

Well, as long as there don't exist some lattice perturbative results, we can't check, right?

Don't take me for granted, but I think there are 1-loop lattice results which, for low energy phenomena, agree with standard 1-loop results. In any case, the book by Montvay and Munster is a very good reference. If you take some time to read it, you will see that the difference between perturbative lattice QFT and ordinary perturbative QFT can, to a large extent, reduce to the difference between Fourier sum and Fourier integral.

 

[A. Neumaier]

So from the fact that nobody did it so far, you conclude that it cannot be done?

No, but for two other reasons:

1. There is the general feeling in the community that the continuum limit of lattice QED is likely to be trivial (I gave a reference for that), which would impose a rigid limit on the accuracy achievable).

2. What has been done anywhere in lattice QFT has always lead to very low accuracy (a few percent at best). Accuracy increases like $N^{-1/2}$ with the number of lattice points per dimension, and work increases like $N^7$. Extrapolating to what is needed to get 10 significant did accuracy would require of the order of $10^{20}$ lattice points per dimension. Thus of the order of $10^{80}$ lattice points and of the order of $10^{140}$ floating point operations would be needed. This doesn't prove that it cannot be done. But surely it cannot be done during the time any of those participating in this discussion will live.

 

[Demystifier]

Well, I don't believe a calculation that doesn't exist.

When I see a big picture that this calculation entails, I do.

 

[A. Neumaier]

I'm not aware of a single paper about lattice QED.

There is a little trickle of them. Nothing at all in their results is inviting to work on the topic.

the book by Montvay and Munster is a very good reference.

Of the 442 pages of main text excluding references, only 12 pages are on lattice QED (Section 4.5), and much of it summarizes results from continuum QED (e.g., Pauli-Villars regularization on p.227). The last 2 1/2 pages (pp.228-230) contain information on nonperturbative studies of lattice QED in very simplified scenarios (e.g., the quenched approximation). They conclude by stating that

the fermion always decouples in the continuum limit, leading to a trivial bosonic theory. These non-perturbative studies of renormalization suggest that the continuum limit of lattice QED is trivial.

This conclusion by experts on the subject is diametrically opposite to atyy's claims that lattice QED is a good approximation of QED, or a good starting point for low energy QED.

 

[A. Neumaier]

I think there are 1-loop lattice results which, for low energy phenomena, agree with standard 1-loop results.

Please share your thinking in enough detail that others can check its cogency.

 

[atyy]

This has nowhere be done; it is wishful thinking.

It is no more wishful that asserting that Poincare invariant QED exists - no one has shown that it exists at all energies, and if it does not exist at all energies, then it cannot be Poincare invariant.

 

[atyy]

As I said, I consider it also not very advantageous to use the lattice regularization to define QFT. There are better ways to regularize perturbative QFT like Pauli Villars or dim. reg. or a stupid old cutoff (in Euclidean QFT maybe the most simple idea). Conceptually regularization is even a bit unintuitive for my taste, and my great "aha feeling" came when I learned the BPHZ formalism, which uses always physical masses, couplings and fields order by order in PT (either loop-wise, i.e., in powers of $\hbar$ or in some coupling constant(s) or any other counting scheme appropriate to the definite model under investigation) and the counter-term approach. Then you simply subtract the divergences, introducing the renormalization scale etc. Then you get the RG equation from the independence of S-matrix elements on the choice of the renormalization scale (and even the renormalization scheme). This more conventional approaches are all well defined ways to work in renormalized perturbation theory with only finite quantities at any step of the calculation. For me it's way more intuitive than an artificial space-time lattice with quite nasty properties (fermion doublers and other kinds of artifacts, the lost Poincare, Lorentz, rotation invariance etc. ect.) and very delicate convergence issues if you want to get to the continuum limit. I'm also not familiar with the calculational techniques to evaluate even a simple one-loop diagram as vacuum polarization and electron self-energy on the lattice. With the other techniques it's not such a big deal. For a nice introduction also to calculational techniques see P. Ramon, QFT - A Modern Primer.

In my opinion, the best way to understand the quite involved ideas behind renormalized perturbation theory is to sit down and do calculations like the one-loop structure of QED to the end. For a beginner, I'd recommend to use the modern approach, using dim. reg. as regulator and then discussing various renormalization schemes like minimal subtraction and/or modified minimal subtraction, the usual on-shell scheme.

There is not much difference between a stupid old cutoff and lattice QED - both mean explicitly that the theory does not exist at all energies. If QED does not exists at all energies, then it cannot be Poincare invariant.

 

[vanhees71]

Nobody claims that QED is valid at all energy scales (nor any other QFT for that matter). It is very likely, although not strictly proven, that QED has a Landau pole, where it breaks down, but it's at very high energies irrelevant for all practical purposes. Whether there exists some more comprehensive theory than relativistic QFT that is valid at all energy scales, is unknown today.

For the so modified interpretation of QFTs as low-energy effective theories, it is totally irrelevant which regularization you use. The regularized theory is just a step in the calculation for the S-matrix elements in terms of well-defined finite renormalized quantities.

It's not even necessary to use regularization at all. You can as well directly renormalize using the BPHZ formalism, it's only technical sometimes a bit less convenient than modern regularization techniques like dim reg or the heat-kernel methods or even Pauli-Villars (in some cases where dim. reg is not easily applicable). Finally, all that counts and all that's comparable to experiment are the results of renormalized perturbative QED.

 

[atyy]

Nobody claims that QED is valid at all energy scales (nor any other QFT for that matter). It is very likely, although not strictly proven, that QED has a Landau pole, where it breaks down, but it's at very high energies irrelevant for all practical purposes. Whether there exists some more comprehensive theory than relativistic QFT that is valid at all energy scales, is unknown today.

For the so modified interpretation of QFTs as low-energy effective theories, it is totally irrelevant which regularization you use. The regularized theory is just a step in the calculation for the S-matrix elements in terms of well-defined finite renormalized quantities.

It's not even necessary to use regularization at all. You can as well directly renormalize using the BPHZ formalism, it's only technical sometimes a bit less convenient than modern regularization techniques like dim reg or the heat-kernel methods or even Pauli-Villars (in some cases where dim. reg is not easily applicable). Finally, all that counts and all that's comparable to experiment are the results of renormalized perturbative QED.

If we don't know whether QED exists at all energies, then doesn't it mean we don't know whether it is truly Poincare invariant (since an energy cutoff spoils Poincare invariance)?

 

[Demystifier]

This conclusion by experts on the subject is diametrically opposite to atyy's claims that lattice QED is a good approximation of QED, or a good starting point for low energy QED.

Again, the claim is that it is a good approximation at large distances. As I already said, I don't give a damn what any theory without quantum gravity say in the continuum limit.